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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Quasi_Borel_Spaces/0011_Measure_as_QuasiBorel_Measure/prob_00200_008604__15506914_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2019 ( 440 unt; 736 typ;   0 def)
%            Number of atoms       : 4286 ( 758 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 15138 ( 216   ~;  41   |; 437   &;12663   @)
%                                         (   0 <=>;1781  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   8 avg)
%            Number of types       :  198 ( 197 usr)
%            Number of type conns  : 2506 (2506   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  542 ( 539 usr;  48 con; 0-4 aty)
%            Number of variables   : 4626 ( 868   ^;3740   !;  18   ?;4626   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:11:50.313
%------------------------------------------------------------------------------
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Binary__Product__Measure_Opair__measure_001tf__b_001tf__b,type,
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thf(sy_c_Borel__Space_Ois__borel_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
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thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Real__Oereal,type,
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thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
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thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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    borel_8800728737916236534l_real: sigma_8004542664021053421l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
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thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    borel_1964376316847149668l_real: sigma_2490040489844529243l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    borel_9155112475215227991l_real: sigma_2308072346491277622l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__a_J,type,
    borel_8855829479337404329real_a: sigma_4670575602351775008real_a ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mtf__b_J,type,
    borel_8855829483640633130real_b: sigma_4741609642398120993real_b ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    borel_8354383352383190528l_real: sigma_5020773068096799327l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__a_Mt__Real__Oreal_J,type,
    borel_8338562977070067131a_real: sigma_2262136186458356274a_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    borel_8051966402323383621od_a_a: sigma_5711748576726957348od_a_a ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    borel_8051966406626612422od_a_b: sigma_5782782616773303333od_a_b ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__b_Mt__Real__Oreal_J,type,
    borel_4711409555403422012b_real: sigma_8706803915453109555b_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__b_Mtf__a_J,type,
    borel_5264510820350707588od_b_a: sigma_5076099576121970659od_b_a ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    borel_5264510824653936389od_b_b: sigma_5147133616168316644od_b_b ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
    borel_5078946678739801102l_real: sigma_measure_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__a,type,
    borel_5459123734250506524orel_a: sigma_measure_a ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001tf__b,type,
    borel_5459123734250506525orel_b: sigma_measure_b ).

thf(sy_c_Euclidean__Space_Oeuclidean__space__class_OBasis_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    euclid5187495680784233478l_real: set_Pr6218003697084177305l_real ).

thf(sy_c_Euclidean__Space_Oeuclidean__space__class_OBasis_001t__Real__Oreal,type,
    euclid1305858884100475807s_real: set_real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P2601594796862164940b_real: ( produc7364210939229353703b_real > produc7929879615859924995b_real ) > ( produc7929879615859924995b_real > produc7364210939229353703b_real ) > produc7929879615859924995b_real > produc7929879615859924995b_real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P850362950206298357b_real: ( produc2738766212015491856l_real > produc7929879615859924995b_real ) > ( produc7929879615859924995b_real > produc2738766212015491856l_real ) > produc7929879615859924995b_real > produc7929879615859924995b_real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P3498366258136572190b_real: ( produc2738766212015491856l_real > real ) > ( produc7929879615859924995b_real > produc2738766212015491856l_real ) > produc7929879615859924995b_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P2581140431950109038b_real: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > produc7929879615859924995b_real > produc7929879615859924995b_real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P8510425873491595589b_real: ( produc7929879615859924995b_real > product_prod_a_b ) > ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > produc7929879615859924995b_real > product_prod_a_b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001t__Real__Oreal,type,
    comp_P7944386915558755310b_real: ( produc7929879615859924995b_real > product_prod_a_b ) > ( real > produc7929879615859924995b_real ) > real > product_prod_a_b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P5041851192982496151b_real: ( produc7929879615859924995b_real > real ) > ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > produc7929879615859924995b_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_P6701117116028920384l_real: ( produc7929879615859924995b_real > real ) > ( real > produc7929879615859924995b_real ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P5119746620826145525b_real: ( produc5010650472090879248b_real > produc7929879615859924995b_real ) > ( produc7929879615859924995b_real > produc5010650472090879248b_real ) > produc7929879615859924995b_real > produc7929879615859924995b_real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_J_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P146849012575345438b_real: ( produc5010650472090879248b_real > real ) > ( produc7929879615859924995b_real > produc5010650472090879248b_real ) > produc7929879615859924995b_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_P7068956567367034951b_real: ( produc2422161461964618553l_real > real ) > ( produc7929879615859924995b_real > produc2422161461964618553l_real ) > produc7929879615859924995b_real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_P4894738870877013744l_real: ( produc2422161461964618553l_real > real ) > ( real > produc2422161461964618553l_real ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__a_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P2648956292766268207od_a_b: ( product_prod_a_b > a ) > ( product_prod_a_b > product_prod_a_b ) > product_prod_a_b > a ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__a_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P2009515992434452078od_a_b: ( product_prod_a_b > b ) > ( product_prod_a_b > product_prod_a_b ) > product_prod_a_b > b ).

thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    comp_P5715990135234274861od_a_b: ( product_prod_b_b > b ) > ( product_prod_a_b > product_prod_b_b ) > product_prod_a_b > b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_r312201211331252503b_real: ( real > produc7929879615859924995b_real ) > ( produc7929879615859924995b_real > real ) > produc7929879615859924995b_real > produc7929879615859924995b_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_r6448350170807567469b_real: ( real > produc2422161461964618553l_real ) > ( produc7929879615859924995b_real > real ) > produc7929879615859924995b_real > produc2422161461964618553l_real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J,type,
    comp_r6460197204725972288b_real: ( real > real ) > ( produc7929879615859924995b_real > real ) > produc7929879615859924995b_real > real ).

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

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    member_real: real > set_real > $o ).

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    member566263741820371442b_real: set_Pr1027596265373204187b_real > set_se6676801767088899473b_real > $o ).

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    member3057847817020379547l_real: set_Pr3917002265660371076l_real > set_se359666411984959674l_real > $o ).

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    member4198857394546769929_a_b_b: set_Pr3674455588107860082_a_b_b > set_se5425259496033807912_a_b_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    member_set_real_real2: set_real_real > set_set_real_real > $o ).

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    member1180967512111262191od_b_a: set_Product_prod_b_a > set_se8386471407611763750od_b_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member1252001552157608176od_b_b: set_Product_prod_b_b > set_se5865286889322728103od_b_b > $o ).

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

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

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

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

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

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

thf(sy_v_A____,type,
    a2: set_real ).

% Relevant facts (1275)
thf(fact_0_h,axiom,
    member_set_real @ a2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% h
thf(fact_1_measurable__snd,axiom,
    ! [M1: sigma_measure_real,M2: sigma_measure_real] : ( member6699615393305559423l_real @ produc3484788084999411615l_real @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) @ M2 ) ) ).

% measurable_snd
thf(fact_2_measurable__snd,axiom,
    ! [M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] : ( member5467075257862553189l_real @ produc7037151529651111887b_real @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ M1 @ M2 ) @ M2 ) ) ).

% measurable_snd
thf(fact_3_measurable__snd,axiom,
    ! [M1: sigma_measure_a,M2: sigma_measure_b] : ( member1080921170058542931_a_b_b @ product_snd_a_b @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ M1 @ M2 ) @ M2 ) ) ).

% measurable_snd
thf(fact_4_measurable__snd_H_H,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,N: sigma_4174442854635832540b_real,P: sigma_5782782616773303333od_a_b] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ N ) )
     => ( member1118978200981565500b_real
        @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc7037151529651111887b_real @ X ) )
        @ ( sigma_1108951568718238936b_real @ ( binary496794562119935756b_real @ P @ M ) @ N ) ) ) ).

% measurable_snd''
thf(fact_5_measurable__snd_H_H,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,P: sigma_measure_a] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( member1080921170058542931_a_b_b
        @ ^ [X: product_prod_a_b] : ( F @ ( product_snd_a_b @ X ) )
        @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ P @ M ) @ N ) ) ) ).

% measurable_snd''
thf(fact_6_measurable__snd_H_H,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( member6699615393305559423l_real
        @ ^ [X: produc2422161461964618553l_real] : ( F @ ( produc3484788084999411615l_real @ X ) )
        @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ P @ M ) @ N ) ) ) ).

% measurable_snd''
thf(fact_7_measurable__snd_H_H,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: sigma_5782782616773303333od_a_b] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc7037151529651111887b_real @ X ) )
        @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ P @ M ) @ N ) ) ) ).

% measurable_snd''
thf(fact_8_measurable__snd_H,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,N: sigma_5782782616773303333od_a_b,P: sigma_measure_real] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ N @ P ) ) )
     => ( member_real_real
        @ ^ [X: real] : ( produc7037151529651111887b_real @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ P ) ) ) ).

% measurable_snd'
thf(fact_9_measurable__snd_H,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_a,P: sigma_measure_b] :
      ( ( member2755110357185872764od_a_b @ F @ ( sigma_5476089079233491280od_a_b @ M @ ( binary867438762418767561re_a_b @ N @ P ) ) )
     => ( member1080921170058542931_a_b_b
        @ ^ [X: product_prod_a_b] : ( product_snd_a_b @ ( F @ X ) )
        @ ( sigma_7225552367035416487_a_b_b @ M @ P ) ) ) ).

% measurable_snd'
thf(fact_10_measurable__snd_H,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ ( binary6478037234023840930l_real @ N @ P ) ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( produc3484788084999411615l_real @ ( F @ X ) )
        @ ( sigma_2402725360379274625l_real @ M @ P ) ) ) ).

% measurable_snd'
thf(fact_11_measurable__snd_H,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ N @ P ) ) )
     => ( member_real_real
        @ ^ [X: real] : ( produc3484788084999411615l_real @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ P ) ) ) ).

% measurable_snd'
thf(fact_12_measurable__snd_H,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_5782782616773303333od_a_b,P: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ N @ P ) ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( produc7037151529651111887b_real @ ( F @ X ) )
        @ ( sigma_2402725360379274625l_real @ M @ P ) ) ) ).

% measurable_snd'
thf(fact_13_borel__measurable__const,axiom,
    ! [C: produc7929879615859924995b_real,M: sigma_4174442854635832540b_real] :
      ( member1118978200981565500b_real
      @ ^ [X: produc7929879615859924995b_real] : C
      @ ( sigma_1108951568718238936b_real @ M @ borel_3131472315568871781b_real ) ) ).

% borel_measurable_const
thf(fact_14_borel__measurable__const,axiom,
    ! [C: b,M: sigma_5782782616773303333od_a_b] :
      ( member1080921170058542931_a_b_b
      @ ^ [X: product_prod_a_b] : C
      @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ).

% borel_measurable_const
thf(fact_15_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_real] :
      ( member_real_real
      @ ^ [X: real] : C
      @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_16_borel__measurable__const,axiom,
    ! [C: real,M: sigma_4174442854635832540b_real] :
      ( member5467075257862553189l_real
      @ ^ [X: produc7929879615859924995b_real] : C
      @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_17_borel__prod,axiom,
    ( ( binary2460265453143672003l_real @ borel_5078946678739801102l_real @ borel_9155112475215227991l_real )
    = borel_1964376316847149668l_real ) ).

% borel_prod
thf(fact_18_borel__prod,axiom,
    ( ( binary2464900950464885149l_real @ borel_9155112475215227991l_real @ borel_5078946678739801102l_real )
    = borel_8800728737916236534l_real ) ).

% borel_prod
thf(fact_19_borel__prod,axiom,
    ( ( binary2427233776965711752l_real @ borel_9155112475215227991l_real @ borel_9155112475215227991l_real )
    = borel_1362741036457552005l_real ) ).

% borel_prod
thf(fact_20_borel__prod,axiom,
    ( ( binary6478037234023840930l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real )
    = borel_9155112475215227991l_real ) ).

% borel_prod
thf(fact_21_measurable__id,axiom,
    ! [M: sigma_measure_real] :
      ( member_real_real
      @ ^ [X: real] : X
      @ ( sigma_5267869275261027754l_real @ M @ M ) ) ).

% measurable_id
thf(fact_22_measurable__id,axiom,
    ! [M: sigma_4174442854635832540b_real] :
      ( member1118978200981565500b_real
      @ ^ [X: produc7929879615859924995b_real] : X
      @ ( sigma_1108951568718238936b_real @ M @ M ) ) ).

% measurable_id
thf(fact_23_measurable__compose,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,L: sigma_measure_real] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ N @ L ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( F @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_24_measurable__compose,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b,N: sigma_5782782616773303333od_a_b,G: product_prod_a_b > b,L: sigma_measure_b] :
      ( ( member2755110357185872764od_a_b @ F @ ( sigma_5476089079233491280od_a_b @ M @ N ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ N @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( G @ ( F @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_25_measurable__compose,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,G: real > produc7929879615859924995b_real,L: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ N ) )
     => ( ( member7738959517937940581b_real @ G @ ( sigma_5526981039175570177b_real @ N @ L ) )
       => ( member1118978200981565500b_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( F @ X ) )
          @ ( sigma_1108951568718238936b_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_26_measurable__compose,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_b,G: b > b,L: sigma_measure_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( G @ ( F @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_27_measurable__compose,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( F @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_28_measurable__compose,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,L: sigma_4174442854635832540b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ N ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ N @ L ) )
       => ( member1118978200981565500b_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( F @ X ) )
          @ ( sigma_1108951568718238936b_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_29_measurable__compose,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,L: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ N @ L ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( F @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_30_measurable__compose,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( F @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_31_measurable__compose__rev,axiom,
    ! [F: real > produc7929879615859924995b_real,L: sigma_measure_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ L @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ L ) )
       => ( member1118978200981565500b_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( G @ X ) )
          @ ( sigma_1108951568718238936b_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_32_measurable__compose__rev,axiom,
    ! [F: b > b,L: sigma_measure_b,N: sigma_measure_b,G: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ L @ N ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_33_measurable__compose__rev,axiom,
    ! [F: produc7929879615859924995b_real > real,L: sigma_4174442854635832540b_real,N: sigma_measure_real,G: real > produc7929879615859924995b_real,M: sigma_measure_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ L @ N ) )
     => ( ( member7738959517937940581b_real @ G @ ( sigma_5526981039175570177b_real @ M @ L ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_34_measurable__compose__rev,axiom,
    ! [F: produc7929879615859924995b_real > real,L: sigma_4174442854635832540b_real,N: sigma_measure_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ L @ N ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ L ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_35_measurable__compose__rev,axiom,
    ! [F: product_prod_a_b > b,L: sigma_5782782616773303333od_a_b,N: sigma_measure_b,G: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ L @ N ) )
     => ( ( member2755110357185872764od_a_b @ G @ ( sigma_5476089079233491280od_a_b @ M @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_36_measurable__compose__rev,axiom,
    ! [F: real > real,L: sigma_measure_real,N: sigma_measure_real,G: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ L ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_37_measurable__compose__rev,axiom,
    ! [F: real > real,L: sigma_measure_real,N: sigma_measure_real,G: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_38_measurable__compose__rev,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,L: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ L @ N ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ L ) )
       => ( member1118978200981565500b_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( G @ X ) )
          @ ( sigma_1108951568718238936b_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_39_is__borel__def,axiom,
    ( borel_3333619307305864900b_real
    = ( ^ [F2: produc7929879615859924995b_real > produc7929879615859924995b_real,M3: sigma_4174442854635832540b_real] : ( member1118978200981565500b_real @ F2 @ ( sigma_1108951568718238936b_real @ M3 @ borel_3131472315568871781b_real ) ) ) ) ).

% is_borel_def
thf(fact_40_is__borel__def,axiom,
    ( borel_484038283607786387_a_b_b
    = ( ^ [F2: product_prod_a_b > b,M3: sigma_5782782616773303333od_a_b] : ( member1080921170058542931_a_b_b @ F2 @ ( sigma_7225552367035416487_a_b_b @ M3 @ borel_5459123734250506525orel_b ) ) ) ) ).

% is_borel_def
thf(fact_41_is__borel__def,axiom,
    ( borel_1821451471680550765l_real
    = ( ^ [F2: produc7929879615859924995b_real > real,M3: sigma_4174442854635832540b_real] : ( member5467075257862553189l_real @ F2 @ ( sigma_2402725360379274625l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_42_is__borel__def,axiom,
    ( borel_236569967776329622l_real
    = ( ^ [F2: real > real,M3: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_43_borel__measurable__scaleR,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V1485227260804924795R_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_scaleR
thf(fact_44_borel__measurable__scaleR,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( real_V1485227260804924795R_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_scaleR
thf(fact_45_borel__measurable__scaleR,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc2422161461964618553l_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member174722968998780374l_real @ G @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member174722968998780374l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V8334778624115123498l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_scaleR
thf(fact_46_borel__measurable__scaleR,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > produc2422161461964618553l_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( real_V8334778624115123498l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_scaleR
thf(fact_47_sets__pair__measure__cong,axiom,
    ! [M1: sigma_5782782616773303333od_a_b,M12: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,M22: sigma_measure_real] :
      ( ( ( sigma_377313353287227655od_a_b @ M1 )
        = ( sigma_377313353287227655od_a_b @ M12 ) )
     => ( ( ( sigma_sets_real @ M2 )
          = ( sigma_sets_real @ M22 ) )
       => ( ( sigma_5719578828308484196b_real @ ( binary496794562119935756b_real @ M1 @ M2 ) )
          = ( sigma_5719578828308484196b_real @ ( binary496794562119935756b_real @ M12 @ M22 ) ) ) ) ) ).

% sets_pair_measure_cong
thf(fact_48_sets__pair__measure__cong,axiom,
    ! [M1: sigma_measure_a,M12: sigma_measure_a,M2: sigma_measure_b,M22: sigma_measure_b] :
      ( ( ( sigma_sets_a @ M1 )
        = ( sigma_sets_a @ M12 ) )
     => ( ( ( sigma_sets_b @ M2 )
          = ( sigma_sets_b @ M22 ) )
       => ( ( sigma_377313353287227655od_a_b @ ( binary867438762418767561re_a_b @ M1 @ M2 ) )
          = ( sigma_377313353287227655od_a_b @ ( binary867438762418767561re_a_b @ M12 @ M22 ) ) ) ) ) ).

% sets_pair_measure_cong
thf(fact_49_sets__pair__measure__cong,axiom,
    ! [M1: sigma_measure_real,M12: sigma_measure_real,M2: sigma_measure_real,M22: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M1 )
        = ( sigma_sets_real @ M12 ) )
     => ( ( ( sigma_sets_real @ M2 )
          = ( sigma_sets_real @ M22 ) )
       => ( ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) )
          = ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ M12 @ M22 ) ) ) ) ) ).

% sets_pair_measure_cong
thf(fact_50_measurable__cong__sets,axiom,
    ! [M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_5719578828308484196b_real @ M )
        = ( sigma_5719578828308484196b_real @ M4 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_2402725360379274625l_real @ M @ N )
          = ( sigma_2402725360379274625l_real @ M4 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_51_measurable__cong__sets,axiom,
    ! [M: sigma_5782782616773303333od_a_b,M4: sigma_5782782616773303333od_a_b,N: sigma_measure_b,N2: sigma_measure_b] :
      ( ( ( sigma_377313353287227655od_a_b @ M )
        = ( sigma_377313353287227655od_a_b @ M4 ) )
     => ( ( ( sigma_sets_b @ N )
          = ( sigma_sets_b @ N2 ) )
       => ( ( sigma_7225552367035416487_a_b_b @ M @ N )
          = ( sigma_7225552367035416487_a_b_b @ M4 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_52_measurable__cong__sets,axiom,
    ! [M: sigma_measure_real,M4: sigma_measure_real,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M4 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_5267869275261027754l_real @ M @ N )
          = ( sigma_5267869275261027754l_real @ M4 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_53_measurable__cong__sets,axiom,
    ! [M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,N2: sigma_4174442854635832540b_real] :
      ( ( ( sigma_5719578828308484196b_real @ M )
        = ( sigma_5719578828308484196b_real @ M4 ) )
     => ( ( ( sigma_5719578828308484196b_real @ N )
          = ( sigma_5719578828308484196b_real @ N2 ) )
       => ( ( sigma_1108951568718238936b_real @ M @ N )
          = ( sigma_1108951568718238936b_real @ M4 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_54_measurable__ident__sets,axiom,
    ! [M: sigma_measure_real,M4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M4 ) )
     => ( member_real_real
        @ ^ [X: real] : X
        @ ( sigma_5267869275261027754l_real @ M @ M4 ) ) ) ).

% measurable_ident_sets
thf(fact_55_measurable__ident__sets,axiom,
    ! [M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real] :
      ( ( ( sigma_5719578828308484196b_real @ M )
        = ( sigma_5719578828308484196b_real @ M4 ) )
     => ( member1118978200981565500b_real
        @ ^ [X: produc7929879615859924995b_real] : X
        @ ( sigma_1108951568718238936b_real @ M @ M4 ) ) ) ).

% measurable_ident_sets
thf(fact_56_borel__measurable__const__scaleR,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,B: real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( real_V1485227260804924795R_real @ B @ ( F @ X ) )
        @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_const_scaleR
thf(fact_57_borel__measurable__const__scaleR,axiom,
    ! [F: real > real,M: sigma_measure_real,B: real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_real_real
        @ ^ [X: real] : ( real_V1485227260804924795R_real @ B @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_const_scaleR
thf(fact_58_borel__measurable__indicator,axiom,
    ! [A: set_Pr6157794097399793337b_real,M: sigma_4174442854635832540b_real] :
      ( ( member2587080422514930128b_real @ A @ ( sigma_5719578828308484196b_real @ M ) )
     => ( member5467075257862553189l_real @ ( indica5149351975077090117l_real @ A ) @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator
thf(fact_59_borel__measurable__indicator,axiom,
    ! [A: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
     => ( member_real_real @ ( indica3952125700972739438l_real @ A ) @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator
thf(fact_60_borel__measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member174722968998780374l_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_61_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_62_borel__measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc2422161461964618553l_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member174722968998780374l_real @ G @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member7847863067707020219l_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc2480879129069238452l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5673202184642345047l_real @ M @ borel_1964376316847149668l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_63_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > produc2422161461964618553l_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member5395584588768555876l_real
          @ ^ [X: real] : ( produc2480879129069238452l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_4503530189580602880l_real @ M @ borel_1964376316847149668l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_64_borel__measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4138993205028768589l_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc2485514626390451598l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_3286182568856656105l_real @ M @ borel_8800728737916236534l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_65_borel__measurable__Pair,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member1686714726090304246l_real
          @ ^ [X: real] : ( produc2485514626390451598l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2116510573794913938l_real @ M @ borel_8800728737916236534l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_66_set__borel__measurable__def,axiom,
    ( set_se6409052079846015710l_real
    = ( ^ [M3: sigma_4174442854635832540b_real,A2: set_Pr6157794097399793337b_real,F2: produc7929879615859924995b_real > real] :
          ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V1485227260804924795R_real @ ( indica5149351975077090117l_real @ A2 @ X ) @ ( F2 @ X ) )
          @ ( sigma_2402725360379274625l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).

% set_borel_measurable_def
thf(fact_67_set__borel__measurable__def,axiom,
    ( set_se6090954689868541191l_real
    = ( ^ [M3: sigma_measure_real,A2: set_real,F2: real > real] :
          ( member_real_real
          @ ^ [X: real] : ( real_V1485227260804924795R_real @ ( indica3952125700972739438l_real @ A2 @ X ) @ ( F2 @ X ) )
          @ ( sigma_5267869275261027754l_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).

% set_borel_measurable_def
thf(fact_68_set__borel__measurable__def,axiom,
    ( set_se3278990456008795294l_real
    = ( ^ [M3: sigma_measure_real,A2: set_real,F2: real > produc2422161461964618553l_real] :
          ( member9086635009091248365l_real
          @ ^ [X: real] : ( real_V8334778624115123498l_real @ ( indica3952125700972739438l_real @ A2 @ X ) @ ( F2 @ X ) )
          @ ( sigma_7998147297565726139l_real @ M3 @ borel_9155112475215227991l_real ) ) ) ) ).

% set_borel_measurable_def
thf(fact_69_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_Pr6157794097399793337b_real,F: produc7929879615859924995b_real > real] :
      ( ( member2587080422514930128b_real @ A @ ( sigma_5719578828308484196b_real @ borel_3131472315568871781b_real ) )
     => ( ( topolo6634679532018074415l_real @ A @ F )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V1485227260804924795R_real @ ( indica5149351975077090117l_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_2402725360379274625l_real @ borel_3131472315568871781b_real @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_70_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_a,F: a > real] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( topolo210349010034097452a_real @ A @ F )
       => ( member_a_real
          @ ^ [X: a] : ( real_V1485227260804924795R_real @ ( indica7750551289776754582a_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_9116425665531756122a_real @ borel_5459123734250506524orel_a @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_71_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_a,F: a > produc2422161461964618553l_real] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( topolo7011970341015082169l_real @ A @ F )
       => ( member2229928074028245815l_real
          @ ^ [X: a] : ( real_V8334778624115123498l_real @ ( indica7750551289776754582a_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_414277600898586891l_real @ borel_5459123734250506524orel_a @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_72_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_b,F: b > real] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
     => ( ( topolo5806567625222228141b_real @ A @ F )
       => ( member_b_real
          @ ^ [X: b] : ( real_V1485227260804924795R_real @ ( indica4123397868110109463b_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_5489272243865111003b_real @ borel_5459123734250506525orel_b @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_73_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_b,F: b > produc2422161461964618553l_real] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
     => ( ( topolo1495072446960129144l_real @ A @ F )
       => ( member5936402216828068598l_real
          @ ^ [X: b] : ( real_V8334778624115123498l_real @ ( indica4123397868110109463b_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_4120751743698409674l_real @ borel_5459123734250506525orel_b @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_74_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_real,F: real > real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( topolo5044208981011980120l_real @ A @ F )
       => ( member_real_real
          @ ^ [X: real] : ( real_V1485227260804924795R_real @ ( indica3952125700972739438l_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_75_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_real,F: real > produc2422161461964618553l_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( topolo5689633929180701069l_real @ A @ F )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( real_V8334778624115123498l_real @ ( indica3952125700972739438l_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_7998147297565726139l_real @ borel_5078946678739801102l_real @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_76_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_Pr6218003697084177305l_real,F: produc2422161461964618553l_real > real] :
      ( ( member6767226683953100098l_real @ A @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) )
     => ( ( topolo5694269426501914215l_real @ A @ F )
       => ( member6699615393305559423l_real
          @ ^ [X: produc2422161461964618553l_real] : ( real_V1485227260804924795R_real @ ( indica3611827895086754257l_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_8002782794886939285l_real @ borel_9155112475215227991l_real @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_77_borel__measurable__continuous__on__indicator,axiom,
    ! [A: set_Pr6218003697084177305l_real,F: produc2422161461964618553l_real > produc2422161461964618553l_real] :
      ( ( member6767226683953100098l_real @ A @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) )
     => ( ( topolo506878965529049918l_real @ A @ F )
       => ( member1176752664076369724l_real
          @ ^ [X: produc2422161461964618553l_real] : ( real_V8334778624115123498l_real @ ( indica3611827895086754257l_real @ A @ X ) @ ( F @ X ) )
          @ ( sigma_484132685431196816l_real @ borel_9155112475215227991l_real @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_continuous_on_indicator
thf(fact_78_borel__measurable__indicator_H,axiom,
    ! [M: sigma_measure_real,F: real > real,A: real > set_real] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( member_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_real_real
        @ ^ [X: real] : ( indica3952125700972739438l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_79_borel__measurable__indicator_H,axiom,
    ! [M: sigma_measure_real,F: real > set_real,A: real > set_set_real] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( member_set_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_real_real
        @ ^ [X: real] : ( indica2804201007932554404l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_80_borel__measurable__indicator_H,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > real,A: set_real > set_real] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( member_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_real
        @ ^ [X: set_real] : ( indica3952125700972739438l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_81_borel__measurable__indicator_H,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > set_real,A: set_real > set_set_real] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( member_set_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_real
        @ ^ [X: set_real] : ( indica2804201007932554404l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_82_borel__measurable__indicator_H,axiom,
    ! [M: sigma_measure_real,F: real > real > real,A: real > set_real_real] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( member_real_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_real_real
        @ ^ [X: real] : ( indica1619667925609444025l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_83_borel__measurable__indicator_H,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > real,A: ( real > real ) > set_real] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( member_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( member5749659578190367193l_real
        @ ^ [X: real > real] : ( indica3952125700972739438l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_84_borel__measurable__indicator_H,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > real > real,A: set_real > set_real_real] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( member_real_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_real
        @ ^ [X: set_real] : ( indica1619667925609444025l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_85_borel__measurable__indicator_H,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > set_real,A: ( real > real ) > set_set_real] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( member_set_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( member5749659578190367193l_real
        @ ^ [X: real > real] : ( indica2804201007932554404l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_86_borel__measurable__indicator_H,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > real > real,A: ( real > real ) > set_real_real] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( member_real_real @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( member5749659578190367193l_real
        @ ^ [X: real > real] : ( indica1619667925609444025l_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_87_borel__measurable__indicator_H,axiom,
    ! [M: sigma_measure_real,F: real > product_prod_a_b > b,A: real > set_Pr3674455588107860082_a_b_b] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( member1080921170058542931_a_b_b @ ( F @ X ) @ ( A @ X ) ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_real_real
        @ ^ [X: real] : ( indica6911144680950340606b_real @ ( A @ X ) @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_indicator'
thf(fact_88_borel__measurable__inner,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( inner_5569486634795291694r_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inner
thf(fact_89_borel__measurable__inner,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( inner_5569486634795291694r_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inner
thf(fact_90_borel__measurable__inner,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc2422161461964618553l_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member174722968998780374l_real @ G @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( inner_2340953980025574455l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inner
thf(fact_91_borel__measurable__inner,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > produc2422161461964618553l_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( inner_2340953980025574455l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inner
thf(fact_92_borel__measurable__dist,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V975177566351809787t_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_dist
thf(fact_93_borel__measurable__dist,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( real_V975177566351809787t_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_dist
thf(fact_94_borel__measurable__dist,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc2422161461964618553l_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member174722968998780374l_real @ G @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V3631300063480491434l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_dist
thf(fact_95_borel__measurable__dist,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > produc2422161461964618553l_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( real_V3631300063480491434l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_dist
thf(fact_96_borel__Times,axiom,
    ! [A: set_a,B2: set_a] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( member_set_a @ B2 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
       => ( member1816616512716248880od_a_a
          @ ( product_Sigma_a_a @ A
            @ ^ [Uu: a] : B2 )
          @ ( sigma_377313348983998854od_a_a @ borel_8051966402323383621od_a_a ) ) ) ) ).

% borel_Times
thf(fact_97_borel__Times,axiom,
    ! [A: set_a,B2: set_b] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
       => ( member1887650552762594865od_a_b
          @ ( product_Sigma_a_b @ A
            @ ^ [Uu: a] : B2 )
          @ ( sigma_377313353287227655od_a_b @ borel_8051966406626612422od_a_b ) ) ) ) ).

% borel_Times
thf(fact_98_borel__Times,axiom,
    ! [A: set_a,B2: set_real] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( member_set_real @ B2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( member4629841302104275750a_real
          @ ( product_Sigma_a_real @ A
            @ ^ [Uu: a] : B2 )
          @ ( sigma_7086255002324679610a_real @ borel_8338562977070067131a_real ) ) ) ) ).

% borel_Times
thf(fact_99_borel__Times,axiom,
    ! [A: set_b,B2: set_a] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
     => ( ( member_set_a @ B2 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
       => ( member1180967512111262191od_b_a
          @ ( product_Sigma_b_a @ A
            @ ^ [Uu: b] : B2 )
          @ ( sigma_6813229803866098629od_b_a @ borel_5264510820350707588od_b_a ) ) ) ) ).

% borel_Times
thf(fact_100_borel__Times,axiom,
    ! [A: set_b,B2: set_b] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
       => ( member1252001552157608176od_b_b
          @ ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : B2 )
          @ ( sigma_6813229808169327430od_b_b @ borel_5264510824653936389od_b_b ) ) ) ) ).

% borel_Times
thf(fact_101_borel__Times,axiom,
    ! [A: set_b,B2: set_real] :
      ( ( member_set_b @ A @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
     => ( ( member_set_real @ B2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( member1851136994244253223b_real
          @ ( product_Sigma_b_real @ A
            @ ^ [Uu: b] : B2 )
          @ ( sigma_3459101580658034491b_real @ borel_4711409555403422012b_real ) ) ) ) ).

% borel_Times
thf(fact_102_borel__Times,axiom,
    ! [A: set_real,B2: set_a] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( member_set_a @ B2 @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
       => ( member7038280717997694484real_a
          @ ( product_Sigma_real_a @ A
            @ ^ [Uu: real] : B2 )
          @ ( sigma_7603521504592016808real_a @ borel_8855829479337404329real_a ) ) ) ) ).

% borel_Times
thf(fact_103_borel__Times,axiom,
    ! [A: set_real,B2: set_b] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( member_set_b @ B2 @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) )
       => ( member7109314758044040469real_b
          @ ( product_Sigma_real_b @ A
            @ ^ [Uu: real] : B2 )
          @ ( sigma_7603521508895245609real_b @ borel_8855829483640633130real_b ) ) ) ) ).

% borel_Times
thf(fact_104_borel__Times,axiom,
    ! [A: set_real,B2: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real @ B2 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( member6767226683953100098l_real
          @ ( produc1020309166918817442l_real @ A
            @ ^ [Uu: real] : B2 )
          @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_Times
thf(fact_105_borel__Times,axiom,
    ! [A: set_a,B2: set_Pr6218003697084177305l_real] :
      ( ( member_set_a @ A @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) )
     => ( ( member6767226683953100098l_real @ B2 @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) )
       => ( member8280623040949385835l_real
          @ ( produc5859231248517886467l_real @ A
            @ ^ [Uu: a] : B2 )
          @ ( sigma_7466920334632516801l_real @ borel_8354383352383190528l_real ) ) ) ) ).

% borel_Times
thf(fact_106_measurable__Pair2__compose,axiom,
    ! [F: real > a,G: real > b,M: sigma_measure_real,M1: sigma_measure_a,M2: sigma_measure_b,H: real > real,N: sigma_measure_real] :
      ( ( member4641211897452655452od_a_b
        @ ^ [X: real] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5261090283068522538od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_b
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_523072396149930113real_b @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_107_measurable__Pair2__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_108_measurable__Pair2__compose,axiom,
    ! [F: real > product_prod_a_b,G: real > real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member7738959517937940581b_real
        @ ^ [X: real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_109_measurable__Pair2__compose,axiom,
    ! [F: b > a,G: b > b,M: sigma_measure_b,M1: sigma_measure_a,M2: sigma_measure_b,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member1389369442121846245od_a_b
        @ ^ [X: b] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1961069564447577785od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( G @ ( H @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_110_measurable__Pair2__compose,axiom,
    ! [F: b > real,G: b > real,M: sigma_measure_b,M1: sigma_measure_real,M2: sigma_measure_real,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member5936402216828068598l_real
        @ ^ [X: b] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4120751743698409674l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member1942095430938869870b_real
          @ ^ [X: product_prod_a_b] : ( G @ ( H @ X ) )
          @ ( sigma_8022152630303495172b_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_111_measurable__Pair2__compose,axiom,
    ! [F: b > product_prod_a_b,G: b > real,M: sigma_measure_b,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member7735767025119871772b_real
        @ ^ [X: b] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_178895490645910834b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member1942095430938869870b_real
          @ ^ [X: product_prod_a_b] : ( G @ ( H @ X ) )
          @ ( sigma_8022152630303495172b_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_112_measurable__Pair2__compose,axiom,
    ! [F: real > a,G: real > b,M: sigma_measure_real,M1: sigma_measure_a,M2: sigma_measure_b,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member4641211897452655452od_a_b
        @ ^ [X: real] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5261090283068522538od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member3001133527305069980real_b
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( H @ X ) )
          @ ( sigma_3928014871867360490real_b @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_113_measurable__Pair2__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( H @ X ) )
          @ ( sigma_2402725360379274625l_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_114_measurable__Pair2__compose,axiom,
    ! [F: real > product_prod_a_b,G: real > real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member7738959517937940581b_real
        @ ^ [X: real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( G @ ( H @ X ) )
          @ ( sigma_2402725360379274625l_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_115_measurable__Pair2__compose,axiom,
    ! [F: produc7929879615859924995b_real > product_prod_a_b,G: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: real > produc7929879615859924995b_real,N: sigma_measure_real] :
      ( ( member1118978200981565500b_real
        @ ^ [X: produc7929879615859924995b_real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member7738959517937940581b_real @ H @ ( sigma_5526981039175570177b_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( G @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M2 ) ) ) ) ).

% measurable_Pair2_compose
thf(fact_116_measurable__Pair1__compose,axiom,
    ! [F: real > a,G: real > b,M: sigma_measure_real,M1: sigma_measure_a,M2: sigma_measure_b,H: real > real,N: sigma_measure_real] :
      ( ( member4641211897452655452od_a_b
        @ ^ [X: real] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5261090283068522538od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_a
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_523072396149930112real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_117_measurable__Pair1__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5267869275261027754l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_118_measurable__Pair1__compose,axiom,
    ! [F: real > product_prod_a_b,G: real > real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: real > real,N: sigma_measure_real] :
      ( ( member7738959517937940581b_real
        @ ^ [X: real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member_real_real @ H @ ( sigma_5267869275261027754l_real @ N @ M ) )
       => ( member4641211897452655452od_a_b
          @ ^ [X: real] : ( F @ ( H @ X ) )
          @ ( sigma_5261090283068522538od_a_b @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_119_measurable__Pair1__compose,axiom,
    ! [F: b > a,G: b > b,M: sigma_measure_b,M1: sigma_measure_a,M2: sigma_measure_b,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member1389369442121846245od_a_b
        @ ^ [X: b] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_1961069564447577785od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member1080921165755314130_a_b_a
          @ ^ [X: product_prod_a_b] : ( F @ ( H @ X ) )
          @ ( sigma_7225552367035416486_a_b_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_120_measurable__Pair1__compose,axiom,
    ! [F: b > real,G: b > real,M: sigma_measure_b,M1: sigma_measure_real,M2: sigma_measure_real,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member5936402216828068598l_real
        @ ^ [X: b] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_4120751743698409674l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member1942095430938869870b_real
          @ ^ [X: product_prod_a_b] : ( F @ ( H @ X ) )
          @ ( sigma_8022152630303495172b_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_121_measurable__Pair1__compose,axiom,
    ! [F: b > product_prod_a_b,G: b > real,M: sigma_measure_b,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: product_prod_a_b > b,N: sigma_5782782616773303333od_a_b] :
      ( ( member7735767025119871772b_real
        @ ^ [X: b] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_178895490645910834b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member1080921170058542931_a_b_b @ H @ ( sigma_7225552367035416487_a_b_b @ N @ M ) )
       => ( member2755110357185872764od_a_b
          @ ^ [X: product_prod_a_b] : ( F @ ( H @ X ) )
          @ ( sigma_5476089079233491280od_a_b @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_122_measurable__Pair1__compose,axiom,
    ! [F: real > a,G: real > b,M: sigma_measure_real,M1: sigma_measure_a,M2: sigma_measure_b,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member4641211897452655452od_a_b
        @ ^ [X: real] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5261090283068522538od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member3001133523001841179real_a
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( H @ X ) )
          @ ( sigma_3928014871867360489real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_123_measurable__Pair1__compose,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( H @ X ) )
          @ ( sigma_2402725360379274625l_real @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_124_measurable__Pair1__compose,axiom,
    ! [F: real > product_prod_a_b,G: real > real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,H: produc7929879615859924995b_real > real,N: sigma_4174442854635832540b_real] :
      ( ( member7738959517937940581b_real
        @ ^ [X: real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
     => ( ( member5467075257862553189l_real @ H @ ( sigma_2402725360379274625l_real @ N @ M ) )
       => ( member2224156091608774853od_a_b
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( H @ X ) )
          @ ( sigma_8670534473570876307od_a_b @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_125_measurable__Pair1__compose,axiom,
    ! [F: produc7929879615859924995b_real > a,G: produc7929879615859924995b_real > b,M: sigma_4174442854635832540b_real,M1: sigma_measure_a,M2: sigma_measure_b,H: produc7929879615859924995b_real > produc7929879615859924995b_real,N: sigma_4174442854635832540b_real] :
      ( ( member2224156091608774853od_a_b
        @ ^ [X: produc7929879615859924995b_real] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
        @ ( sigma_8670534473570876307od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
     => ( ( member1118978200981565500b_real @ H @ ( sigma_1108951568718238936b_real @ N @ M ) )
       => ( member3001133523001841179real_a
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( H @ X ) )
          @ ( sigma_3928014871867360489real_a @ N @ M1 ) ) ) ) ).

% measurable_Pair1_compose
thf(fact_126_measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,M2: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ M2 ) )
       => ( member1006085877915592749b_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc425996204919967944b_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_184416647750077563b_real @ M @ ( binary1215910559743187449b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_127_measurable__Pair,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_b,G: product_prod_a_b > b,M2: sigma_measure_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ M1 ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) )
       => ( member2119461356580886075od_b_b
          @ ^ [X: product_prod_a_b] : ( product_Pair_b_b @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_2688633497260815247od_b_b @ M @ ( binary7303355217300867336re_b_b @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_128_measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,M2: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member284797873430130221l_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc6525112562978448200l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7135904424529465979l_real @ M @ ( binary7315026917801667705l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_129_measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,M2: sigma_4174442854635832540b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ M2 ) )
       => ( member5942386789700278916b_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc876450409233290399b_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_371502731946915922b_real @ M @ ( binary388897294610017232b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_130_measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > product_prod_a_b,M: sigma_4174442854635832540b_real,M1: sigma_5782782616773303333od_a_b,G: produc7929879615859924995b_real > real,M2: sigma_measure_real] :
      ( ( member2224156091608774853od_a_b @ F @ ( sigma_8670534473570876307od_a_b @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member1118978200981565500b_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_131_measurable__Pair,axiom,
    ! [F: real > product_prod_a_b,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,G: real > real,M2: sigma_measure_real] :
      ( ( member4641211897452655452od_a_b @ F @ ( sigma_5261090283068522538od_a_b @ M @ M1 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
       => ( member7738959517937940581b_real
          @ ^ [X: real] : ( produc9021675809588086653b_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_132_measurable__Pair,axiom,
    ! [F: product_prod_a_b > a,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_a,G: product_prod_a_b > b,M2: sigma_measure_b] :
      ( ( member1080921165755314130_a_b_a @ F @ ( sigma_7225552367035416486_a_b_a @ M @ M1 ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) )
       => ( member2755110357185872764od_a_b
          @ ^ [X: product_prod_a_b] : ( product_Pair_a_b @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_5476089079233491280od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_133_measurable__Pair,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,G: produc7929879615859924995b_real > real,M2: sigma_measure_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member174722968998780374l_real
          @ ^ [X: produc7929879615859924995b_real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_3866086195517720740l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_134_measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,M1: sigma_measure_real,G: real > real,M2: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M1 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
       => ( member9086635009091248365l_real
          @ ^ [X: real] : ( produc4511245868158468465l_real @ ( F @ X ) @ ( G @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_Pair
thf(fact_135_measurable__fst_H,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_5782782616773303333od_a_b,P: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ N @ P ) ) )
     => ( member2224156091608774853od_a_b
        @ ^ [X: produc7929879615859924995b_real] : ( produc7266368915318857873b_real @ ( F @ X ) )
        @ ( sigma_8670534473570876307od_a_b @ M @ N ) ) ) ).

% measurable_fst'
thf(fact_136_measurable__fst_H,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ ( binary6478037234023840930l_real @ N @ P ) ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( produc5828954698716094813l_real @ ( F @ X ) )
        @ ( sigma_2402725360379274625l_real @ M @ N ) ) ) ).

% measurable_fst'
thf(fact_137_measurable__fst_H,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ N @ P ) ) )
     => ( member_real_real
        @ ^ [X: real] : ( produc5828954698716094813l_real @ ( F @ X ) )
        @ ( sigma_5267869275261027754l_real @ M @ N ) ) ) ).

% measurable_fst'
thf(fact_138_sets_Otop,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.top
thf(fact_139_measurable__fst,axiom,
    ! [M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] : ( member2224156091608774853od_a_b @ produc7266368915318857873b_real @ ( sigma_8670534473570876307od_a_b @ ( binary496794562119935756b_real @ M1 @ M2 ) @ M1 ) ) ).

% measurable_fst
thf(fact_140_measurable__fst,axiom,
    ! [M1: sigma_measure_a,M2: sigma_measure_b] : ( member1080921165755314130_a_b_a @ product_fst_a_b @ ( sigma_7225552367035416486_a_b_a @ ( binary867438762418767561re_a_b @ M1 @ M2 ) @ M1 ) ) ).

% measurable_fst
thf(fact_141_measurable__fst,axiom,
    ! [M1: sigma_measure_real,M2: sigma_measure_real] : ( member6699615393305559423l_real @ produc5828954698716094813l_real @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) @ M1 ) ) ).

% measurable_fst
thf(fact_142_pair__measureI,axiom,
    ! [X2: set_Product_prod_a_b,A: sigma_5782782616773303333od_a_b,Y: set_real,B2: sigma_measure_real] :
      ( ( member1887650552762594865od_a_b @ X2 @ ( sigma_377313353287227655od_a_b @ A ) )
     => ( ( member_set_real @ Y @ ( sigma_sets_real @ B2 ) )
       => ( member2587080422514930128b_real
          @ ( produc7302604275931811084b_real @ X2
            @ ^ [Uu: product_prod_a_b] : Y )
          @ ( sigma_5719578828308484196b_real @ ( binary496794562119935756b_real @ A @ B2 ) ) ) ) ) ).

% pair_measureI
thf(fact_143_pair__measureI,axiom,
    ! [X2: set_a,A: sigma_measure_a,Y: set_b,B2: sigma_measure_b] :
      ( ( member_set_a @ X2 @ ( sigma_sets_a @ A ) )
     => ( ( member_set_b @ Y @ ( sigma_sets_b @ B2 ) )
       => ( member1887650552762594865od_a_b
          @ ( product_Sigma_a_b @ X2
            @ ^ [Uu: a] : Y )
          @ ( sigma_377313353287227655od_a_b @ ( binary867438762418767561re_a_b @ A @ B2 ) ) ) ) ) ).

% pair_measureI
thf(fact_144_pair__measureI,axiom,
    ! [X2: set_real,A: sigma_measure_real,Y: set_real,B2: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_sets_real @ A ) )
     => ( ( member_set_real @ Y @ ( sigma_sets_real @ B2 ) )
       => ( member6767226683953100098l_real
          @ ( produc1020309166918817442l_real @ X2
            @ ^ [Uu: real] : Y )
          @ ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ A @ B2 ) ) ) ) ) ).

% pair_measureI
thf(fact_145_space__pair__measure,axiom,
    ! [A: sigma_5782782616773303333od_a_b,B2: sigma_measure_real] :
      ( ( sigma_709238742955535483b_real @ ( binary496794562119935756b_real @ A @ B2 ) )
      = ( produc7302604275931811084b_real @ ( sigma_1564438922714704432od_a_b @ A )
        @ ^ [Uu: product_prod_a_b] : ( sigma_space_real @ B2 ) ) ) ).

% space_pair_measure
thf(fact_146_space__pair__measure,axiom,
    ! [A: sigma_measure_a,B2: sigma_measure_b] :
      ( ( sigma_1564438922714704432od_a_b @ ( binary867438762418767561re_a_b @ A @ B2 ) )
      = ( product_Sigma_a_b @ ( sigma_space_a @ A )
        @ ^ [Uu: a] : ( sigma_space_b @ B2 ) ) ) ).

% space_pair_measure
thf(fact_147_space__pair__measure,axiom,
    ! [A: sigma_measure_real,B2: sigma_measure_real] :
      ( ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ A @ B2 ) )
      = ( produc1020309166918817442l_real @ ( sigma_space_real @ A )
        @ ^ [Uu: real] : ( sigma_space_real @ B2 ) ) ) ).

% space_pair_measure
thf(fact_148_SIGMA__Collect__eq,axiom,
    ! [M: sigma_5782782616773303333od_a_b,N: sigma_measure_real,P: product_prod_a_b > real > $o] :
      ( ( produc7302604275931811084b_real @ ( sigma_1564438922714704432od_a_b @ M )
        @ ^ [X: product_prod_a_b] :
            ( collect_real
            @ ^ [Y2: real] :
                ( ( member_real @ Y2 @ ( sigma_space_real @ N ) )
                & ( P @ X @ Y2 ) ) ) )
      = ( collec4671965459240937304b_real
        @ ^ [X: produc7929879615859924995b_real] :
            ( ( member6125958737618052890b_real @ X @ ( sigma_709238742955535483b_real @ ( binary496794562119935756b_real @ M @ N ) ) )
            & ( P @ ( produc7266368915318857873b_real @ X ) @ ( produc7037151529651111887b_real @ X ) ) ) ) ) ).

% SIGMA_Collect_eq
thf(fact_149_SIGMA__Collect__eq,axiom,
    ! [M: sigma_measure_a,N: sigma_measure_b,P: a > b > $o] :
      ( ( product_Sigma_a_b @ ( sigma_space_a @ M )
        @ ^ [X: a] :
            ( collect_b
            @ ^ [Y2: b] :
                ( ( member_b @ Y2 @ ( sigma_space_b @ N ) )
                & ( P @ X @ Y2 ) ) ) )
      = ( collec3336397801687681299od_a_b
        @ ^ [X: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X @ ( sigma_1564438922714704432od_a_b @ ( binary867438762418767561re_a_b @ M @ N ) ) )
            & ( P @ ( product_fst_a_b @ X ) @ ( product_snd_a_b @ X ) ) ) ) ) ).

% SIGMA_Collect_eq
thf(fact_150_SIGMA__Collect__eq,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,P: real > real > $o] :
      ( ( produc1020309166918817442l_real @ ( sigma_space_real @ M )
        @ ^ [X: real] :
            ( collect_real
            @ ^ [Y2: real] :
                ( ( member_real @ Y2 @ ( sigma_space_real @ N ) )
                & ( P @ X @ Y2 ) ) ) )
      = ( collec3799799289383736868l_real
        @ ^ [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ ( sigma_2519298556039103681l_real @ ( binary6478037234023840930l_real @ M @ N ) ) )
            & ( P @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) ) ) ) ).

% SIGMA_Collect_eq
thf(fact_151_measurable__Pair1_H,axiom,
    ! [X2: product_prod_a_b,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] :
      ( ( member1426531481828664017od_a_b @ X2 @ ( sigma_1564438922714704432od_a_b @ M1 ) )
     => ( member7738959517937940581b_real @ ( produc9021675809588086653b_real @ X2 ) @ ( sigma_5526981039175570177b_real @ M2 @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1'
thf(fact_152_measurable__Pair1_H,axiom,
    ! [X2: a,M1: sigma_measure_a,M2: sigma_measure_b] :
      ( ( member_a @ X2 @ ( sigma_space_a @ M1 ) )
     => ( member1389369442121846245od_a_b @ ( product_Pair_a_b @ X2 ) @ ( sigma_1961069564447577785od_a_b @ M2 @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1'
thf(fact_153_measurable__Pair1_H,axiom,
    ! [X2: real,M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
     => ( member9086635009091248365l_real @ ( produc4511245868158468465l_real @ X2 ) @ ( sigma_7998147297565726139l_real @ M2 @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair1'
thf(fact_154_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_real,M4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M4 ) )
     => ( ( sigma_space_real @ M )
        = ( sigma_space_real @ M4 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_155_measurable__cong,axiom,
    ! [M: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > real,G: produc7929879615859924995b_real > real,M4: sigma_measure_real] :
      ( ! [W: produc7929879615859924995b_real] :
          ( ( member6125958737618052890b_real @ W @ ( sigma_709238742955535483b_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
        = ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M4 ) ) ) ) ).

% measurable_cong
thf(fact_156_measurable__cong,axiom,
    ! [M: sigma_5782782616773303333od_a_b,F: product_prod_a_b > b,G: product_prod_a_b > b,M4: sigma_measure_b] :
      ( ! [W: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ W @ ( sigma_1564438922714704432od_a_b @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
        = ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) ) ) ) ).

% measurable_cong
thf(fact_157_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real,M4: sigma_measure_real] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
        = ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M4 ) ) ) ) ).

% measurable_cong
thf(fact_158_measurable__cong,axiom,
    ! [M: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > produc7929879615859924995b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,M4: sigma_4174442854635832540b_real] :
      ( ! [W: produc7929879615859924995b_real] :
          ( ( member6125958737618052890b_real @ W @ ( sigma_709238742955535483b_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
        = ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ M4 ) ) ) ) ).

% measurable_cong
thf(fact_159_measurable__space,axiom,
    ! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,X2: real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
       => ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_160_measurable__space,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,A: sigma_measure_real,X2: set_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ A ) )
     => ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
       => ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_161_measurable__space,axiom,
    ! [F: real > set_real,M: sigma_measure_real,A: sigma_3733394171116455995t_real,X2: real] :
      ( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ A ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
       => ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).

% measurable_space
thf(fact_162_measurable__space,axiom,
    ! [F: set_real > set_real,M: sigma_3733394171116455995t_real,A: sigma_3733394171116455995t_real,X2: set_real] :
      ( ( member8820419758626733818t_real @ F @ ( sigma_8759315257289043734t_real @ M @ A ) )
     => ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
       => ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).

% measurable_space
thf(fact_163_measurable__space,axiom,
    ! [F: real > real > real,M: sigma_measure_real,A: sigma_4258434043392614480l_real,X2: real] :
      ( ( member8878224140454985689l_real @ F @ ( sigma_5187198232267106421l_real @ M @ A ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
       => ( member_real_real @ ( F @ X2 ) @ ( sigma_3619470280215722479l_real @ A ) ) ) ) ).

% measurable_space
thf(fact_164_measurable__space,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,A: sigma_measure_real,X2: real > real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ A ) )
     => ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M ) )
       => ( member_real @ ( F @ X2 ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_165_measurable__space,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,A: sigma_measure_b,X2: product_prod_a_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ A ) )
     => ( ( member1426531481828664017od_a_b @ X2 @ ( sigma_1564438922714704432od_a_b @ M ) )
       => ( member_b @ ( F @ X2 ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_166_measurable__space,axiom,
    ! [F: set_real > real > real,M: sigma_3733394171116455995t_real,A: sigma_4258434043392614480l_real,X2: set_real] :
      ( ( member4134752328433604879l_real @ F @ ( sigma_6541816678006044075l_real @ M @ A ) )
     => ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M ) )
       => ( member_real_real @ ( F @ X2 ) @ ( sigma_3619470280215722479l_real @ A ) ) ) ) ).

% measurable_space
thf(fact_167_measurable__space,axiom,
    ! [F: ( real > real ) > set_real,M: sigma_4258434043392614480l_real,A: sigma_3733394171116455995t_real,X2: real > real] :
      ( ( member3724334074564800271t_real @ F @ ( sigma_1826694118076045611t_real @ M @ A ) )
     => ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M ) )
       => ( member_set_real @ ( F @ X2 ) @ ( sigma_space_set_real @ A ) ) ) ) ).

% measurable_space
thf(fact_168_measurable__space,axiom,
    ! [F: real > product_prod_a_b > b,M: sigma_measure_real,A: sigma_7535330959625612565_a_b_b,X2: real] :
      ( ( member3163235428394349598_a_b_b @ F @ ( sigma_7394143108149223354_a_b_b @ M @ A ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M ) )
       => ( member1080921170058542931_a_b_b @ ( F @ X2 ) @ ( sigma_8376800615221029940_a_b_b @ A ) ) ) ) ).

% measurable_space
thf(fact_169_mem__Collect__eq,axiom,
    ! [A3: produc7929879615859924995b_real > real,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member5467075257862553189l_real @ A3 @ ( collec869437573433150115l_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_170_mem__Collect__eq,axiom,
    ! [A3: set_real,P: set_real > $o] :
      ( ( member_set_real @ A3 @ ( collect_set_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_171_mem__Collect__eq,axiom,
    ! [A3: real,P: real > $o] :
      ( ( member_real @ A3 @ ( collect_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_172_mem__Collect__eq,axiom,
    ! [A3: product_prod_a_b > b,P: ( product_prod_a_b > b ) > $o] :
      ( ( member1080921170058542931_a_b_b @ A3 @ ( collec2250730345169186961_a_b_b @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_173_mem__Collect__eq,axiom,
    ! [A3: real > real,P: ( real > real ) > $o] :
      ( ( member_real_real @ A3 @ ( collect_real_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_174_mem__Collect__eq,axiom,
    ! [A3: produc7929879615859924995b_real > produc7929879615859924995b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member1118978200981565500b_real @ A3 @ ( collec5735648035525910650b_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_175_Collect__mem__eq,axiom,
    ! [A: set_Pr3917002265660371076l_real] :
      ( ( collec869437573433150115l_real
        @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
    ! [A: set_set_real] :
      ( ( collect_set_real
        @ ^ [X: set_real] : ( member_set_real @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_177_Collect__mem__eq,axiom,
    ! [A: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_178_Collect__mem__eq,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b] :
      ( ( collec2250730345169186961_a_b_b
        @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_179_Collect__mem__eq,axiom,
    ! [A: set_real_real] :
      ( ( collect_real_real
        @ ^ [X: real > real] : ( member_real_real @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_180_Collect__mem__eq,axiom,
    ! [A: set_Pr1027596265373204187b_real] :
      ( ( collec5735648035525910650b_real
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_181_measurable__cong__simp,axiom,
    ! [M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,M4: sigma_measure_real,N2: sigma_measure_real,F: produc7929879615859924995b_real > real,G: produc7929879615859924995b_real > real] :
      ( ( M = N )
     => ( ( M4 = N2 )
       => ( ! [W: produc7929879615859924995b_real] :
              ( ( member6125958737618052890b_real @ W @ ( sigma_709238742955535483b_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
            = ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_182_measurable__cong__simp,axiom,
    ! [M: sigma_5782782616773303333od_a_b,N: sigma_5782782616773303333od_a_b,M4: sigma_measure_b,N2: sigma_measure_b,F: product_prod_a_b > b,G: product_prod_a_b > b] :
      ( ( M = N )
     => ( ( M4 = N2 )
       => ( ! [W: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ W @ ( sigma_1564438922714704432od_a_b @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
            = ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_183_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M4: sigma_measure_real,N2: sigma_measure_real,F: real > real,G: real > real] :
      ( ( M = N )
     => ( ( M4 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
            = ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_184_measurable__cong__simp,axiom,
    ! [M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real,N2: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > produc7929879615859924995b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real] :
      ( ( M = N )
     => ( ( M4 = N2 )
       => ( ! [W: produc7929879615859924995b_real] :
              ( ( member6125958737618052890b_real @ W @ ( sigma_709238742955535483b_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
            = ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_185_measurable__Pair2_H,axiom,
    ! [Y: real,M2: sigma_measure_real,M1: sigma_5782782616773303333od_a_b] :
      ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
     => ( member3452904153566040133b_real
        @ ^ [X: product_prod_a_b] : ( produc9021675809588086653b_real @ X @ Y )
        @ ( sigma_6988460731143032155b_real @ M1 @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair2'
thf(fact_186_measurable__Pair2_H,axiom,
    ! [Y: b,M2: sigma_measure_b,M1: sigma_measure_a] :
      ( ( member_b @ Y @ ( sigma_space_b @ M2 ) )
     => ( member2028809742453662374od_a_b
        @ ^ [X: a] : ( product_Pair_a_b @ X @ Y )
        @ ( sigma_2600509864779393914od_a_b @ M1 @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) ) ) ).

% measurable_Pair2'
thf(fact_187_measurable__Pair2_H,axiom,
    ! [Y: real,M2: sigma_measure_real,M1: sigma_measure_real] :
      ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
     => ( member9086635009091248365l_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ Y )
        @ ( sigma_7998147297565726139l_real @ M1 @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ).

% measurable_Pair2'
thf(fact_188_sets_Osets__Collect__const,axiom,
    ! [M: sigma_1368263781377809575l_real,P: $o] :
      ( member3057847817020379547l_real
      @ ( collec869437573433150115l_real
        @ ^ [X: produc7929879615859924995b_real > real] :
            ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
            & P ) )
      @ ( sigma_7475115973638204335l_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_189_sets_Osets__Collect__const,axiom,
    ! [M: sigma_3733394171116455995t_real,P: $o] :
      ( member_set_set_real
      @ ( collect_set_real
        @ ^ [X: set_real] :
            ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
            & P ) )
      @ ( sigma_sets_set_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_190_sets_Osets__Collect__const,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,P: $o] :
      ( member4198857394546769929_a_b_b
      @ ( collec2250730345169186961_a_b_b
        @ ^ [X: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
            & P ) )
      @ ( sigma_7736500445245453213_a_b_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_191_sets_Osets__Collect__const,axiom,
    ! [M: sigma_4258434043392614480l_real,P: $o] :
      ( member_set_real_real2
      @ ( collect_real_real
        @ ^ [X: real > real] :
            ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
            & P ) )
      @ ( sigma_sets_real_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_192_sets_Osets__Collect__const,axiom,
    ! [M: sigma_7931231546270528766b_real,P: $o] :
      ( member566263741820371442b_real
      @ ( collec5735648035525910650b_real
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
            ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
            & P ) )
      @ ( sigma_1595522527309588870b_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_193_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real,P: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( sigma_space_real @ M ) )
            & P ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_194_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_1368263781377809575l_real,P: ( produc7929879615859924995b_real > real ) > $o,Q: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member3057847817020379547l_real
        @ ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7475115973638204335l_real @ M ) )
     => ( ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_195_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_set_real @ M ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_196_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,P: ( product_prod_a_b > b ) > $o,Q: ( product_prod_a_b > b ) > $o] :
      ( ( member4198857394546769929_a_b_b
        @ ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7736500445245453213_a_b_b @ M ) )
     => ( ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_197_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_real @ M ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_198_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_7931231546270528766b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o,Q: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member566263741820371442b_real
        @ ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_1595522527309588870b_real @ M ) )
     => ( ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) )
       => ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_199_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( Q @ X )
                  | ( P @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_200_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_1368263781377809575l_real,P: ( produc7929879615859924995b_real > real ) > $o,Q: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member3057847817020379547l_real
        @ ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7475115973638204335l_real @ M ) )
     => ( ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_201_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_set_real @ M ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_202_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,P: ( product_prod_a_b > b ) > $o,Q: ( product_prod_a_b > b ) > $o] :
      ( ( member4198857394546769929_a_b_b
        @ ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7736500445245453213_a_b_b @ M ) )
     => ( ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_203_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_real @ M ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_204_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_7931231546270528766b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o,Q: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member566263741820371442b_real
        @ ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_1595522527309588870b_real @ M ) )
     => ( ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) )
       => ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_205_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X )
                & ( P @ X ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_206_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_1368263781377809575l_real,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member3057847817020379547l_real
        @ ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7475115973638204335l_real @ M ) )
     => ( member3057847817020379547l_real
        @ ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_7475115973638204335l_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_207_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_3733394171116455995t_real,P: set_real > $o] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_set_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_208_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,P: ( product_prod_a_b > b ) > $o] :
      ( ( member4198857394546769929_a_b_b
        @ ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7736500445245453213_a_b_b @ M ) )
     => ( member4198857394546769929_a_b_b
        @ ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_209_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_210_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_7931231546270528766b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member566263741820371442b_real
        @ ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_1595522527309588870b_real @ M ) )
     => ( member566263741820371442b_real
        @ ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_1595522527309588870b_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_211_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real,P: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ~ ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_212_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_1368263781377809575l_real,P: ( produc7929879615859924995b_real > real ) > $o,Q: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member3057847817020379547l_real
        @ ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7475115973638204335l_real @ M ) )
     => ( ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_213_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_3733394171116455995t_real,P: set_real > $o,Q: set_real > $o] :
      ( ( member_set_set_real
        @ ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_set_real @ M ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_set_real @ M ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_214_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,P: ( product_prod_a_b > b ) > $o,Q: ( product_prod_a_b > b ) > $o] :
      ( ( member4198857394546769929_a_b_b
        @ ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_7736500445245453213_a_b_b @ M ) )
     => ( ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_215_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_4258434043392614480l_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
      ( ( member_set_real_real2
        @ ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real_real @ M ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real_real @ M ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_216_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_7931231546270528766b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o,Q: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member566263741820371442b_real
        @ ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_1595522527309588870b_real @ M ) )
     => ( ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) )
       => ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_217_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ ( sigma_space_real @ M ) )
              & ( P @ X ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( Q @ X ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( Q @ X )
                 => ( P @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_218_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_1368263781377809575l_real,Pb: $o] :
      ( member3057847817020379547l_real
      @ ( collec869437573433150115l_real
        @ ^ [X: produc7929879615859924995b_real > real] :
            ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
            & Pb ) )
      @ ( sigma_7475115973638204335l_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_219_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_3733394171116455995t_real,Pb: $o] :
      ( member_set_set_real
      @ ( collect_set_real
        @ ^ [X: set_real] :
            ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_set_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_220_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_7535330959625612565_a_b_b,Pb: $o] :
      ( member4198857394546769929_a_b_b
      @ ( collec2250730345169186961_a_b_b
        @ ^ [X: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
            & Pb ) )
      @ ( sigma_7736500445245453213_a_b_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_221_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_4258434043392614480l_real,Pb: $o] :
      ( member_set_real_real2
      @ ( collect_real_real
        @ ^ [X: real > real] :
            ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_222_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_7931231546270528766b_real,Pb: $o] :
      ( member566263741820371442b_real
      @ ( collec5735648035525910650b_real
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
            ( ( member1118978200981565500b_real @ X @ ( sigma_6346777283556531613b_real @ M ) )
            & Pb ) )
      @ ( sigma_1595522527309588870b_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_223_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real,Pb: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ ( sigma_space_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_224_measurable__const,axiom,
    ! [C: real,M4: sigma_measure_real,M: sigma_4174442854635832540b_real] :
      ( ( member_real @ C @ ( sigma_space_real @ M4 ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : C
        @ ( sigma_2402725360379274625l_real @ M @ M4 ) ) ) ).

% measurable_const
thf(fact_225_measurable__const,axiom,
    ! [C: b,M4: sigma_measure_b,M: sigma_5782782616773303333od_a_b] :
      ( ( member_b @ C @ ( sigma_space_b @ M4 ) )
     => ( member1080921170058542931_a_b_b
        @ ^ [X: product_prod_a_b] : C
        @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) ) ) ).

% measurable_const
thf(fact_226_measurable__const,axiom,
    ! [C: real,M4: sigma_measure_real,M: sigma_measure_real] :
      ( ( member_real @ C @ ( sigma_space_real @ M4 ) )
     => ( member_real_real
        @ ^ [X: real] : C
        @ ( sigma_5267869275261027754l_real @ M @ M4 ) ) ) ).

% measurable_const
thf(fact_227_measurable__const,axiom,
    ! [C: produc7929879615859924995b_real,M4: sigma_4174442854635832540b_real,M: sigma_4174442854635832540b_real] :
      ( ( member6125958737618052890b_real @ C @ ( sigma_709238742955535483b_real @ M4 ) )
     => ( member1118978200981565500b_real
        @ ^ [X: produc7929879615859924995b_real] : C
        @ ( sigma_1108951568718238936b_real @ M @ M4 ) ) ) ).

% measurable_const
thf(fact_228_measurable__Pair1,axiom,
    ! [F: product_prod_a_b > b,M1: sigma_measure_a,M2: sigma_measure_b,M: sigma_measure_b,Y: b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ M1 @ M2 ) @ M ) )
     => ( ( member_b @ Y @ ( sigma_space_b @ M2 ) )
       => ( member_a_b
          @ ^ [X: a] : ( F @ ( product_Pair_a_b @ X @ Y ) )
          @ ( sigma_measurable_a_b @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_229_measurable__Pair1,axiom,
    ! [F: produc2422161461964618553l_real > real,M1: sigma_measure_real,M2: sigma_measure_real,M: sigma_measure_real,Y: real] :
      ( ( member6699615393305559423l_real @ F @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( produc4511245868158468465l_real @ X @ Y ) )
          @ ( sigma_5267869275261027754l_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_230_measurable__Pair1,axiom,
    ! [F: produc1875881916783206511t_real > real,M1: sigma_measure_real,M2: sigma_3733394171116455995t_real,M: sigma_measure_real,Y: set_real] :
      ( ( member4682917391315569589l_real @ F @ ( sigma_666548255234693323l_real @ ( binary2139931705814519000t_real @ M1 @ M2 ) @ M ) )
     => ( ( member_set_real @ Y @ ( sigma_space_set_real @ M2 ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( produc3307717890874201511t_real @ X @ Y ) )
          @ ( sigma_5267869275261027754l_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_231_measurable__Pair1,axiom,
    ! [F: produc9105736867229962372l_real > real,M1: sigma_measure_real,M2: sigma_4258434043392614480l_real,M: sigma_measure_real,Y: real > real] :
      ( ( member1507221849960417610l_real @ F @ ( sigma_7070337421754014816l_real @ ( binary4810646286260424557l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real_real @ Y @ ( sigma_3619470280215722479l_real @ M2 ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( produc7028217496045009468l_real @ X @ Y ) )
          @ ( sigma_5267869275261027754l_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_232_measurable__Pair1,axiom,
    ! [F: produc7929879615859924995b_real > b,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,M: sigma_measure_b,Y: real] :
      ( ( member3001133527305069980real_b @ F @ ( sigma_3928014871867360490real_b @ ( binary496794562119935756b_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( produc9021675809588086653b_real @ X @ Y ) )
          @ ( sigma_7225552367035416487_a_b_b @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_233_measurable__Pair1,axiom,
    ! [F: produc7929879615859924995b_real > real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,M: sigma_measure_real,Y: real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
       => ( member1942095430938869870b_real
          @ ^ [X: product_prod_a_b] : ( F @ ( produc9021675809588086653b_real @ X @ Y ) )
          @ ( sigma_8022152630303495172b_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_234_measurable__Pair1,axiom,
    ! [F: produc3115013810655550777t_real > b,M1: sigma_5782782616773303333od_a_b,M2: sigma_3733394171116455995t_real,M: sigma_measure_b,Y: set_real] :
      ( ( member7504260724026431974real_b @ F @ ( sigma_1968366296916378932real_b @ ( binary4158155687072845634t_real @ M1 @ M2 ) @ M ) )
     => ( ( member_set_real @ Y @ ( sigma_space_set_real @ M2 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( produc350744015223090099t_real @ X @ Y ) )
          @ ( sigma_7225552367035416487_a_b_b @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_235_measurable__Pair1,axiom,
    ! [F: produc2738766212015491856l_real > real,M1: sigma_4174442854635832540b_real,M2: sigma_measure_real,M: sigma_measure_real,Y: real] :
      ( ( member1512972631563433430l_real @ F @ ( sigma_7595990971189275884l_real @ ( binary7315026917801667705l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ Y @ ( sigma_space_real @ M2 ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc6525112562978448200l_real @ X @ Y ) )
          @ ( sigma_2402725360379274625l_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_236_measurable__Pair1,axiom,
    ! [F: produc4020472289826027854l_real > b,M1: sigma_5782782616773303333od_a_b,M2: sigma_4258434043392614480l_real,M: sigma_measure_b,Y: real > real] :
      ( ( member1848875981206096145real_b @ F @ ( sigma_358066140431186527real_b @ ( binary6999594221180239319l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real_real @ Y @ ( sigma_3619470280215722479l_real @ M2 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( produc5247457396530362440l_real @ X @ Y ) )
          @ ( sigma_7225552367035416487_a_b_b @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_237_measurable__Pair1,axiom,
    ! [F: produc2732119482649539401_a_b_b > real,M1: sigma_measure_real,M2: sigma_7535330959625612565_a_b_b,M: sigma_measure_real,Y: product_prod_a_b > b] :
      ( ( member414368567675257999b_real @ F @ ( sigma_8034178502852740389b_real @ ( binary3228914569849170610_a_b_b @ M1 @ M2 ) @ M ) )
     => ( ( member1080921170058542931_a_b_b @ Y @ ( sigma_8376800615221029940_a_b_b @ M2 ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( produc4396699455692612097_a_b_b @ X @ Y ) )
          @ ( sigma_5267869275261027754l_real @ M1 @ M ) ) ) ) ).

% measurable_Pair1
thf(fact_238_measurable__Pair2,axiom,
    ! [F: product_prod_a_b > b,M1: sigma_measure_a,M2: sigma_measure_b,M: sigma_measure_b,X2: a] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ M1 @ M2 ) @ M ) )
     => ( ( member_a @ X2 @ ( sigma_space_a @ M1 ) )
       => ( member_b_b
          @ ^ [Y2: b] : ( F @ ( product_Pair_a_b @ X2 @ Y2 ) )
          @ ( sigma_measurable_b_b @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_239_measurable__Pair2,axiom,
    ! [F: produc2422161461964618553l_real > real,M1: sigma_measure_real,M2: sigma_measure_real,M: sigma_measure_real,X2: real] :
      ( ( member6699615393305559423l_real @ F @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( F @ ( produc4511245868158468465l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_240_measurable__Pair2,axiom,
    ! [F: produc2329993357307243631l_real > real,M1: sigma_3733394171116455995t_real,M2: sigma_measure_real,M: sigma_measure_real,X2: set_real] :
      ( ( member5462418605983881141l_real @ F @ ( sigma_1446049469903004875l_real @ ( binary5154340633480048088l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M1 ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( F @ ( produc6322126818539730599l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_241_measurable__Pair2,axiom,
    ! [F: produc1405624045518934769od_a_b > b,M1: sigma_measure_real,M2: sigma_5782782616773303333od_a_b,M: sigma_measure_b,X2: real] :
      ( ( member8926022635040246318_a_b_b @ F @ ( sigma_629531942747761020_a_b_b @ ( binary6959104251739738930od_a_b @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( F @ ( produc6260613462353114019od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_242_measurable__Pair2,axiom,
    ! [F: produc5977172304965343876l_real > real,M1: sigma_4258434043392614480l_real,M2: sigma_measure_real,M: sigma_measure_real,X2: real > real] :
      ( ( member9087738007144076106l_real @ F @ ( sigma_5427481542082897504l_real @ ( binary9161046941930879981l_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M1 ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( F @ ( produc2155246114860689084l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_243_measurable__Pair2,axiom,
    ! [F: produc7929879615859924995b_real > real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real,M: sigma_measure_real,X2: product_prod_a_b] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ M1 @ M2 ) @ M ) )
     => ( ( member1426531481828664017od_a_b @ X2 @ ( sigma_1564438922714704432od_a_b @ M1 ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( F @ ( produc9021675809588086653b_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_244_measurable__Pair2,axiom,
    ! [F: produc7395750855005580859od_a_b > b,M1: sigma_3733394171116455995t_real,M2: sigma_5782782616773303333od_a_b,M: sigma_measure_b,X2: set_real] :
      ( ( member1535311032450272612_a_b_b @ F @ ( sigma_5222788642194995378_a_b_b @ ( binary5539011404815609468od_a_b @ M1 @ M2 ) @ M ) )
     => ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M1 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( F @ ( produc1731599732965853933od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_245_measurable__Pair2,axiom,
    ! [F: produc5010650472090879248b_real > real,M1: sigma_measure_real,M2: sigma_4174442854635832540b_real,M: sigma_measure_real,X2: real] :
      ( ( member5854058990004940758l_real @ F @ ( sigma_2713705292776007404l_real @ ( binary1215910559743187449b_real @ M1 @ M2 ) @ M ) )
     => ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
       => ( member5467075257862553189l_real
          @ ^ [Y2: produc7929879615859924995b_real] : ( F @ ( produc425996204919967944b_real @ X2 @ Y2 ) )
          @ ( sigma_2402725360379274625l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_246_measurable__Pair2,axiom,
    ! [F: produc8974495979853700582od_a_b > b,M1: sigma_4258434043392614480l_real,M2: sigma_5782782616773303333od_a_b,M: sigma_measure_b,X2: real > real] :
      ( ( member7766758707578714745_a_b_b @ F @ ( sigma_6275948866803805127_a_b_b @ ( binary4749032278539451943od_a_b @ M1 @ M2 ) @ M ) )
     => ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M1 ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( F @ ( produc2996895453889575064od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_247_measurable__Pair2,axiom,
    ! [F: produc5421563176480731593b_real > real,M1: sigma_7535330959625612565_a_b_b,M2: sigma_measure_real,M: sigma_measure_real,X2: product_prod_a_b > b] :
      ( ( member4880972771787981583l_real @ F @ ( sigma_3277410670110688165l_real @ ( binary6662343259877833522b_real @ M1 @ M2 ) @ M ) )
     => ( ( member1080921170058542931_a_b_b @ X2 @ ( sigma_8376800615221029940_a_b_b @ M1 ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( F @ ( produc7830128145721275009b_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ M ) ) ) ) ).

% measurable_Pair2
thf(fact_248_measurable__compose__Pair1,axiom,
    ! [X2: a,M1: sigma_measure_a,G: product_prod_a_b > b,M2: sigma_measure_b,L: sigma_measure_b] :
      ( ( member_a @ X2 @ ( sigma_space_a @ M1 ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ M1 @ M2 ) @ L ) )
       => ( member_b_b
          @ ^ [Y2: b] : ( G @ ( product_Pair_a_b @ X2 @ Y2 ) )
          @ ( sigma_measurable_b_b @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_249_measurable__compose__Pair1,axiom,
    ! [X2: real,M1: sigma_measure_real,G: produc2422161461964618553l_real > real,M2: sigma_measure_real,L: sigma_measure_real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
     => ( ( member6699615393305559423l_real @ G @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M1 @ M2 ) @ L ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( G @ ( produc4511245868158468465l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_250_measurable__compose__Pair1,axiom,
    ! [X2: set_real,M1: sigma_3733394171116455995t_real,G: produc2329993357307243631l_real > real,M2: sigma_measure_real,L: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M1 ) )
     => ( ( member5462418605983881141l_real @ G @ ( sigma_1446049469903004875l_real @ ( binary5154340633480048088l_real @ M1 @ M2 ) @ L ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( G @ ( produc6322126818539730599l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_251_measurable__compose__Pair1,axiom,
    ! [X2: real,M1: sigma_measure_real,G: produc1405624045518934769od_a_b > b,M2: sigma_5782782616773303333od_a_b,L: sigma_measure_b] :
      ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
     => ( ( member8926022635040246318_a_b_b @ G @ ( sigma_629531942747761020_a_b_b @ ( binary6959104251739738930od_a_b @ M1 @ M2 ) @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( G @ ( produc6260613462353114019od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_252_measurable__compose__Pair1,axiom,
    ! [X2: real > real,M1: sigma_4258434043392614480l_real,G: produc5977172304965343876l_real > real,M2: sigma_measure_real,L: sigma_measure_real] :
      ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M1 ) )
     => ( ( member9087738007144076106l_real @ G @ ( sigma_5427481542082897504l_real @ ( binary9161046941930879981l_real @ M1 @ M2 ) @ L ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( G @ ( produc2155246114860689084l_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_253_measurable__compose__Pair1,axiom,
    ! [X2: product_prod_a_b,M1: sigma_5782782616773303333od_a_b,G: produc7929879615859924995b_real > real,M2: sigma_measure_real,L: sigma_measure_real] :
      ( ( member1426531481828664017od_a_b @ X2 @ ( sigma_1564438922714704432od_a_b @ M1 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ M1 @ M2 ) @ L ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( G @ ( produc9021675809588086653b_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_254_measurable__compose__Pair1,axiom,
    ! [X2: set_real,M1: sigma_3733394171116455995t_real,G: produc7395750855005580859od_a_b > b,M2: sigma_5782782616773303333od_a_b,L: sigma_measure_b] :
      ( ( member_set_real @ X2 @ ( sigma_space_set_real @ M1 ) )
     => ( ( member1535311032450272612_a_b_b @ G @ ( sigma_5222788642194995378_a_b_b @ ( binary5539011404815609468od_a_b @ M1 @ M2 ) @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( G @ ( produc1731599732965853933od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_255_measurable__compose__Pair1,axiom,
    ! [X2: real,M1: sigma_measure_real,G: produc5010650472090879248b_real > real,M2: sigma_4174442854635832540b_real,L: sigma_measure_real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ M1 ) )
     => ( ( member5854058990004940758l_real @ G @ ( sigma_2713705292776007404l_real @ ( binary1215910559743187449b_real @ M1 @ M2 ) @ L ) )
       => ( member5467075257862553189l_real
          @ ^ [Y2: produc7929879615859924995b_real] : ( G @ ( produc425996204919967944b_real @ X2 @ Y2 ) )
          @ ( sigma_2402725360379274625l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_256_measurable__compose__Pair1,axiom,
    ! [X2: real > real,M1: sigma_4258434043392614480l_real,G: produc8974495979853700582od_a_b > b,M2: sigma_5782782616773303333od_a_b,L: sigma_measure_b] :
      ( ( member_real_real @ X2 @ ( sigma_3619470280215722479l_real @ M1 ) )
     => ( ( member7766758707578714745_a_b_b @ G @ ( sigma_6275948866803805127_a_b_b @ ( binary4749032278539451943od_a_b @ M1 @ M2 ) @ L ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [Y2: product_prod_a_b] : ( G @ ( produc2996895453889575064od_a_b @ X2 @ Y2 ) )
          @ ( sigma_7225552367035416487_a_b_b @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_257_measurable__compose__Pair1,axiom,
    ! [X2: product_prod_a_b > b,M1: sigma_7535330959625612565_a_b_b,G: produc5421563176480731593b_real > real,M2: sigma_measure_real,L: sigma_measure_real] :
      ( ( member1080921170058542931_a_b_b @ X2 @ ( sigma_8376800615221029940_a_b_b @ M1 ) )
     => ( ( member4880972771787981583l_real @ G @ ( sigma_3277410670110688165l_real @ ( binary6662343259877833522b_real @ M1 @ M2 ) @ L ) )
       => ( member_real_real
          @ ^ [Y2: real] : ( G @ ( produc7830128145721275009b_real @ X2 @ Y2 ) )
          @ ( sigma_5267869275261027754l_real @ M2 @ L ) ) ) ) ).

% measurable_compose_Pair1
thf(fact_258_measurable__If,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,M4: sigma_measure_real,G: produc7929879615859924995b_real > real,P: produc7929879615859924995b_real > $o] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
       => ( ( member2587080422514930128b_real
            @ ( collec4671965459240937304b_real
              @ ^ [X: produc7929879615859924995b_real] :
                  ( ( member6125958737618052890b_real @ X @ ( sigma_709238742955535483b_real @ M ) )
                  & ( P @ X ) ) )
            @ ( sigma_5719578828308484196b_real @ M ) )
         => ( member5467075257862553189l_real
            @ ^ [X: produc7929879615859924995b_real] : ( if_real @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_2402725360379274625l_real @ M @ M4 ) ) ) ) ) ).

% measurable_If
thf(fact_259_measurable__If,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,M4: sigma_measure_b,G: product_prod_a_b > b,P: product_prod_a_b > $o] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
       => ( ( member1887650552762594865od_a_b
            @ ( collec3336397801687681299od_a_b
              @ ^ [X: product_prod_a_b] :
                  ( ( member1426531481828664017od_a_b @ X @ ( sigma_1564438922714704432od_a_b @ M ) )
                  & ( P @ X ) ) )
            @ ( sigma_377313353287227655od_a_b @ M ) )
         => ( member1080921170058542931_a_b_b
            @ ^ [X: product_prod_a_b] : ( if_b @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) ) ) ) ) ).

% measurable_If
thf(fact_260_measurable__If,axiom,
    ! [F: real > real,M: sigma_measure_real,M4: sigma_measure_real,G: real > real,P: real > $o] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( member_real @ X @ ( sigma_space_real @ M ) )
                  & ( P @ X ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_real
            @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ M4 ) ) ) ) ) ).

% measurable_If
thf(fact_261_measurable__If,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,P: produc7929879615859924995b_real > $o] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
       => ( ( member2587080422514930128b_real
            @ ( collec4671965459240937304b_real
              @ ^ [X: produc7929879615859924995b_real] :
                  ( ( member6125958737618052890b_real @ X @ ( sigma_709238742955535483b_real @ M ) )
                  & ( P @ X ) ) )
            @ ( sigma_5719578828308484196b_real @ M ) )
         => ( member1118978200981565500b_real
            @ ^ [X: produc7929879615859924995b_real] : ( if_Pro2237973481812034749b_real @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_1108951568718238936b_real @ M @ M4 ) ) ) ) ) ).

% measurable_If
thf(fact_262_measurable__sets__Collect,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: real > $o] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_263_measurable__sets__Collect,axiom,
    ! [F: real > set_real,M: sigma_measure_real,N: sigma_3733394171116455995t_real,P: set_real > $o] :
      ( ( member_real_set_real @ F @ ( sigma_6606012509476713952t_real @ M @ N ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_set_real @ N ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_264_measurable__sets__Collect,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,N: sigma_measure_real,P: real > $o] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_265_measurable__sets__Collect,axiom,
    ! [F: set_real > set_real,M: sigma_3733394171116455995t_real,N: sigma_3733394171116455995t_real,P: set_real > $o] :
      ( ( member8820419758626733818t_real @ F @ ( sigma_8759315257289043734t_real @ M @ N ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_set_real @ N ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_266_measurable__sets__Collect,axiom,
    ! [F: real > real > real,M: sigma_measure_real,N: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
      ( ( member8878224140454985689l_real @ F @ ( sigma_5187198232267106421l_real @ M @ N ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_real @ N ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_267_measurable__sets__Collect,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,N: sigma_measure_real,P: real > $o] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_268_measurable__sets__Collect,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_b,P: b > $o] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ ( sigma_space_b @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member1887650552762594865od_a_b
          @ ( collec3336397801687681299od_a_b
            @ ^ [X: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ X @ ( sigma_1564438922714704432od_a_b @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_377313353287227655od_a_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_269_measurable__sets__Collect,axiom,
    ! [F: ( real > real ) > set_real,M: sigma_4258434043392614480l_real,N: sigma_3733394171116455995t_real,P: set_real > $o] :
      ( ( member3724334074564800271t_real @ F @ ( sigma_1826694118076045611t_real @ M @ N ) )
     => ( ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_set_real @ N ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_270_measurable__sets__Collect,axiom,
    ! [F: set_real > real > real,M: sigma_3733394171116455995t_real,N: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
      ( ( member4134752328433604879l_real @ F @ ( sigma_6541816678006044075l_real @ M @ N ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_real @ N ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_271_measurable__sets__Collect,axiom,
    ! [F: ( real > real ) > real > real,M: sigma_4258434043392614480l_real,N: sigma_4258434043392614480l_real,P: ( real > real ) > $o] :
      ( ( member1867607144002920228l_real @ F @ ( sigma_5559617972653766080l_real @ M @ N ) )
     => ( ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ N ) )
                & ( P @ X ) ) )
          @ ( sigma_sets_real_real @ N ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( P @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_272_borel__measurable__eq,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real,G: ( produc7929879615859924995b_real > real ) > real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member497696658880670256l_real @ G @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [W2: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_273_borel__measurable__eq,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_274_borel__measurable__eq,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b,G: ( product_prod_a_b > b ) > real] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5852679122225541790b_real @ G @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [W2: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_275_borel__measurable__eq,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [W2: real > real] :
                ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_276_borel__measurable__eq,axiom,
    ! [F: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real,M: sigma_7931231546270528766b_real,G: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real] :
      ( ( member5779234595813729927l_real @ F @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5779234595813729927l_real @ G @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [W2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ W2 @ ( sigma_6346777283556531613b_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_277_borel__measurable__eq,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member2587080422514930128b_real
          @ ( collec4671965459240937304b_real
            @ ^ [W2: produc7929879615859924995b_real] :
                ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_278_borel__measurable__eq,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ( F @ W2 )
                  = ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_eq
thf(fact_279_borel__measurable__neq,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real,G: ( produc7929879615859924995b_real > real ) > real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member497696658880670256l_real @ G @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [W2: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_280_borel__measurable__neq,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_281_borel__measurable__neq,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b,G: ( product_prod_a_b > b ) > real] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5852679122225541790b_real @ G @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [W2: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_282_borel__measurable__neq,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [W2: real > real] :
                ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_283_borel__measurable__neq,axiom,
    ! [F: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real,M: sigma_7931231546270528766b_real,G: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real] :
      ( ( member5779234595813729927l_real @ F @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5779234595813729927l_real @ G @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member566263741820371442b_real
          @ ( collec5735648035525910650b_real
            @ ^ [W2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ W2 @ ( sigma_6346777283556531613b_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_284_borel__measurable__neq,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member2587080422514930128b_real
          @ ( collec4671965459240937304b_real
            @ ^ [W2: produc7929879615859924995b_real] :
                ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_285_borel__measurable__neq,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ( F @ W2 )
                 != ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_neq
thf(fact_286_measurable__equality__set,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_287_measurable__equality__set,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_288_measurable__equality__set,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real,G: ( real > real ) > real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5749659578190367193l_real @ G @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_289_measurable__equality__set,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > produc2422161461964618553l_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_290_measurable__equality__set,axiom,
    ! [F: set_real > produc2422161461964618553l_real,M: sigma_3733394171116455995t_real,G: set_real > produc2422161461964618553l_real] :
      ( ( member1965154327494239031l_real @ F @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member1965154327494239031l_real @ G @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_291_measurable__equality__set,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b,G: ( product_prod_a_b > b ) > real] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5852679122225541790b_real @ G @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_292_measurable__equality__set,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member2587080422514930128b_real
          @ ( collec4671965459240937304b_real
            @ ^ [X: produc7929879615859924995b_real] :
                ( ( member6125958737618052890b_real @ X @ ( sigma_709238742955535483b_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_293_measurable__equality__set,axiom,
    ! [F: ( real > real ) > produc2422161461964618553l_real,M: sigma_4258434043392614480l_real,G: ( real > real ) > produc2422161461964618553l_real] :
      ( ( member4094776897846995810l_real @ F @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member4094776897846995810l_real @ G @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_294_measurable__equality__set,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real,G: ( produc7929879615859924995b_real > real ) > real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member497696658880670256l_real @ G @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member3057847817020379547l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ ( sigma_1202137500299886022l_real @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_295_measurable__equality__set,axiom,
    ! [F: ( product_prod_a_b > b ) > produc2422161461964618553l_real,M: sigma_7535330959625612565_a_b_b,G: ( product_prod_a_b > b ) > produc2422161461964618553l_real] :
      ( ( member861154087952452317l_real @ F @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member861154087952452317l_real @ G @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
       => ( member4198857394546769929_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ ( sigma_8376800615221029940_a_b_b @ M ) )
                & ( ( F @ X )
                  = ( G @ X ) ) ) )
          @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% measurable_equality_set
thf(fact_296_measurable__fst_H_H,axiom,
    ! [F: product_prod_a_b > real,M: sigma_5782782616773303333od_a_b,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member1942095430938869870b_real @ F @ ( sigma_8022152630303495172b_real @ M @ N ) )
     => ( member5467075257862553189l_real
        @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc7266368915318857873b_real @ X ) )
        @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ M @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_297_measurable__fst_H_H,axiom,
    ! [F: product_prod_a_b > produc7929879615859924995b_real,M: sigma_5782782616773303333od_a_b,N: sigma_4174442854635832540b_real,P: sigma_measure_real] :
      ( ( member3452904153566040133b_real @ F @ ( sigma_6988460731143032155b_real @ M @ N ) )
     => ( member1118978200981565500b_real
        @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc7266368915318857873b_real @ X ) )
        @ ( sigma_1108951568718238936b_real @ ( binary496794562119935756b_real @ M @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_298_measurable__fst_H_H,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_b,P: sigma_measure_real] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ N ) )
     => ( member3001133527305069980real_b
        @ ^ [X: produc7929879615859924995b_real] : ( F @ ( produc7266368915318857873b_real @ X ) )
        @ ( sigma_3928014871867360490real_b @ ( binary496794562119935756b_real @ M @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_299_measurable__fst_H_H,axiom,
    ! [F: a > b,M: sigma_measure_a,N: sigma_measure_b,P: sigma_measure_b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ N ) )
     => ( member1080921170058542931_a_b_b
        @ ^ [X: product_prod_a_b] : ( F @ ( product_fst_a_b @ X ) )
        @ ( sigma_7225552367035416487_a_b_b @ ( binary867438762418767561re_a_b @ M @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_300_measurable__fst_H_H,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( member6699615393305559423l_real
        @ ^ [X: produc2422161461964618553l_real] : ( F @ ( produc5828954698716094813l_real @ X ) )
        @ ( sigma_8002782794886939285l_real @ ( binary6478037234023840930l_real @ M @ P ) @ N ) ) ) ).

% measurable_fst''
thf(fact_301_prod_Ocollapse,axiom,
    ! [Prod: produc7929879615859924995b_real] :
      ( ( produc9021675809588086653b_real @ ( produc7266368915318857873b_real @ Prod ) @ ( produc7037151529651111887b_real @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_302_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_b] :
      ( ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_303_mem__Sigma__iff,axiom,
    ! [A3: real,B: real,A: set_real,B2: real > set_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A3 @ B ) @ ( produc1020309166918817442l_real @ A @ B2 ) )
      = ( ( member_real @ A3 @ A )
        & ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_304_mem__Sigma__iff,axiom,
    ! [A3: set_real,B: real,A: set_set_real,B2: set_real > set_real] :
      ( ( member2879216344176166424l_real @ ( produc6322126818539730599l_real @ A3 @ B ) @ ( produc7442713998644955096l_real @ A @ B2 ) )
      = ( ( member_set_real @ A3 @ A )
        & ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_305_mem__Sigma__iff,axiom,
    ! [A3: real,B: set_real,A: set_real,B2: real > set_set_real] :
      ( ( member2425104903652129304t_real @ ( produc3307717890874201511t_real @ A3 @ B ) @ ( produc4428305070979426008t_real @ A @ B2 ) )
      = ( ( member_real @ A3 @ A )
        & ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_306_mem__Sigma__iff,axiom,
    ! [A3: set_real,B: set_real,A: set_set_real,B2: set_real > set_set_real] :
      ( ( member7630365606615838542t_real @ ( produc6377492392003646173t_real @ A3 @ B ) @ ( produc1310512389015486478t_real @ A @ B2 ) )
      = ( ( member_set_real @ A3 @ A )
        & ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_307_mem__Sigma__iff,axiom,
    ! [A3: real,B: real > real,A: set_real,B2: real > set_real_real] :
      ( ( member3840528292457551917l_real @ ( produc7028217496045009468l_real @ A3 @ B ) @ ( produc7569734811729095021l_real @ A @ B2 ) )
      = ( ( member_real @ A3 @ A )
        & ( member_real_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_308_mem__Sigma__iff,axiom,
    ! [A3: real > real,B: real,A: set_real_real,B2: ( real > real ) > set_real] :
      ( ( member711963730192933421l_real @ ( produc2155246114860689084l_real @ A3 @ B ) @ ( produc2696763430544774637l_real @ A @ B2 ) )
      = ( ( member_real_real @ A3 @ A )
        & ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_309_mem__Sigma__iff,axiom,
    ! [A3: set_real,B: real > real,A: set_set_real,B2: set_real > set_real_real] :
      ( ( member4278869433197060451l_real @ ( produc4720449797562444658l_real @ A3 @ B ) @ ( produc5248674476412606115l_real @ A @ B2 ) )
      = ( ( member_set_real @ A3 @ A )
        & ( member_real_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_310_mem__Sigma__iff,axiom,
    ! [A3: real > real,B: set_real,A: set_real_real,B2: ( real > real ) > set_set_real] :
      ( ( member3868451179328255843t_real @ ( produc5327237632446194t_real @ A3 @ B ) @ ( produc533551916482607651t_real @ A @ B2 ) )
      = ( ( member_real_real @ A3 @ A )
        & ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_311_mem__Sigma__iff,axiom,
    ! [A3: real,B: product_prod_a_b > b,A: set_real,B2: real > set_Pr3674455588107860082_a_b_b] :
      ( ( member5558031216899592818_a_b_b @ ( produc4396699455692612097_a_b_b @ A3 @ B ) @ ( produc6879959742073472178_a_b_b @ A @ B2 ) )
      = ( ( member_real @ A3 @ A )
        & ( member1080921170058542931_a_b_b @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_312_mem__Sigma__iff,axiom,
    ! [A3: product_prod_a_b > b,B: real,A: set_Pr3674455588107860082_a_b_b,B2: ( product_prod_a_b > b ) > set_real] :
      ( ( member8247474910730785010b_real @ ( produc7830128145721275009b_real @ A3 @ B ) @ ( produc1090016395247359282b_real @ A @ B2 ) )
      = ( ( member1080921170058542931_a_b_b @ A3 @ A )
        & ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_313_SigmaI,axiom,
    ! [A3: real,A: set_real,B: real,B2: real > set_real] :
      ( ( member_real @ A3 @ A )
     => ( ( member_real @ B @ ( B2 @ A3 ) )
       => ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A3 @ B ) @ ( produc1020309166918817442l_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_314_SigmaI,axiom,
    ! [A3: set_real,A: set_set_real,B: real,B2: set_real > set_real] :
      ( ( member_set_real @ A3 @ A )
     => ( ( member_real @ B @ ( B2 @ A3 ) )
       => ( member2879216344176166424l_real @ ( produc6322126818539730599l_real @ A3 @ B ) @ ( produc7442713998644955096l_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_315_SigmaI,axiom,
    ! [A3: real,A: set_real,B: set_real,B2: real > set_set_real] :
      ( ( member_real @ A3 @ A )
     => ( ( member_set_real @ B @ ( B2 @ A3 ) )
       => ( member2425104903652129304t_real @ ( produc3307717890874201511t_real @ A3 @ B ) @ ( produc4428305070979426008t_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_316_SigmaI,axiom,
    ! [A3: set_real,A: set_set_real,B: set_real,B2: set_real > set_set_real] :
      ( ( member_set_real @ A3 @ A )
     => ( ( member_set_real @ B @ ( B2 @ A3 ) )
       => ( member7630365606615838542t_real @ ( produc6377492392003646173t_real @ A3 @ B ) @ ( produc1310512389015486478t_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_317_SigmaI,axiom,
    ! [A3: real,A: set_real,B: real > real,B2: real > set_real_real] :
      ( ( member_real @ A3 @ A )
     => ( ( member_real_real @ B @ ( B2 @ A3 ) )
       => ( member3840528292457551917l_real @ ( produc7028217496045009468l_real @ A3 @ B ) @ ( produc7569734811729095021l_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_318_SigmaI,axiom,
    ! [A3: real > real,A: set_real_real,B: real,B2: ( real > real ) > set_real] :
      ( ( member_real_real @ A3 @ A )
     => ( ( member_real @ B @ ( B2 @ A3 ) )
       => ( member711963730192933421l_real @ ( produc2155246114860689084l_real @ A3 @ B ) @ ( produc2696763430544774637l_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_319_SigmaI,axiom,
    ! [A3: set_real,A: set_set_real,B: real > real,B2: set_real > set_real_real] :
      ( ( member_set_real @ A3 @ A )
     => ( ( member_real_real @ B @ ( B2 @ A3 ) )
       => ( member4278869433197060451l_real @ ( produc4720449797562444658l_real @ A3 @ B ) @ ( produc5248674476412606115l_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_320_SigmaI,axiom,
    ! [A3: real > real,A: set_real_real,B: set_real,B2: ( real > real ) > set_set_real] :
      ( ( member_real_real @ A3 @ A )
     => ( ( member_set_real @ B @ ( B2 @ A3 ) )
       => ( member3868451179328255843t_real @ ( produc5327237632446194t_real @ A3 @ B ) @ ( produc533551916482607651t_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_321_SigmaI,axiom,
    ! [A3: real,A: set_real,B: product_prod_a_b > b,B2: real > set_Pr3674455588107860082_a_b_b] :
      ( ( member_real @ A3 @ A )
     => ( ( member1080921170058542931_a_b_b @ B @ ( B2 @ A3 ) )
       => ( member5558031216899592818_a_b_b @ ( produc4396699455692612097_a_b_b @ A3 @ B ) @ ( produc6879959742073472178_a_b_b @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_322_SigmaI,axiom,
    ! [A3: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B: real,B2: ( product_prod_a_b > b ) > set_real] :
      ( ( member1080921170058542931_a_b_b @ A3 @ A )
     => ( ( member_real @ B @ ( B2 @ A3 ) )
       => ( member8247474910730785010b_real @ ( produc7830128145721275009b_real @ A3 @ B ) @ ( produc1090016395247359282b_real @ A @ B2 ) ) ) ) ).

% SigmaI
thf(fact_323_mem__Times__iff,axiom,
    ! [X2: produc2422161461964618553l_real,A: set_real,B2: set_real] :
      ( ( member7849222048561428706l_real @ X2
        @ ( produc1020309166918817442l_real @ A
          @ ^ [Uu: real] : B2 ) )
      = ( ( member_real @ ( produc5828954698716094813l_real @ X2 ) @ A )
        & ( member_real @ ( produc3484788084999411615l_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_324_mem__Times__iff,axiom,
    ! [X2: product_prod_a_b,A: set_a,B2: set_b] :
      ( ( member1426531481828664017od_a_b @ X2
        @ ( product_Sigma_a_b @ A
          @ ^ [Uu: a] : B2 ) )
      = ( ( member_a @ ( product_fst_a_b @ X2 ) @ A )
        & ( member_b @ ( product_snd_a_b @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_325_mem__Times__iff,axiom,
    ! [X2: produc2329993357307243631l_real,A: set_set_real,B2: set_real] :
      ( ( member2879216344176166424l_real @ X2
        @ ( produc7442713998644955096l_real @ A
          @ ^ [Uu: set_real] : B2 ) )
      = ( ( member_set_real @ ( produc2701064676077905555l_real @ X2 ) @ A )
        & ( member_real @ ( produc3944167683213677781l_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_326_mem__Times__iff,axiom,
    ! [X2: produc1875881916783206511t_real,A: set_real,B2: set_set_real] :
      ( ( member2425104903652129304t_real @ X2
        @ ( produc4428305070979426008t_real @ A
          @ ^ [Uu: real] : B2 ) )
      = ( ( member_real @ ( produc8910027785267152275t_real @ X2 ) @ A )
        & ( member_set_real @ ( produc929758755548148693t_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_327_mem__Times__iff,axiom,
    ! [X2: produc2118043781569182117t_real,A: set_set_real,B2: set_set_real] :
      ( ( member7630365606615838542t_real @ X2
        @ ( produc1310512389015486478t_real @ A
          @ ^ [Uu: set_real] : B2 ) )
      = ( ( member_set_real @ ( produc5066390771641299657t_real @ X2 ) @ A )
        & ( member_set_real @ ( produc325895082611923211t_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_328_mem__Times__iff,axiom,
    ! [X2: produc9105736867229962372l_real,A: set_real,B2: set_real_real] :
      ( ( member3840528292457551917l_real @ X2
        @ ( produc7569734811729095021l_real @ A
          @ ^ [Uu: real] : B2 ) )
      = ( ( member_real @ ( produc8378123679107813928l_real @ X2 ) @ A )
        & ( member_real_real @ ( produc7571918366874909802l_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_329_mem__Times__iff,axiom,
    ! [X2: produc5977172304965343876l_real,A: set_real_real,B2: set_real] :
      ( ( member711963730192933421l_real @ X2
        @ ( produc2696763430544774637l_real @ A
          @ ^ [Uu: real > real] : B2 ) )
      = ( ( member_real_real @ ( produc3505152297923493544l_real @ X2 ) @ A )
        & ( member_real @ ( produc2698946985690589418l_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_330_mem__Times__iff,axiom,
    ! [X2: produc7929879615859924995b_real,A: set_Product_prod_a_b,B2: set_real] :
      ( ( member6125958737618052890b_real @ X2
        @ ( produc7302604275931811084b_real @ A
          @ ^ [Uu: product_prod_a_b] : B2 ) )
      = ( ( member1426531481828664017od_a_b @ ( produc7266368915318857873b_real @ X2 ) @ A )
        & ( member_real @ ( produc7037151529651111887b_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_331_mem__Times__iff,axiom,
    ! [X2: produc5114776888151014842l_real,A: set_set_real,B2: set_real_real] :
      ( ( member4278869433197060451l_real @ X2
        @ ( produc5248674476412606115l_real @ A
          @ ^ [Uu: set_real] : B2 ) )
      = ( ( member_set_real @ ( produc7954759493867295582l_real @ X2 ) @ A )
        & ( member_real_real @ ( produc8332554227345356192l_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_332_mem__Times__iff,axiom,
    ! [X2: produc4704358634282210234t_real,A: set_real_real,B2: set_set_real] :
      ( ( member3868451179328255843t_real @ X2
        @ ( produc533551916482607651t_real @ A
          @ ^ [Uu: real > real] : B2 ) )
      = ( ( member_real_real @ ( produc3239636933937297118t_real @ X2 ) @ A )
        & ( member_set_real @ ( produc3617431667415357728t_real @ X2 ) @ B2 ) ) ) ).

% mem_Times_iff
thf(fact_333_prod_Oexhaust__sel,axiom,
    ! [Prod: produc7929879615859924995b_real] :
      ( Prod
      = ( produc9021675809588086653b_real @ ( produc7266368915318857873b_real @ Prod ) @ ( produc7037151529651111887b_real @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_334_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_b] :
      ( Prod
      = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_335_exI__realizer,axiom,
    ! [P: real > product_prod_a_b > $o,Y: real,X2: product_prod_a_b] :
      ( ( P @ Y @ X2 )
     => ( P @ ( produc7037151529651111887b_real @ ( produc9021675809588086653b_real @ X2 @ Y ) ) @ ( produc7266368915318857873b_real @ ( produc9021675809588086653b_real @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_336_exI__realizer,axiom,
    ! [P: b > a > $o,Y: b,X2: a] :
      ( ( P @ Y @ X2 )
     => ( P @ ( product_snd_a_b @ ( product_Pair_a_b @ X2 @ Y ) ) @ ( product_fst_a_b @ ( product_Pair_a_b @ X2 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_337_split__pairs,axiom,
    ! [A: product_prod_a_b,B2: real,X3: produc7929879615859924995b_real] :
      ( ( ( produc9021675809588086653b_real @ A @ B2 )
        = X3 )
      = ( ( ( produc7266368915318857873b_real @ X3 )
          = A )
        & ( ( produc7037151529651111887b_real @ X3 )
          = B2 ) ) ) ).

% split_pairs
thf(fact_338_split__pairs,axiom,
    ! [A: a,B2: b,X3: product_prod_a_b] :
      ( ( ( product_Pair_a_b @ A @ B2 )
        = X3 )
      = ( ( ( product_fst_a_b @ X3 )
          = A )
        & ( ( product_snd_a_b @ X3 )
          = B2 ) ) ) ).

% split_pairs
thf(fact_339_conjI__realizer,axiom,
    ! [P: product_prod_a_b > $o,P2: product_prod_a_b,Q: real > $o,Q2: real] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc7266368915318857873b_real @ ( produc9021675809588086653b_real @ P2 @ Q2 ) ) )
          & ( Q @ ( produc7037151529651111887b_real @ ( produc9021675809588086653b_real @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_340_conjI__realizer,axiom,
    ! [P: a > $o,P2: a,Q: b > $o,Q2: b] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_a_b @ ( product_Pair_a_b @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_a_b @ ( product_Pair_a_b @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_341_surjective__pairing,axiom,
    ! [T: produc7929879615859924995b_real] :
      ( T
      = ( produc9021675809588086653b_real @ ( produc7266368915318857873b_real @ T ) @ ( produc7037151529651111887b_real @ T ) ) ) ).

% surjective_pairing
thf(fact_342_surjective__pairing,axiom,
    ! [T: product_prod_a_b] :
      ( T
      = ( product_Pair_a_b @ ( product_fst_a_b @ T ) @ ( product_snd_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_343_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: product_prod_a_b > real > $o,X2: product_prod_a_b,Y: real,A3: produc7929879615859924995b_real] :
      ( ( P @ X2 @ Y )
     => ( ( A3
          = ( produc9021675809588086653b_real @ X2 @ Y ) )
       => ( P @ ( produc7266368915318857873b_real @ A3 ) @ ( produc7037151529651111887b_real @ A3 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_344_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: a > b > $o,X2: a,Y: b,A3: product_prod_a_b] :
      ( ( P @ X2 @ Y )
     => ( ( A3
          = ( product_Pair_a_b @ X2 @ Y ) )
       => ( P @ ( product_fst_a_b @ A3 ) @ ( product_snd_a_b @ A3 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_345_snd__conv,axiom,
    ! [X1: product_prod_a_b,X22: real] :
      ( ( produc7037151529651111887b_real @ ( produc9021675809588086653b_real @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_346_snd__conv,axiom,
    ! [X1: a,X22: b] :
      ( ( product_snd_a_b @ ( product_Pair_a_b @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_347_snd__eqD,axiom,
    ! [X2: product_prod_a_b,Y: real,A3: real] :
      ( ( ( produc7037151529651111887b_real @ ( produc9021675809588086653b_real @ X2 @ Y ) )
        = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
thf(fact_348_snd__eqD,axiom,
    ! [X2: a,Y: b,A3: b] :
      ( ( ( product_snd_a_b @ ( product_Pair_a_b @ X2 @ Y ) )
        = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
thf(fact_349_SigmaE,axiom,
    ! [C: produc2422161461964618553l_real,A: set_real,B2: real > set_real] :
      ( ( member7849222048561428706l_real @ C @ ( produc1020309166918817442l_real @ A @ B2 ) )
     => ~ ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ! [Y3: real] :
                ( ( member_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc4511245868158468465l_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_350_SigmaE,axiom,
    ! [C: produc2329993357307243631l_real,A: set_set_real,B2: set_real > set_real] :
      ( ( member2879216344176166424l_real @ C @ ( produc7442713998644955096l_real @ A @ B2 ) )
     => ~ ! [X4: set_real] :
            ( ( member_set_real @ X4 @ A )
           => ! [Y3: real] :
                ( ( member_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc6322126818539730599l_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_351_SigmaE,axiom,
    ! [C: produc1875881916783206511t_real,A: set_real,B2: real > set_set_real] :
      ( ( member2425104903652129304t_real @ C @ ( produc4428305070979426008t_real @ A @ B2 ) )
     => ~ ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ! [Y3: set_real] :
                ( ( member_set_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc3307717890874201511t_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_352_SigmaE,axiom,
    ! [C: produc2118043781569182117t_real,A: set_set_real,B2: set_real > set_set_real] :
      ( ( member7630365606615838542t_real @ C @ ( produc1310512389015486478t_real @ A @ B2 ) )
     => ~ ! [X4: set_real] :
            ( ( member_set_real @ X4 @ A )
           => ! [Y3: set_real] :
                ( ( member_set_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc6377492392003646173t_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_353_SigmaE,axiom,
    ! [C: produc9105736867229962372l_real,A: set_real,B2: real > set_real_real] :
      ( ( member3840528292457551917l_real @ C @ ( produc7569734811729095021l_real @ A @ B2 ) )
     => ~ ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ! [Y3: real > real] :
                ( ( member_real_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc7028217496045009468l_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_354_SigmaE,axiom,
    ! [C: produc5977172304965343876l_real,A: set_real_real,B2: ( real > real ) > set_real] :
      ( ( member711963730192933421l_real @ C @ ( produc2696763430544774637l_real @ A @ B2 ) )
     => ~ ! [X4: real > real] :
            ( ( member_real_real @ X4 @ A )
           => ! [Y3: real] :
                ( ( member_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc2155246114860689084l_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_355_SigmaE,axiom,
    ! [C: produc5114776888151014842l_real,A: set_set_real,B2: set_real > set_real_real] :
      ( ( member4278869433197060451l_real @ C @ ( produc5248674476412606115l_real @ A @ B2 ) )
     => ~ ! [X4: set_real] :
            ( ( member_set_real @ X4 @ A )
           => ! [Y3: real > real] :
                ( ( member_real_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc4720449797562444658l_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_356_SigmaE,axiom,
    ! [C: produc4704358634282210234t_real,A: set_real_real,B2: ( real > real ) > set_set_real] :
      ( ( member3868451179328255843t_real @ C @ ( produc533551916482607651t_real @ A @ B2 ) )
     => ~ ! [X4: real > real] :
            ( ( member_real_real @ X4 @ A )
           => ! [Y3: set_real] :
                ( ( member_set_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc5327237632446194t_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_357_SigmaE,axiom,
    ! [C: produc2732119482649539401_a_b_b,A: set_real,B2: real > set_Pr3674455588107860082_a_b_b] :
      ( ( member5558031216899592818_a_b_b @ C @ ( produc6879959742073472178_a_b_b @ A @ B2 ) )
     => ~ ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ! [Y3: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc4396699455692612097_a_b_b @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_358_SigmaE,axiom,
    ! [C: produc5421563176480731593b_real,A: set_Pr3674455588107860082_a_b_b,B2: ( product_prod_a_b > b ) > set_real] :
      ( ( member8247474910730785010b_real @ C @ ( produc1090016395247359282b_real @ A @ B2 ) )
     => ~ ! [X4: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X4 @ A )
           => ! [Y3: real] :
                ( ( member_real @ Y3 @ ( B2 @ X4 ) )
               => ( C
                 != ( produc7830128145721275009b_real @ X4 @ Y3 ) ) ) ) ) ).

% SigmaE
thf(fact_359_SigmaE2,axiom,
    ! [A3: real,B: real,A: set_real,B2: real > set_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A3 @ B ) @ ( produc1020309166918817442l_real @ A @ B2 ) )
     => ~ ( ( member_real @ A3 @ A )
         => ~ ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_360_SigmaE2,axiom,
    ! [A3: set_real,B: real,A: set_set_real,B2: set_real > set_real] :
      ( ( member2879216344176166424l_real @ ( produc6322126818539730599l_real @ A3 @ B ) @ ( produc7442713998644955096l_real @ A @ B2 ) )
     => ~ ( ( member_set_real @ A3 @ A )
         => ~ ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_361_SigmaE2,axiom,
    ! [A3: real,B: set_real,A: set_real,B2: real > set_set_real] :
      ( ( member2425104903652129304t_real @ ( produc3307717890874201511t_real @ A3 @ B ) @ ( produc4428305070979426008t_real @ A @ B2 ) )
     => ~ ( ( member_real @ A3 @ A )
         => ~ ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_362_SigmaE2,axiom,
    ! [A3: set_real,B: set_real,A: set_set_real,B2: set_real > set_set_real] :
      ( ( member7630365606615838542t_real @ ( produc6377492392003646173t_real @ A3 @ B ) @ ( produc1310512389015486478t_real @ A @ B2 ) )
     => ~ ( ( member_set_real @ A3 @ A )
         => ~ ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_363_SigmaE2,axiom,
    ! [A3: real,B: real > real,A: set_real,B2: real > set_real_real] :
      ( ( member3840528292457551917l_real @ ( produc7028217496045009468l_real @ A3 @ B ) @ ( produc7569734811729095021l_real @ A @ B2 ) )
     => ~ ( ( member_real @ A3 @ A )
         => ~ ( member_real_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_364_SigmaE2,axiom,
    ! [A3: real > real,B: real,A: set_real_real,B2: ( real > real ) > set_real] :
      ( ( member711963730192933421l_real @ ( produc2155246114860689084l_real @ A3 @ B ) @ ( produc2696763430544774637l_real @ A @ B2 ) )
     => ~ ( ( member_real_real @ A3 @ A )
         => ~ ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_365_SigmaE2,axiom,
    ! [A3: set_real,B: real > real,A: set_set_real,B2: set_real > set_real_real] :
      ( ( member4278869433197060451l_real @ ( produc4720449797562444658l_real @ A3 @ B ) @ ( produc5248674476412606115l_real @ A @ B2 ) )
     => ~ ( ( member_set_real @ A3 @ A )
         => ~ ( member_real_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_366_SigmaE2,axiom,
    ! [A3: real > real,B: set_real,A: set_real_real,B2: ( real > real ) > set_set_real] :
      ( ( member3868451179328255843t_real @ ( produc5327237632446194t_real @ A3 @ B ) @ ( produc533551916482607651t_real @ A @ B2 ) )
     => ~ ( ( member_real_real @ A3 @ A )
         => ~ ( member_set_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_367_SigmaE2,axiom,
    ! [A3: real,B: product_prod_a_b > b,A: set_real,B2: real > set_Pr3674455588107860082_a_b_b] :
      ( ( member5558031216899592818_a_b_b @ ( produc4396699455692612097_a_b_b @ A3 @ B ) @ ( produc6879959742073472178_a_b_b @ A @ B2 ) )
     => ~ ( ( member_real @ A3 @ A )
         => ~ ( member1080921170058542931_a_b_b @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_368_SigmaE2,axiom,
    ! [A3: product_prod_a_b > b,B: real,A: set_Pr3674455588107860082_a_b_b,B2: ( product_prod_a_b > b ) > set_real] :
      ( ( member8247474910730785010b_real @ ( produc7830128145721275009b_real @ A3 @ B ) @ ( produc1090016395247359282b_real @ A @ B2 ) )
     => ~ ( ( member1080921170058542931_a_b_b @ A3 @ A )
         => ~ ( member_real @ B @ ( B2 @ A3 ) ) ) ) ).

% SigmaE2
thf(fact_369_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc7929879615859924995b_real,Z: produc7929879615859924995b_real] : ( Y4 = Z ) )
    = ( ^ [S: produc7929879615859924995b_real,T2: produc7929879615859924995b_real] :
          ( ( ( produc7266368915318857873b_real @ S )
            = ( produc7266368915318857873b_real @ T2 ) )
          & ( ( produc7037151529651111887b_real @ S )
            = ( produc7037151529651111887b_real @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_370_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_a_b,Z: product_prod_a_b] : ( Y4 = Z ) )
    = ( ^ [S: product_prod_a_b,T2: product_prod_a_b] :
          ( ( ( product_fst_a_b @ S )
            = ( product_fst_a_b @ T2 ) )
          & ( ( product_snd_a_b @ S )
            = ( product_snd_a_b @ T2 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_371_exE__realizer_H,axiom,
    ! [P: real > product_prod_a_b > $o,P2: produc7929879615859924995b_real] :
      ( ( P @ ( produc7037151529651111887b_real @ P2 ) @ ( produc7266368915318857873b_real @ P2 ) )
     => ~ ! [X4: product_prod_a_b,Y3: real] :
            ~ ( P @ Y3 @ X4 ) ) ).

% exE_realizer'
thf(fact_372_exE__realizer_H,axiom,
    ! [P: b > a > $o,P2: product_prod_a_b] :
      ( ( P @ ( product_snd_a_b @ P2 ) @ ( product_fst_a_b @ P2 ) )
     => ~ ! [X4: a,Y3: b] :
            ~ ( P @ Y3 @ X4 ) ) ).

% exE_realizer'
thf(fact_373_prod__eqI,axiom,
    ! [P2: produc7929879615859924995b_real,Q2: produc7929879615859924995b_real] :
      ( ( ( produc7266368915318857873b_real @ P2 )
        = ( produc7266368915318857873b_real @ Q2 ) )
     => ( ( ( produc7037151529651111887b_real @ P2 )
          = ( produc7037151529651111887b_real @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_374_prod__eqI,axiom,
    ! [P2: product_prod_a_b,Q2: product_prod_a_b] :
      ( ( ( product_fst_a_b @ P2 )
        = ( product_fst_a_b @ Q2 ) )
     => ( ( ( product_snd_a_b @ P2 )
          = ( product_snd_a_b @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_375_prod_Oexpand,axiom,
    ! [Prod: produc7929879615859924995b_real,Prod2: produc7929879615859924995b_real] :
      ( ( ( ( produc7266368915318857873b_real @ Prod )
          = ( produc7266368915318857873b_real @ Prod2 ) )
        & ( ( produc7037151529651111887b_real @ Prod )
          = ( produc7037151529651111887b_real @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_376_prod_Oexpand,axiom,
    ! [Prod: product_prod_a_b,Prod2: product_prod_a_b] :
      ( ( ( ( product_fst_a_b @ Prod )
          = ( product_fst_a_b @ Prod2 ) )
        & ( ( product_snd_a_b @ Prod )
          = ( product_snd_a_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_377_eq__snd__iff,axiom,
    ! [B: real,P2: produc7929879615859924995b_real] :
      ( ( B
        = ( produc7037151529651111887b_real @ P2 ) )
      = ( ? [A4: product_prod_a_b] :
            ( P2
            = ( produc9021675809588086653b_real @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_378_eq__snd__iff,axiom,
    ! [B: b,P2: product_prod_a_b] :
      ( ( B
        = ( product_snd_a_b @ P2 ) )
      = ( ? [A4: a] :
            ( P2
            = ( product_Pair_a_b @ A4 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_379_sndI,axiom,
    ! [X2: produc7929879615859924995b_real,Y: product_prod_a_b,Z2: real] :
      ( ( X2
        = ( produc9021675809588086653b_real @ Y @ Z2 ) )
     => ( ( produc7037151529651111887b_real @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_380_sndI,axiom,
    ! [X2: product_prod_a_b,Y: a,Z2: b] :
      ( ( X2
        = ( product_Pair_a_b @ Y @ Z2 ) )
     => ( ( product_snd_a_b @ X2 )
        = Z2 ) ) ).

% sndI
thf(fact_381_measurable__entropy__density,axiom,
    ! [B: real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real] : ( member5467075257862553189l_real @ ( entrop4636206444604467707b_real @ B @ M @ N ) @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ).

% measurable_entropy_density
thf(fact_382_measurable__entropy__density,axiom,
    ! [B: real,M: sigma_measure_real,N: sigma_measure_real] : ( member_real_real @ ( entropy_density_real @ B @ M @ N ) @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ).

% measurable_entropy_density
thf(fact_383_borel__measurable__euclidean__space,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ( member5467075257862553189l_real
              @ ^ [Y2: produc7929879615859924995b_real] : ( inner_5569486634795291694r_real @ ( F @ Y2 ) @ X )
              @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_euclidean_space
thf(fact_384_borel__measurable__euclidean__space,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ( member_real_real
              @ ^ [Y2: real] : ( inner_5569486634795291694r_real @ ( F @ Y2 ) @ X )
              @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_euclidean_space
thf(fact_385_borel__measurable__euclidean__space,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ( member5467075257862553189l_real
              @ ^ [Y2: produc7929879615859924995b_real] : ( inner_2340953980025574455l_real @ ( F @ Y2 ) @ X )
              @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_euclidean_space
thf(fact_386_borel__measurable__euclidean__space,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ( member_real_real
              @ ^ [Y2: real] : ( inner_2340953980025574455l_real @ ( F @ Y2 ) @ X )
              @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_euclidean_space
thf(fact_387_borel__measurable__indicator__iff,axiom,
    ! [A: set_Pr6157794097399793337b_real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ ( indica5149351975077090117l_real @ A ) @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( member2587080422514930128b_real @ ( inf_in1453021003782806603b_real @ A @ ( sigma_709238742955535483b_real @ M ) ) @ ( sigma_5719578828308484196b_real @ M ) ) ) ).

% borel_measurable_indicator_iff
thf(fact_388_borel__measurable__indicator__iff,axiom,
    ! [A: set_real,M: sigma_measure_real] :
      ( ( member_real_real @ ( indica3952125700972739438l_real @ A ) @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( member_set_real @ ( inf_inf_set_real @ A @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) ) ) ).

% borel_measurable_indicator_iff
thf(fact_389_measurable__pair,axiom,
    ! [F: produc7929879615859924995b_real > produc5010650472090879248b_real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,M2: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ ( comp_P146849012575345438b_real @ produc8280286207298640052b_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
     => ( ( member1118978200981565500b_real @ ( comp_P5119746620826145525b_real @ produc6974232290649910006b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M2 ) )
       => ( member1006085877915592749b_real @ F @ ( sigma_184416647750077563b_real @ M @ ( binary1215910559743187449b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_390_measurable__pair,axiom,
    ! [F: product_prod_a_b > product_prod_b_b,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_b,M2: sigma_measure_b] :
      ( ( member1080921170058542931_a_b_b @ ( comp_P5715990135234274861od_a_b @ product_fst_b_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M1 ) )
     => ( ( member1080921170058542931_a_b_b @ ( comp_P5715990135234274861od_a_b @ product_snd_b_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) )
       => ( member2119461356580886075od_b_b @ F @ ( sigma_2688633497260815247od_b_b @ M @ ( binary7303355217300867336re_b_b @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_391_measurable__pair,axiom,
    ! [F: produc7929879615859924995b_real > produc2738766212015491856l_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,M2: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ ( comp_P850362950206298357b_real @ produc5156030528502344500l_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ ( comp_P3498366258136572190b_real @ produc3849976611853614454l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member284797873430130221l_real @ F @ ( sigma_7135904424529465979l_real @ M @ ( binary7315026917801667705l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_392_measurable__pair,axiom,
    ! [F: produc7929879615859924995b_real > produc7364210939229353703b_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,M2: sigma_4174442854635832540b_real] :
      ( ( member1118978200981565500b_real @ ( comp_P2601594796862164940b_real @ produc403517537962017931b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
     => ( ( member1118978200981565500b_real @ ( comp_P2601594796862164940b_real @ produc4283488707444060877b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M2 ) )
       => ( member5942386789700278916b_real @ F @ ( sigma_371502731946915922b_real @ M @ ( binary388897294610017232b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_393_measurable__pair,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] :
      ( ( member2224156091608774853od_a_b @ ( comp_P8510425873491595589b_real @ produc7266368915318857873b_real @ F ) @ ( sigma_8670534473570876307od_a_b @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ ( comp_P5041851192982496151b_real @ produc7037151529651111887b_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_394_measurable__pair,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] :
      ( ( member4641211897452655452od_a_b @ ( comp_P7944386915558755310b_real @ produc7266368915318857873b_real @ F ) @ ( sigma_5261090283068522538od_a_b @ M @ M1 ) )
     => ( ( member_real_real @ ( comp_P6701117116028920384l_real @ produc7037151529651111887b_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
       => ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_395_measurable__pair,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_a,M2: sigma_measure_b] :
      ( ( member1080921165755314130_a_b_a @ ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ F ) @ ( sigma_7225552367035416486_a_b_a @ M @ M1 ) )
     => ( ( member1080921170058542931_a_b_b @ ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) )
       => ( member2755110357185872764od_a_b @ F @ ( sigma_5476089079233491280od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_396_measurable__pair,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( member5467075257862553189l_real @ ( comp_P7068956567367034951b_real @ produc5828954698716094813l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
     => ( ( member5467075257862553189l_real @ ( comp_P7068956567367034951b_real @ produc3484788084999411615l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) )
       => ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_397_measurable__pair,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( member_real_real @ ( comp_P4894738870877013744l_real @ produc5828954698716094813l_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M1 ) )
     => ( ( member_real_real @ ( comp_P4894738870877013744l_real @ produc3484788084999411615l_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
       => ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) ) ) ) ).

% measurable_pair
thf(fact_398_measurable__pair__iff,axiom,
    ! [F: produc7929879615859924995b_real > produc5010650472090879248b_real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,M2: sigma_4174442854635832540b_real] :
      ( ( member1006085877915592749b_real @ F @ ( sigma_184416647750077563b_real @ M @ ( binary1215910559743187449b_real @ M1 @ M2 ) ) )
      = ( ( member5467075257862553189l_real @ ( comp_P146849012575345438b_real @ produc8280286207298640052b_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
        & ( member1118978200981565500b_real @ ( comp_P5119746620826145525b_real @ produc6974232290649910006b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_399_measurable__pair__iff,axiom,
    ! [F: product_prod_a_b > product_prod_b_b,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_b,M2: sigma_measure_b] :
      ( ( member2119461356580886075od_b_b @ F @ ( sigma_2688633497260815247od_b_b @ M @ ( binary7303355217300867336re_b_b @ M1 @ M2 ) ) )
      = ( ( member1080921170058542931_a_b_b @ ( comp_P5715990135234274861od_a_b @ product_fst_b_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M1 ) )
        & ( member1080921170058542931_a_b_b @ ( comp_P5715990135234274861od_a_b @ product_snd_b_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_400_measurable__pair__iff,axiom,
    ! [F: produc7929879615859924995b_real > produc2738766212015491856l_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,M2: sigma_measure_real] :
      ( ( member284797873430130221l_real @ F @ ( sigma_7135904424529465979l_real @ M @ ( binary7315026917801667705l_real @ M1 @ M2 ) ) )
      = ( ( member1118978200981565500b_real @ ( comp_P850362950206298357b_real @ produc5156030528502344500l_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
        & ( member5467075257862553189l_real @ ( comp_P3498366258136572190b_real @ produc3849976611853614454l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_401_measurable__pair__iff,axiom,
    ! [F: produc7929879615859924995b_real > produc7364210939229353703b_real,M: sigma_4174442854635832540b_real,M1: sigma_4174442854635832540b_real,M2: sigma_4174442854635832540b_real] :
      ( ( member5942386789700278916b_real @ F @ ( sigma_371502731946915922b_real @ M @ ( binary388897294610017232b_real @ M1 @ M2 ) ) )
      = ( ( member1118978200981565500b_real @ ( comp_P2601594796862164940b_real @ produc403517537962017931b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M1 ) )
        & ( member1118978200981565500b_real @ ( comp_P2601594796862164940b_real @ produc4283488707444060877b_real @ F ) @ ( sigma_1108951568718238936b_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_402_measurable__pair__iff,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
      = ( ( member4641211897452655452od_a_b @ ( comp_P7944386915558755310b_real @ produc7266368915318857873b_real @ F ) @ ( sigma_5261090283068522538od_a_b @ M @ M1 ) )
        & ( member_real_real @ ( comp_P6701117116028920384l_real @ produc7037151529651111887b_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_403_measurable__pair__iff,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M1: sigma_5782782616773303333od_a_b,M2: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ ( binary496794562119935756b_real @ M1 @ M2 ) ) )
      = ( ( member2224156091608774853od_a_b @ ( comp_P8510425873491595589b_real @ produc7266368915318857873b_real @ F ) @ ( sigma_8670534473570876307od_a_b @ M @ M1 ) )
        & ( member5467075257862553189l_real @ ( comp_P5041851192982496151b_real @ produc7037151529651111887b_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_404_measurable__pair__iff,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b,M1: sigma_measure_a,M2: sigma_measure_b] :
      ( ( member2755110357185872764od_a_b @ F @ ( sigma_5476089079233491280od_a_b @ M @ ( binary867438762418767561re_a_b @ M1 @ M2 ) ) )
      = ( ( member1080921165755314130_a_b_a @ ( comp_P2648956292766268207od_a_b @ product_fst_a_b @ F ) @ ( sigma_7225552367035416486_a_b_a @ M @ M1 ) )
        & ( member1080921170058542931_a_b_b @ ( comp_P2009515992434452078od_a_b @ product_snd_a_b @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_405_measurable__pair__iff,axiom,
    ! [F: produc7929879615859924995b_real > produc2422161461964618553l_real,M: sigma_4174442854635832540b_real,M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( member174722968998780374l_real @ F @ ( sigma_3866086195517720740l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
      = ( ( member5467075257862553189l_real @ ( comp_P7068956567367034951b_real @ produc5828954698716094813l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M1 ) )
        & ( member5467075257862553189l_real @ ( comp_P7068956567367034951b_real @ produc3484788084999411615l_real @ F ) @ ( sigma_2402725360379274625l_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_406_measurable__pair__iff,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,M1: sigma_measure_real,M2: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ ( binary6478037234023840930l_real @ M1 @ M2 ) ) )
      = ( ( member_real_real @ ( comp_P4894738870877013744l_real @ produc5828954698716094813l_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M1 ) )
        & ( member_real_real @ ( comp_P4894738870877013744l_real @ produc3484788084999411615l_real @ F ) @ ( sigma_5267869275261027754l_real @ M @ M2 ) ) ) ) ).

% measurable_pair_iff
thf(fact_407_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real,H2: real > real > a] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo1359147121410913923real_a @ top_to1175844992842649833l_real
            @ ^ [X: produc2422161461964618553l_real] : ( H2 @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) )
         => ( member_real_a
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_408_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real,H2: real > real > b] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo1359147121410913924real_b @ top_to1175844992842649833l_real
            @ ^ [X: produc2422161461964618553l_real] : ( H2 @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) )
         => ( member_real_b
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_409_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real,H2: real > real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo5694269426501914215l_real @ top_to1175844992842649833l_real
            @ ^ [X: produc2422161461964618553l_real] : ( H2 @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) )
         => ( member_real_real
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_410_borel__measurable__continuous__Pair,axiom,
    ! [F: product_prod_a_b > real,M: sigma_5782782616773303333od_a_b,G: product_prod_a_b > real,H2: real > real > b] :
      ( ( member1942095430938869870b_real @ F @ ( sigma_8022152630303495172b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member1942095430938869870b_real @ G @ ( sigma_8022152630303495172b_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo1359147121410913924real_b @ top_to1175844992842649833l_real
            @ ^ [X: produc2422161461964618553l_real] : ( H2 @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) )
         => ( member1080921170058542931_a_b_b
            @ ^ [X: product_prod_a_b] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_411_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real,H2: real > real > produc2422161461964618553l_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo506878965529049918l_real @ top_to1175844992842649833l_real
            @ ^ [X: produc2422161461964618553l_real] : ( H2 @ ( produc5828954698716094813l_real @ X ) @ ( produc3484788084999411615l_real @ X ) ) )
         => ( member9086635009091248365l_real
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_412_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > produc2422161461964618553l_real,H2: real > produc2422161461964618553l_real > a] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( ( topolo5703465612353636092real_a @ top_to1855813389712021096l_real
            @ ^ [X: produc322147437362262530l_real] : ( H2 @ ( produc2651074646626695112l_real @ X ) @ ( produc4189857760027666438l_real @ X ) ) )
         => ( member_real_a
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_413_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > produc2422161461964618553l_real,H2: real > produc2422161461964618553l_real > b] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( ( topolo5703465612353636093real_b @ top_to1855813389712021096l_real
            @ ^ [X: produc322147437362262530l_real] : ( H2 @ ( produc2651074646626695112l_real @ X ) @ ( produc4189857760027666438l_real @ X ) ) )
         => ( member_real_b
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_414_borel__measurable__continuous__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > produc2422161461964618553l_real,H2: real > produc2422161461964618553l_real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member9086635009091248365l_real @ G @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
       => ( ( topolo3541141988540419630l_real @ top_to1855813389712021096l_real
            @ ^ [X: produc322147437362262530l_real] : ( H2 @ ( produc2651074646626695112l_real @ X ) @ ( produc4189857760027666438l_real @ X ) ) )
         => ( member_real_real
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_415_borel__measurable__continuous__Pair,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > real,H2: produc2422161461964618553l_real > real > a] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo2335561341866929898real_a @ top_to7370315563888545274l_real
            @ ^ [X: produc7158499858431349396l_real] : ( H2 @ ( produc2655710143947908258l_real @ X ) @ ( produc4194493257348879584l_real @ X ) ) )
         => ( member_real_a
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_416_borel__measurable__continuous__Pair,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real,G: real > real,H2: produc2422161461964618553l_real > real > b] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( ( topolo2335561341866929899real_b @ top_to7370315563888545274l_real
            @ ^ [X: produc7158499858431349396l_real] : ( H2 @ ( produc2655710143947908258l_real @ X ) @ ( produc4194493257348879584l_real @ X ) ) )
         => ( member_real_b
            @ ^ [X: real] : ( H2 @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_continuous_Pair
thf(fact_417_sets__pair__in__sets,axiom,
    ! [A: sigma_5782782616773303333od_a_b,B2: sigma_measure_real,N: sigma_4174442854635832540b_real] :
      ( ! [A5: set_Product_prod_a_b,B3: set_real] :
          ( ( member1887650552762594865od_a_b @ A5 @ ( sigma_377313353287227655od_a_b @ A ) )
         => ( ( member_set_real @ B3 @ ( sigma_sets_real @ B2 ) )
           => ( member2587080422514930128b_real
              @ ( produc7302604275931811084b_real @ A5
                @ ^ [Uu: product_prod_a_b] : B3 )
              @ ( sigma_5719578828308484196b_real @ N ) ) ) )
     => ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ ( binary496794562119935756b_real @ A @ B2 ) ) @ ( sigma_5719578828308484196b_real @ N ) ) ) ).

% sets_pair_in_sets
thf(fact_418_sets__pair__in__sets,axiom,
    ! [A: sigma_measure_a,B2: sigma_measure_b,N: sigma_5782782616773303333od_a_b] :
      ( ! [A5: set_a,B3: set_b] :
          ( ( member_set_a @ A5 @ ( sigma_sets_a @ A ) )
         => ( ( member_set_b @ B3 @ ( sigma_sets_b @ B2 ) )
           => ( member1887650552762594865od_a_b
              @ ( product_Sigma_a_b @ A5
                @ ^ [Uu: a] : B3 )
              @ ( sigma_377313353287227655od_a_b @ N ) ) ) )
     => ( ord_le8697249284497989384od_a_b @ ( sigma_377313353287227655od_a_b @ ( binary867438762418767561re_a_b @ A @ B2 ) ) @ ( sigma_377313353287227655od_a_b @ N ) ) ) ).

% sets_pair_in_sets
thf(fact_419_sets__pair__in__sets,axiom,
    ! [A: sigma_measure_real,B2: sigma_measure_real,N: sigma_2308072346491277622l_real] :
      ( ! [A5: set_real,B3: set_real] :
          ( ( member_set_real @ A5 @ ( sigma_sets_real @ A ) )
         => ( ( member_set_real @ B3 @ ( sigma_sets_real @ B2 ) )
           => ( member6767226683953100098l_real
              @ ( produc1020309166918817442l_real @ A5
                @ ^ [Uu: real] : B3 )
              @ ( sigma_1543982426361076760l_real @ N ) ) ) )
     => ( ord_le7027846394859511705l_real @ ( sigma_1543982426361076760l_real @ ( binary6478037234023840930l_real @ A @ B2 ) ) @ ( sigma_1543982426361076760l_real @ N ) ) ) ).

% sets_pair_in_sets
thf(fact_420_sets_OInt,axiom,
    ! [A3: set_real,M: sigma_measure_real,B: set_real] :
      ( ( member_set_real @ A3 @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( inf_inf_set_real @ A3 @ B ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.Int
thf(fact_421_space__borel,axiom,
    ( ( sigma_space_a @ borel_5459123734250506524orel_a )
    = top_top_set_a ) ).

% space_borel
thf(fact_422_space__borel,axiom,
    ( ( sigma_space_b @ borel_5459123734250506525orel_b )
    = top_top_set_b ) ).

% space_borel
thf(fact_423_space__borel,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% space_borel
thf(fact_424_space__borel,axiom,
    ( ( sigma_2519298556039103681l_real @ borel_9155112475215227991l_real )
    = top_to1175844992842649833l_real ) ).

% space_borel
thf(fact_425_sets_OInt__space__eq1,axiom,
    ! [X2: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
     => ( ( inf_inf_set_real @ ( sigma_space_real @ M ) @ X2 )
        = X2 ) ) ).

% sets.Int_space_eq1
thf(fact_426_sets_OInt__space__eq2,axiom,
    ! [X2: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
     => ( ( inf_inf_set_real @ X2 @ ( sigma_space_real @ M ) )
        = X2 ) ) ).

% sets.Int_space_eq2
thf(fact_427_subset__Collect__iff,axiom,
    ! [B2: set_Pr3917002265660371076l_real,A: set_Pr3917002265660371076l_real,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( ord_le8308227983669428452l_real @ B2 @ A )
     => ( ( ord_le8308227983669428452l_real @ B2
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_428_subset__Collect__iff,axiom,
    ! [B2: set_set_real,A: set_set_real,P: set_real > $o] :
      ( ( ord_le3558479182127378552t_real @ B2 @ A )
     => ( ( ord_le3558479182127378552t_real @ B2
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: set_real] :
              ( ( member_set_real @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_429_subset__Collect__iff,axiom,
    ! [B2: set_real,A: set_real,P: real > $o] :
      ( ( ord_less_eq_set_real @ B2 @ A )
     => ( ( ord_less_eq_set_real @ B2
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: real] :
              ( ( member_real @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_430_subset__Collect__iff,axiom,
    ! [B2: set_Pr3674455588107860082_a_b_b,A: set_Pr3674455588107860082_a_b_b,P: ( product_prod_a_b > b ) > $o] :
      ( ( ord_le8439411419596101842_a_b_b @ B2 @ A )
     => ( ( ord_le8439411419596101842_a_b_b @ B2
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_431_subset__Collect__iff,axiom,
    ! [B2: set_real_real,A: set_real_real,P: ( real > real ) > $o] :
      ( ( ord_le4198349162570665613l_real @ B2 @ A )
     => ( ( ord_le4198349162570665613l_real @ B2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: real > real] :
              ( ( member_real_real @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_432_subset__Collect__iff,axiom,
    ! [B2: set_Pr1027596265373204187b_real,A: set_Pr1027596265373204187b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( ord_le2582152453551361851b_real @ B2 @ A )
     => ( ( ord_le2582152453551361851b_real @ B2
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ B2 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_433_subset__CollectI,axiom,
    ! [B2: set_Pr3917002265660371076l_real,A: set_Pr3917002265660371076l_real,Q: ( produc7929879615859924995b_real > real ) > $o,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( ord_le8308227983669428452l_real @ B2 @ A )
     => ( ! [X4: produc7929879615859924995b_real > real] :
            ( ( member5467075257862553189l_real @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le8308227983669428452l_real
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec869437573433150115l_real
            @ ^ [X: produc7929879615859924995b_real > real] :
                ( ( member5467075257862553189l_real @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_434_subset__CollectI,axiom,
    ! [B2: set_set_real,A: set_set_real,Q: set_real > $o,P: set_real > $o] :
      ( ( ord_le3558479182127378552t_real @ B2 @ A )
     => ( ! [X4: set_real] :
            ( ( member_set_real @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3558479182127378552t_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_435_subset__CollectI,axiom,
    ! [B2: set_real,A: set_real,Q: real > $o,P: real > $o] :
      ( ( ord_less_eq_set_real @ B2 @ A )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_less_eq_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_436_subset__CollectI,axiom,
    ! [B2: set_Pr3674455588107860082_a_b_b,A: set_Pr3674455588107860082_a_b_b,Q: ( product_prod_a_b > b ) > $o,P: ( product_prod_a_b > b ) > $o] :
      ( ( ord_le8439411419596101842_a_b_b @ B2 @ A )
     => ( ! [X4: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le8439411419596101842_a_b_b
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec2250730345169186961_a_b_b
            @ ^ [X: product_prod_a_b > b] :
                ( ( member1080921170058542931_a_b_b @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_437_subset__CollectI,axiom,
    ! [B2: set_real_real,A: set_real_real,Q: ( real > real ) > $o,P: ( real > real ) > $o] :
      ( ( ord_le4198349162570665613l_real @ B2 @ A )
     => ( ! [X4: real > real] :
            ( ( member_real_real @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le4198349162570665613l_real
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_438_subset__CollectI,axiom,
    ! [B2: set_Pr1027596265373204187b_real,A: set_Pr1027596265373204187b_real,Q: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( ord_le2582152453551361851b_real @ B2 @ A )
     => ( ! [X4: produc7929879615859924995b_real > produc7929879615859924995b_real] :
            ( ( member1118978200981565500b_real @ X4 @ B2 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le2582152453551361851b_real
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ B2 )
                & ( Q @ X ) ) )
          @ ( collec5735648035525910650b_real
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                ( ( member1118978200981565500b_real @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_439_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_real,B2: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B2 ) )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B2 ) ) ) ).

% sets_le_imp_space_le
thf(fact_440_measurable__comp,axiom,
    ! [F: real > produc7929879615859924995b_real,M: sigma_measure_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,L: sigma_measure_real] :
      ( ( member7738959517937940581b_real @ F @ ( sigma_5526981039175570177b_real @ M @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ N @ L ) )
       => ( member_real_real @ ( comp_P6701117116028920384l_real @ G @ F ) @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_441_measurable__comp,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,M: sigma_5782782616773303333od_a_b,N: sigma_5782782616773303333od_a_b,G: product_prod_a_b > b,L: sigma_measure_b] :
      ( ( member2755110357185872764od_a_b @ F @ ( sigma_5476089079233491280od_a_b @ M @ N ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ N @ L ) )
       => ( member1080921170058542931_a_b_b @ ( comp_P2009515992434452078od_a_b @ G @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_442_measurable__comp,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,G: real > produc7929879615859924995b_real,L: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ N ) )
     => ( ( member7738959517937940581b_real @ G @ ( sigma_5526981039175570177b_real @ N @ L ) )
       => ( member1118978200981565500b_real @ ( comp_r312201211331252503b_real @ G @ F ) @ ( sigma_1108951568718238936b_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_443_measurable__comp,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
       => ( member5467075257862553189l_real @ ( comp_r6460197204725972288b_real @ G @ F ) @ ( sigma_2402725360379274625l_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_444_measurable__comp,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_b,G: b > b,L: sigma_measure_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ L ) )
       => ( member1080921170058542931_a_b_b @ ( comp_b3886954628874447685od_a_b @ G @ F ) @ ( sigma_7225552367035416487_a_b_b @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_445_measurable__comp,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
       => ( member_real_real @ ( comp_real_real_real @ G @ F ) @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_446_measurable__comp,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real,L: sigma_measure_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ N ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ N @ L ) )
       => ( member5467075257862553189l_real @ ( comp_P5041851192982496151b_real @ G @ F ) @ ( sigma_2402725360379274625l_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_447_measurable__comp,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,L: sigma_4174442854635832540b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ N ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ N @ L ) )
       => ( member1118978200981565500b_real @ ( comp_P2581140431950109038b_real @ G @ F ) @ ( sigma_1108951568718238936b_real @ M @ L ) ) ) ) ).

% measurable_comp
thf(fact_448_sets_Osets__into__space,axiom,
    ! [X2: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
     => ( ord_less_eq_set_real @ X2 @ ( sigma_space_real @ M ) ) ) ).

% sets.sets_into_space
thf(fact_449_space__in__borel,axiom,
    member_set_a @ top_top_set_a @ ( sigma_sets_a @ borel_5459123734250506524orel_a ) ).

% space_in_borel
thf(fact_450_space__in__borel,axiom,
    member_set_b @ top_top_set_b @ ( sigma_sets_b @ borel_5459123734250506525orel_b ) ).

% space_in_borel
thf(fact_451_space__in__borel,axiom,
    member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% space_in_borel
thf(fact_452_space__in__borel,axiom,
    member6767226683953100098l_real @ top_to1175844992842649833l_real @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) ).

% space_in_borel
thf(fact_453_borel__measurable__inf,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( inf_inf_real @ ( G @ X ) @ ( F @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inf
thf(fact_454_borel__measurable__inf,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( inf_inf_real @ ( G @ X ) @ ( F @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_inf
thf(fact_455_borel__measurable__continuous__onI,axiom,
    ! [F: a > a] :
      ( ( topolo2963393042455755902on_a_a @ top_top_set_a @ F )
     => ( member_a_a @ F @ ( sigma_measurable_a_a @ borel_5459123734250506524orel_a @ borel_5459123734250506524orel_a ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_456_borel__measurable__continuous__onI,axiom,
    ! [F: a > b] :
      ( ( topolo2963393042455755903on_a_b @ top_top_set_a @ F )
     => ( member_a_b @ F @ ( sigma_measurable_a_b @ borel_5459123734250506524orel_a @ borel_5459123734250506525orel_b ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_457_borel__measurable__continuous__onI,axiom,
    ! [F: a > real] :
      ( ( topolo210349010034097452a_real @ top_top_set_a @ F )
     => ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ borel_5459123734250506524orel_a @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_458_borel__measurable__continuous__onI,axiom,
    ! [F: b > a] :
      ( ( topolo175937460483079869on_b_a @ top_top_set_b @ F )
     => ( member_b_a @ F @ ( sigma_measurable_b_a @ borel_5459123734250506525orel_b @ borel_5459123734250506524orel_a ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_459_borel__measurable__continuous__onI,axiom,
    ! [F: b > b] :
      ( ( topolo175937460483079870on_b_b @ top_top_set_b @ F )
     => ( member_b_b @ F @ ( sigma_measurable_b_b @ borel_5459123734250506525orel_b @ borel_5459123734250506525orel_b ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_460_borel__measurable__continuous__onI,axiom,
    ! [F: b > real] :
      ( ( topolo5806567625222228141b_real @ top_top_set_b @ F )
     => ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ borel_5459123734250506525orel_b @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_461_borel__measurable__continuous__onI,axiom,
    ! [F: real > a] :
      ( ( topolo840367777507047250real_a @ top_top_set_real @ F )
     => ( member_real_a @ F @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ borel_5459123734250506524orel_a ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_462_borel__measurable__continuous__onI,axiom,
    ! [F: real > b] :
      ( ( topolo840367777507047251real_b @ top_top_set_real @ F )
     => ( member_real_b @ F @ ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ borel_5459123734250506525orel_b ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_463_borel__measurable__continuous__onI,axiom,
    ! [F: real > real] :
      ( ( topolo5044208981011980120l_real @ top_top_set_real @ F )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_464_borel__measurable__continuous__onI,axiom,
    ! [F: product_prod_a_b > b] :
      ( ( topolo7391332748632156757_a_b_b @ top_to8134405472303993176od_a_b @ F )
     => ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ borel_8051966406626612422od_a_b @ borel_5459123734250506525orel_b ) ) ) ).

% borel_measurable_continuous_onI
thf(fact_465_measurable__If__set,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,M4: sigma_measure_real,G: produc7929879615859924995b_real > real,A: set_Pr6157794097399793337b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
     => ( ( member5467075257862553189l_real @ G @ ( sigma_2402725360379274625l_real @ M @ M4 ) )
       => ( ( member2587080422514930128b_real @ ( inf_in1453021003782806603b_real @ A @ ( sigma_709238742955535483b_real @ M ) ) @ ( sigma_5719578828308484196b_real @ M ) )
         => ( member5467075257862553189l_real
            @ ^ [X: produc7929879615859924995b_real] : ( if_real @ ( member6125958737618052890b_real @ X @ A ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_2402725360379274625l_real @ M @ M4 ) ) ) ) ) ).

% measurable_If_set
thf(fact_466_measurable__If__set,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,M4: sigma_measure_b,G: product_prod_a_b > b,A: set_Product_prod_a_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) )
       => ( ( member1887650552762594865od_a_b @ ( inf_in8976041639890736118od_a_b @ A @ ( sigma_1564438922714704432od_a_b @ M ) ) @ ( sigma_377313353287227655od_a_b @ M ) )
         => ( member1080921170058542931_a_b_b
            @ ^ [X: product_prod_a_b] : ( if_b @ ( member1426531481828664017od_a_b @ X @ A ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_7225552367035416487_a_b_b @ M @ M4 ) ) ) ) ) ).

% measurable_If_set
thf(fact_467_measurable__If__set,axiom,
    ! [F: real > real,M: sigma_measure_real,M4: sigma_measure_real,G: real > real,A: set_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M4 ) )
       => ( ( member_set_real @ ( inf_inf_set_real @ A @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
         => ( member_real_real
            @ ^ [X: real] : ( if_real @ ( member_real @ X @ A ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_5267869275261027754l_real @ M @ M4 ) ) ) ) ) ).

% measurable_If_set
thf(fact_468_measurable__If__set,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real,G: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr6157794097399793337b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
     => ( ( member1118978200981565500b_real @ G @ ( sigma_1108951568718238936b_real @ M @ M4 ) )
       => ( ( member2587080422514930128b_real @ ( inf_in1453021003782806603b_real @ A @ ( sigma_709238742955535483b_real @ M ) ) @ ( sigma_5719578828308484196b_real @ M ) )
         => ( member1118978200981565500b_real
            @ ^ [X: produc7929879615859924995b_real] : ( if_Pro2237973481812034749b_real @ ( member6125958737618052890b_real @ X @ A ) @ ( F @ X ) @ ( G @ X ) )
            @ ( sigma_1108951568718238936b_real @ M @ M4 ) ) ) ) ) ).

% measurable_If_set
thf(fact_469_borel__measurable__continuous__on,axiom,
    ! [F: a > real,G: real > a,M: sigma_measure_real] :
      ( ( topolo210349010034097452a_real @ top_top_set_a @ F )
     => ( ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_470_borel__measurable__continuous__on,axiom,
    ! [F: b > real,G: real > b,M: sigma_measure_real] :
      ( ( topolo5806567625222228141b_real @ top_top_set_b @ F )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_471_borel__measurable__continuous__on,axiom,
    ! [F: real > a,G: real > real,M: sigma_measure_real] :
      ( ( topolo840367777507047250real_a @ top_top_set_real @ F )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_a
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_472_borel__measurable__continuous__on,axiom,
    ! [F: real > b,G: real > real,M: sigma_measure_real] :
      ( ( topolo840367777507047251real_b @ top_top_set_real @ F )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_b
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_473_borel__measurable__continuous__on,axiom,
    ! [F: real > real,G: real > real,M: sigma_measure_real] :
      ( ( topolo5044208981011980120l_real @ top_top_set_real @ F )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_real_real
          @ ^ [X: real] : ( F @ ( G @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_474_borel__measurable__continuous__on,axiom,
    ! [F: a > b,G: product_prod_a_b > a,M: sigma_5782782616773303333od_a_b] :
      ( ( topolo2963393042455755903on_a_b @ top_top_set_a @ F )
     => ( ( member1080921165755314130_a_b_a @ G @ ( sigma_7225552367035416486_a_b_a @ M @ borel_5459123734250506524orel_a ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_475_borel__measurable__continuous__on,axiom,
    ! [F: b > a,G: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b] :
      ( ( topolo175937460483079869on_b_a @ top_top_set_b @ F )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) )
       => ( member1080921165755314130_a_b_a
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416486_a_b_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_476_borel__measurable__continuous__on,axiom,
    ! [F: b > b,G: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b] :
      ( ( topolo175937460483079870on_b_b @ top_top_set_b @ F )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_477_borel__measurable__continuous__on,axiom,
    ! [F: b > real,G: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b] :
      ( ( topolo5806567625222228141b_real @ top_top_set_b @ F )
     => ( ( member1080921170058542931_a_b_b @ G @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) )
       => ( member1942095430938869870b_real
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_8022152630303495172b_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_478_borel__measurable__continuous__on,axiom,
    ! [F: real > b,G: product_prod_a_b > real,M: sigma_5782782616773303333od_a_b] :
      ( ( topolo840367777507047251real_b @ top_top_set_real @ F )
     => ( ( member1942095430938869870b_real @ G @ ( sigma_8022152630303495172b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member1080921170058542931_a_b_b
          @ ^ [X: product_prod_a_b] : ( F @ ( G @ X ) )
          @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ).

% borel_measurable_continuous_on
thf(fact_479_borel__measurable__subalgebra,axiom,
    ! [N: sigma_4174442854635832540b_real,M: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > produc7929879615859924995b_real] :
      ( ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ N ) @ ( sigma_5719578828308484196b_real @ M ) )
     => ( ( ( sigma_709238742955535483b_real @ N )
          = ( sigma_709238742955535483b_real @ M ) )
       => ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ N @ borel_3131472315568871781b_real ) )
         => ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ borel_3131472315568871781b_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_480_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > a] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ N @ borel_5459123734250506524orel_a ) )
         => ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ borel_5459123734250506524orel_a ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_481_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > b] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ N @ borel_5459123734250506525orel_b ) )
         => ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_482_borel__measurable__subalgebra,axiom,
    ! [N: sigma_5782782616773303333od_a_b,M: sigma_5782782616773303333od_a_b,F: product_prod_a_b > b] :
      ( ( ord_le8697249284497989384od_a_b @ ( sigma_377313353287227655od_a_b @ N ) @ ( sigma_377313353287227655od_a_b @ M ) )
     => ( ( ( sigma_1564438922714704432od_a_b @ N )
          = ( sigma_1564438922714704432od_a_b @ M ) )
       => ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ N @ borel_5459123734250506525orel_b ) )
         => ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ borel_5459123734250506525orel_b ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_483_borel__measurable__subalgebra,axiom,
    ! [N: sigma_4174442854635832540b_real,M: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > real] :
      ( ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ N ) @ ( sigma_5719578828308484196b_real @ M ) )
     => ( ( ( sigma_709238742955535483b_real @ N )
          = ( sigma_709238742955535483b_real @ M ) )
       => ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ N @ borel_5078946678739801102l_real ) )
         => ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_484_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
         => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_485_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > produc2422161461964618553l_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ N @ borel_9155112475215227991l_real ) )
         => ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_486_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > nat] :
      ( ! [Y3: nat] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_487_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_le
thf(fact_488_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > extended_ereal] :
      ( ! [Y3: extended_ereal] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member7593515600531736322_ereal @ F @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) ) ) ).

% borel_measurableI_le
thf(fact_489_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ! [Y3: real] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_490_borel__measurableI__le,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > nat] :
      ( ! [Y3: nat] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_nat @ F @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_491_borel__measurableI__le,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member7922526604024092368nnreal @ F @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_le
thf(fact_492_borel__measurableI__le,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > extended_ereal] :
      ( ! [Y3: extended_ereal] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member4196616219543705932_ereal @ F @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) ) ) ).

% borel_measurableI_le
thf(fact_493_borel__measurableI__le,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > real] :
      ( ! [Y3: real] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_494_borel__measurableI__le,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > nat] :
      ( ! [Y3: nat] :
          ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real_real @ M ) )
     => ( member_real_real_nat @ F @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_495_borel__measurableI__le,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ Y3 ) ) )
          @ ( sigma_sets_real_real @ M ) )
     => ( member237164684335263077nnreal @ F @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_le
thf(fact_496_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > nat] :
      ( ! [Y3: nat] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_497_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_ge
thf(fact_498_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > extended_ereal] :
      ( ! [Y3: extended_ereal] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member7593515600531736322_ereal @ F @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) ) ) ).

% borel_measurableI_ge
thf(fact_499_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ! [Y3: real] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_500_borel__measurableI__ge,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > nat] :
      ( ! [Y3: nat] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_nat @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_nat @ F @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_501_borel__measurableI__ge,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member7922526604024092368nnreal @ F @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_ge
thf(fact_502_borel__measurableI__ge,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > extended_ereal] :
      ( ! [Y3: extended_ereal] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member4196616219543705932_ereal @ F @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) ) ) ).

% borel_measurableI_ge
thf(fact_503_borel__measurableI__ge,axiom,
    ! [M: sigma_3733394171116455995t_real,F: set_real > real] :
      ( ! [Y3: real] :
          ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_real @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) )
     => ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_504_borel__measurableI__ge,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > nat] :
      ( ! [Y3: nat] :
          ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_less_eq_nat @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) )
     => ( member_real_real_nat @ F @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_505_borel__measurableI__ge,axiom,
    ! [M: sigma_4258434043392614480l_real,F: ( real > real ) > extend8495563244428889912nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ Y3 @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) )
     => ( member237164684335263077nnreal @ F @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).

% borel_measurableI_ge
thf(fact_506_borel__measurable__le,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_507_borel__measurable__le,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_508_borel__measurable__le,axiom,
    ! [F: real > extended_ereal,M: sigma_measure_real,G: real > extended_ereal] :
      ( ( member7593515600531736322_ereal @ F @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member7593515600531736322_ereal @ G @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_509_borel__measurable__le,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_510_borel__measurable__le,axiom,
    ! [F: set_real > nat,M: sigma_3733394171116455995t_real,G: set_real > nat] :
      ( ( member_set_real_nat @ F @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_set_real_nat @ G @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_511_borel__measurable__le,axiom,
    ! [F: set_real > extend8495563244428889912nnreal,M: sigma_3733394171116455995t_real,G: set_real > extend8495563244428889912nnreal] :
      ( ( member7922526604024092368nnreal @ F @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member7922526604024092368nnreal @ G @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_512_borel__measurable__le,axiom,
    ! [F: set_real > extended_ereal,M: sigma_3733394171116455995t_real,G: set_real > extended_ereal] :
      ( ( member4196616219543705932_ereal @ F @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member4196616219543705932_ereal @ G @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_513_borel__measurable__le,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [W2: set_real] :
                ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_514_borel__measurable__le,axiom,
    ! [F: ( real > real ) > nat,M: sigma_4258434043392614480l_real,G: ( real > real ) > nat] :
      ( ( member_real_real_nat @ F @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_real_nat @ G @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [W2: real > real] :
                ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_515_borel__measurable__le,axiom,
    ! [F: ( real > real ) > extend8495563244428889912nnreal,M: sigma_4258434043392614480l_real,G: ( real > real ) > extend8495563244428889912nnreal] :
      ( ( member237164684335263077nnreal @ F @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member237164684335263077nnreal @ G @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [W2: real > real] :
                ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_516_measurable__inequality__set_I1_J,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_517_measurable__inequality__set_I1_J,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_518_measurable__inequality__set_I1_J,axiom,
    ! [F: real > extended_ereal,M: sigma_measure_real,G: real > extended_ereal] :
      ( ( member7593515600531736322_ereal @ F @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member7593515600531736322_ereal @ G @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_519_measurable__inequality__set_I1_J,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_520_measurable__inequality__set_I1_J,axiom,
    ! [F: set_real > nat,M: sigma_3733394171116455995t_real,G: set_real > nat] :
      ( ( member_set_real_nat @ F @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_set_real_nat @ G @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_521_measurable__inequality__set_I1_J,axiom,
    ! [F: set_real > extend8495563244428889912nnreal,M: sigma_3733394171116455995t_real,G: set_real > extend8495563244428889912nnreal] :
      ( ( member7922526604024092368nnreal @ F @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member7922526604024092368nnreal @ G @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_522_measurable__inequality__set_I1_J,axiom,
    ! [F: set_real > extended_ereal,M: sigma_3733394171116455995t_real,G: set_real > extended_ereal] :
      ( ( member4196616219543705932_ereal @ F @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member4196616219543705932_ereal @ G @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_523_measurable__inequality__set_I1_J,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_524_measurable__inequality__set_I1_J,axiom,
    ! [F: ( real > real ) > nat,M: sigma_4258434043392614480l_real,G: ( real > real ) > nat] :
      ( ( member_real_real_nat @ F @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_real_nat @ G @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_525_measurable__inequality__set_I1_J,axiom,
    ! [F: ( real > real ) > extend8495563244428889912nnreal,M: sigma_4258434043392614480l_real,G: ( real > real ) > extend8495563244428889912nnreal] :
      ( ( member237164684335263077nnreal @ F @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member237164684335263077nnreal @ G @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( F @ X ) @ ( G @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_526_measurable__inequality__set_I3_J,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_527_measurable__inequality__set_I3_J,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_528_measurable__inequality__set_I3_J,axiom,
    ! [F: real > extended_ereal,M: sigma_measure_real,G: real > extended_ereal] :
      ( ( member7593515600531736322_ereal @ F @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member7593515600531736322_ereal @ G @ ( sigma_3635921994777574992_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_529_measurable__inequality__set_I3_J,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_530_measurable__inequality__set_I3_J,axiom,
    ! [F: set_real > nat,M: sigma_3733394171116455995t_real,G: set_real > nat] :
      ( ( member_set_real_nat @ F @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_set_real_nat @ G @ ( sigma_5262614976308547076al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_531_measurable__inequality__set_I3_J,axiom,
    ! [F: set_real > extend8495563244428889912nnreal,M: sigma_3733394171116455995t_real,G: set_real > extend8495563244428889912nnreal] :
      ( ( member7922526604024092368nnreal @ F @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member7922526604024092368nnreal @ G @ ( sigma_1593293107762293100nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_532_measurable__inequality__set_I3_J,axiom,
    ! [F: set_real > extended_ereal,M: sigma_3733394171116455995t_real,G: set_real > extended_ereal] :
      ( ( member4196616219543705932_ereal @ F @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
     => ( ( member4196616219543705932_ereal @ G @ ( sigma_4856685845782207642_ereal @ M @ borel_2631802743099733228_ereal ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_le1083603963089353582_ereal @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_533_measurable__inequality__set_I3_J,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real,G: set_real > real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real_real @ G @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_real
          @ ( collect_set_real
            @ ^ [X: set_real] :
                ( ( member_set_real @ X @ ( sigma_space_set_real @ M ) )
                & ( ord_less_eq_real @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_set_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_534_measurable__inequality__set_I3_J,axiom,
    ! [F: ( real > real ) > nat,M: sigma_4258434043392614480l_real,G: ( real > real ) > nat] :
      ( ( member_real_real_nat @ F @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_real_nat @ G @ ( sigma_788105080603002521al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_less_eq_nat @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_535_measurable__inequality__set_I3_J,axiom,
    ! [F: ( real > real ) > extend8495563244428889912nnreal,M: sigma_4258434043392614480l_real,G: ( real > real ) > extend8495563244428889912nnreal] :
      ( ( member237164684335263077nnreal @ F @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
     => ( ( member237164684335263077nnreal @ G @ ( sigma_1117148153383385601nnreal @ M @ borel_6524799422816628122nnreal ) )
       => ( member_set_real_real2
          @ ( collect_real_real
            @ ^ [X: real > real] :
                ( ( member_real_real @ X @ ( sigma_3619470280215722479l_real @ M ) )
                & ( ord_le3935885782089961368nnreal @ ( G @ X ) @ ( F @ X ) ) ) )
          @ ( sigma_sets_real_real @ M ) ) ) ) ).

% measurable_inequality_set(3)
thf(fact_536_inf__top__left,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ X2 )
      = X2 ) ).

% inf_top_left
thf(fact_537_UNIV__I,axiom,
    ! [X2: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X2 @ top_to5286755836160161588l_real ) ).

% UNIV_I
thf(fact_538_UNIV__I,axiom,
    ! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).

% UNIV_I
thf(fact_539_UNIV__I,axiom,
    ! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).

% UNIV_I
thf(fact_540_UNIV__I,axiom,
    ! [X2: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X2 @ top_to8883559732078315298_a_b_b ) ).

% UNIV_I
thf(fact_541_UNIV__I,axiom,
    ! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).

% UNIV_I
thf(fact_542_UNIV__I,axiom,
    ! [X2: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X2 @ top_to1564948194967201163b_real ) ).

% UNIV_I
thf(fact_543_subsetI,axiom,
    ! [A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ! [X4: produc7929879615859924995b_real > real] :
          ( ( member5467075257862553189l_real @ X4 @ A )
         => ( member5467075257862553189l_real @ X4 @ B2 ) )
     => ( ord_le8308227983669428452l_real @ A @ B2 ) ) ).

% subsetI
thf(fact_544_subsetI,axiom,
    ! [A: set_set_real,B2: set_set_real] :
      ( ! [X4: set_real] :
          ( ( member_set_real @ X4 @ A )
         => ( member_set_real @ X4 @ B2 ) )
     => ( ord_le3558479182127378552t_real @ A @ B2 ) ) ).

% subsetI
thf(fact_545_subsetI,axiom,
    ! [A: set_real,B2: set_real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A )
         => ( member_real @ X4 @ B2 ) )
     => ( ord_less_eq_set_real @ A @ B2 ) ) ).

% subsetI
thf(fact_546_subsetI,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ! [X4: product_prod_a_b > b] :
          ( ( member1080921170058542931_a_b_b @ X4 @ A )
         => ( member1080921170058542931_a_b_b @ X4 @ B2 ) )
     => ( ord_le8439411419596101842_a_b_b @ A @ B2 ) ) ).

% subsetI
thf(fact_547_subsetI,axiom,
    ! [A: set_real_real,B2: set_real_real] :
      ( ! [X4: real > real] :
          ( ( member_real_real @ X4 @ A )
         => ( member_real_real @ X4 @ B2 ) )
     => ( ord_le4198349162570665613l_real @ A @ B2 ) ) ).

% subsetI
thf(fact_548_subsetI,axiom,
    ! [A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ! [X4: produc7929879615859924995b_real > produc7929879615859924995b_real] :
          ( ( member1118978200981565500b_real @ X4 @ A )
         => ( member1118978200981565500b_real @ X4 @ B2 ) )
     => ( ord_le2582152453551361851b_real @ A @ B2 ) ) ).

% subsetI
thf(fact_549_Int__iff,axiom,
    ! [C: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ( member5467075257862553189l_real @ C @ ( inf_in3822636991831349782l_real @ A @ B2 ) )
      = ( ( member5467075257862553189l_real @ C @ A )
        & ( member5467075257862553189l_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_550_Int__iff,axiom,
    ! [C: set_real,A: set_set_real,B2: set_set_real] :
      ( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
      = ( ( member_set_real @ C @ A )
        & ( member_set_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_551_Int__iff,axiom,
    ! [C: real,A: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A @ B2 ) )
      = ( ( member_real @ C @ A )
        & ( member_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_552_Int__iff,axiom,
    ! [C: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ( member1080921170058542931_a_b_b @ C @ ( inf_in1278836612478006532_a_b_b @ A @ B2 ) )
      = ( ( member1080921170058542931_a_b_b @ C @ A )
        & ( member1080921170058542931_a_b_b @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_553_Int__iff,axiom,
    ! [C: real > real,A: set_real_real,B2: set_real_real] :
      ( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B2 ) )
      = ( ( member_real_real @ C @ A )
        & ( member_real_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_554_Int__iff,axiom,
    ! [C: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ( member1118978200981565500b_real @ C @ ( inf_in7065115160788051053b_real @ A @ B2 ) )
      = ( ( member1118978200981565500b_real @ C @ A )
        & ( member1118978200981565500b_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_555_IntI,axiom,
    ! [C: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ( member5467075257862553189l_real @ C @ A )
     => ( ( member5467075257862553189l_real @ C @ B2 )
       => ( member5467075257862553189l_real @ C @ ( inf_in3822636991831349782l_real @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_556_IntI,axiom,
    ! [C: set_real,A: set_set_real,B2: set_set_real] :
      ( ( member_set_real @ C @ A )
     => ( ( member_set_real @ C @ B2 )
       => ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_557_IntI,axiom,
    ! [C: real,A: set_real,B2: set_real] :
      ( ( member_real @ C @ A )
     => ( ( member_real @ C @ B2 )
       => ( member_real @ C @ ( inf_inf_set_real @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_558_IntI,axiom,
    ! [C: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ( member1080921170058542931_a_b_b @ C @ A )
     => ( ( member1080921170058542931_a_b_b @ C @ B2 )
       => ( member1080921170058542931_a_b_b @ C @ ( inf_in1278836612478006532_a_b_b @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_559_IntI,axiom,
    ! [C: real > real,A: set_real_real,B2: set_real_real] :
      ( ( member_real_real @ C @ A )
     => ( ( member_real_real @ C @ B2 )
       => ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_560_IntI,axiom,
    ! [C: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ( member1118978200981565500b_real @ C @ A )
     => ( ( member1118978200981565500b_real @ C @ B2 )
       => ( member1118978200981565500b_real @ C @ ( inf_in7065115160788051053b_real @ A @ B2 ) ) ) ) ).

% IntI
thf(fact_561_Pair__le,axiom,
    ! [A3: nat,B: nat,C: nat,D: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A3 @ B ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A3 @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_562_Pair__le,axiom,
    ! [A3: nat,B: real,C: nat,D: real] :
      ( ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ A3 @ B ) @ ( produc7837566107596912789t_real @ C @ D ) )
      = ( ( ord_less_eq_nat @ A3 @ C )
        & ( ord_less_eq_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_563_Pair__le,axiom,
    ! [A3: nat,B: extend8495563244428889912nnreal,C: nat,D: extend8495563244428889912nnreal] :
      ( ( ord_le8638621733772292105nnreal @ ( produc5075389201112886689nnreal @ A3 @ B ) @ ( produc5075389201112886689nnreal @ C @ D ) )
      = ( ( ord_less_eq_nat @ A3 @ C )
        & ( ord_le3935885782089961368nnreal @ B @ D ) ) ) ).

% Pair_le
thf(fact_564_Pair__le,axiom,
    ! [A3: nat,B: extended_ereal,C: nat,D: extended_ereal] :
      ( ( ord_le4672251659283004883_ereal @ ( produc2502934844456986405_ereal @ A3 @ B ) @ ( produc2502934844456986405_ereal @ C @ D ) )
      = ( ( ord_less_eq_nat @ A3 @ C )
        & ( ord_le1083603963089353582_ereal @ B @ D ) ) ) ).

% Pair_le
thf(fact_565_Pair__le,axiom,
    ! [A3: real,B: nat,C: real,D: nat] :
      ( ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ A3 @ B ) @ ( produc3181502643871035669al_nat @ C @ D ) )
      = ( ( ord_less_eq_real @ A3 @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_566_Pair__le,axiom,
    ! [A3: real,B: real,C: real,D: real] :
      ( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ A3 @ B ) @ ( produc4511245868158468465l_real @ C @ D ) )
      = ( ( ord_less_eq_real @ A3 @ C )
        & ( ord_less_eq_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_567_Pair__le,axiom,
    ! [A3: real,B: extend8495563244428889912nnreal,C: real,D: extend8495563244428889912nnreal] :
      ( ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ A3 @ B ) @ ( produc4778015194254607485nnreal @ C @ D ) )
      = ( ( ord_less_eq_real @ A3 @ C )
        & ( ord_le3935885782089961368nnreal @ B @ D ) ) ) ).

% Pair_le
thf(fact_568_Pair__le,axiom,
    ! [A3: real,B: extended_ereal,C: real,D: extended_ereal] :
      ( ( ord_le6751365068476080247_ereal @ ( produc6400082213244922185_ereal @ A3 @ B ) @ ( produc6400082213244922185_ereal @ C @ D ) )
      = ( ( ord_less_eq_real @ A3 @ C )
        & ( ord_le1083603963089353582_ereal @ B @ D ) ) ) ).

% Pair_le
thf(fact_569_Pair__le,axiom,
    ! [A3: extend8495563244428889912nnreal,B: nat,C: extend8495563244428889912nnreal,D: nat] :
      ( ( ord_le6792163879939894281al_nat @ ( produc625717604924970401al_nat @ A3 @ B ) @ ( produc625717604924970401al_nat @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A3 @ C )
        & ( ord_less_eq_nat @ B @ D ) ) ) ).

% Pair_le
thf(fact_570_Pair__le,axiom,
    ! [A3: extend8495563244428889912nnreal,B: real,C: extend8495563244428889912nnreal,D: real] :
      ( ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ A3 @ B ) @ ( produc2810268924804063229l_real @ C @ D ) )
      = ( ( ord_le3935885782089961368nnreal @ A3 @ C )
        & ( ord_less_eq_real @ B @ D ) ) ) ).

% Pair_le
thf(fact_571_inf_Obounded__iff,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A3 @ B )
        & ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_572_inf_Obounded__iff,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B @ C ) )
      = ( ( ord_less_eq_real @ A3 @ B )
        & ( ord_less_eq_real @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_573_inf_Obounded__iff,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( inf_in7439215052339218890nnreal @ B @ C ) )
      = ( ( ord_le3935885782089961368nnreal @ A3 @ B )
        & ( ord_le3935885782089961368nnreal @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_574_inf_Obounded__iff,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ ( inf_in2794916579150040252_ereal @ B @ C ) )
      = ( ( ord_le1083603963089353582_ereal @ A3 @ B )
        & ( ord_le1083603963089353582_ereal @ A3 @ C ) ) ) ).

% inf.bounded_iff
thf(fact_575_le__inf__iff,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_nat @ X2 @ Y )
        & ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_576_le__inf__iff,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ ( inf_inf_real @ Y @ Z2 ) )
      = ( ( ord_less_eq_real @ X2 @ Y )
        & ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_577_le__inf__iff,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ Y @ Z2 ) )
      = ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
        & ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_578_le__inf__iff,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ ( inf_in2794916579150040252_ereal @ Y @ Z2 ) )
      = ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
        & ( ord_le1083603963089353582_ereal @ X2 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_579_inf__top_Oright__neutral,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( inf_in7439215052339218890nnreal @ A3 @ top_to1496364449551166952nnreal )
      = A3 ) ).

% inf_top.right_neutral
thf(fact_580_inf__top_Oneutr__eq__iff,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( top_to1496364449551166952nnreal
        = ( inf_in7439215052339218890nnreal @ A3 @ B ) )
      = ( ( A3 = top_to1496364449551166952nnreal )
        & ( B = top_to1496364449551166952nnreal ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_581_inf__top_Oleft__neutral,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( inf_in7439215052339218890nnreal @ top_to1496364449551166952nnreal @ A3 )
      = A3 ) ).

% inf_top.left_neutral
thf(fact_582_inf__top_Oeq__neutr__iff,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( inf_in7439215052339218890nnreal @ A3 @ B )
        = top_to1496364449551166952nnreal )
      = ( ( A3 = top_to1496364449551166952nnreal )
        & ( B = top_to1496364449551166952nnreal ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_583_top__eq__inf__iff,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( top_to1496364449551166952nnreal
        = ( inf_in7439215052339218890nnreal @ X2 @ Y ) )
      = ( ( X2 = top_to1496364449551166952nnreal )
        & ( Y = top_to1496364449551166952nnreal ) ) ) ).

% top_eq_inf_iff
thf(fact_584_inf__eq__top__iff,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ( inf_in7439215052339218890nnreal @ X2 @ Y )
        = top_to1496364449551166952nnreal )
      = ( ( X2 = top_to1496364449551166952nnreal )
        & ( Y = top_to1496364449551166952nnreal ) ) ) ).

% inf_eq_top_iff
thf(fact_585_inf__top__right,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( inf_in7439215052339218890nnreal @ X2 @ top_to1496364449551166952nnreal )
      = X2 ) ).

% inf_top_right
thf(fact_586_inf__set__def,axiom,
    ( inf_in3822636991831349782l_real
    = ( ^ [A2: set_Pr3917002265660371076l_real,B4: set_Pr3917002265660371076l_real] :
          ( collec869437573433150115l_real
          @ ( inf_in7897423780479368175real_o
            @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ A2 )
            @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_587_inf__set__def,axiom,
    ( inf_inf_set_set_real
    = ( ^ [A2: set_set_real,B4: set_set_real] :
          ( collect_set_real
          @ ( inf_inf_set_real_o
            @ ^ [X: set_real] : ( member_set_real @ X @ A2 )
            @ ^ [X: set_real] : ( member_set_real @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_588_inf__set__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A2: set_real,B4: set_real] :
          ( collect_real
          @ ( inf_inf_real_o
            @ ^ [X: real] : ( member_real @ X @ A2 )
            @ ^ [X: real] : ( member_real @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_589_inf__set__def,axiom,
    ( inf_in1278836612478006532_a_b_b
    = ( ^ [A2: set_Pr3674455588107860082_a_b_b,B4: set_Pr3674455588107860082_a_b_b] :
          ( collec2250730345169186961_a_b_b
          @ ( inf_in9192623926311598209_b_b_o
            @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ A2 )
            @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_590_inf__set__def,axiom,
    ( inf_in5166753670444608447l_real
    = ( ^ [A2: set_real_real,B4: set_real_real] :
          ( collect_real_real
          @ ( inf_inf_real_real_o
            @ ^ [X: real > real] : ( member_real_real @ X @ A2 )
            @ ^ [X: real > real] : ( member_real_real @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_591_inf__set__def,axiom,
    ( inf_in7065115160788051053b_real
    = ( ^ [A2: set_Pr1027596265373204187b_real,B4: set_Pr1027596265373204187b_real] :
          ( collec5735648035525910650b_real
          @ ( inf_in2390107153597142616real_o
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ A2 )
            @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ B4 ) ) ) ) ) ).

% inf_set_def
thf(fact_592_less__eq__set__def,axiom,
    ( ord_le8308227983669428452l_real
    = ( ^ [A2: set_Pr3917002265660371076l_real,B4: set_Pr3917002265660371076l_real] :
          ( ord_le8040096449130281633real_o
          @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ A2 )
          @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_593_less__eq__set__def,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A2: set_set_real,B4: set_set_real] :
          ( ord_le2392157289819280397real_o
          @ ^ [X: set_real] : ( member_set_real @ X @ A2 )
          @ ^ [X: set_real] : ( member_set_real @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_594_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A2: set_real,B4: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A2 )
          @ ^ [X: real] : ( member_real @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_595_less__eq__set__def,axiom,
    ( ord_le8439411419596101842_a_b_b
    = ( ^ [A2: set_Pr3674455588107860082_a_b_b,B4: set_Pr3674455588107860082_a_b_b] :
          ( ord_le3099313929993222707_b_b_o
          @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ A2 )
          @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_596_less__eq__set__def,axiom,
    ( ord_le4198349162570665613l_real
    = ( ^ [A2: set_real_real,B4: set_real_real] :
          ( ord_le5273791883478943800real_o
          @ ^ [X: real > real] : ( member_real_real @ X @ A2 )
          @ ^ [X: real > real] : ( member_real_real @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_597_less__eq__set__def,axiom,
    ( ord_le2582152453551361851b_real
    = ( ^ [A2: set_Pr1027596265373204187b_real,B4: set_Pr1027596265373204187b_real] :
          ( ord_le5987956920600383754real_o
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ A2 )
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_598_snd__diag__snd,axiom,
    ( ( comp_P7068956567367034951b_real @ produc3484788084999411615l_real
      @ ( comp_r6448350170807567469b_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc7037151529651111887b_real ) )
    = produc7037151529651111887b_real ) ).

% snd_diag_snd
thf(fact_599_snd__diag__snd,axiom,
    ( ( comp_P5715990135234274861od_a_b @ product_snd_b_b
      @ ( comp_b6573281471806798941od_a_b
        @ ^ [X: b] : ( product_Pair_b_b @ X @ X )
        @ product_snd_a_b ) )
    = product_snd_a_b ) ).

% snd_diag_snd
thf(fact_600_UNIV__witness,axiom,
    ? [X4: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X4 @ top_to5286755836160161588l_real ) ).

% UNIV_witness
thf(fact_601_UNIV__witness,axiom,
    ? [X4: set_real] : ( member_set_real @ X4 @ top_top_set_set_real ) ).

% UNIV_witness
thf(fact_602_UNIV__witness,axiom,
    ? [X4: real] : ( member_real @ X4 @ top_top_set_real ) ).

% UNIV_witness
thf(fact_603_UNIV__witness,axiom,
    ? [X4: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X4 @ top_to8883559732078315298_a_b_b ) ).

% UNIV_witness
thf(fact_604_UNIV__witness,axiom,
    ? [X4: real > real] : ( member_real_real @ X4 @ top_to2071711978144146653l_real ) ).

% UNIV_witness
thf(fact_605_UNIV__witness,axiom,
    ? [X4: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X4 @ top_to1564948194967201163b_real ) ).

% UNIV_witness
thf(fact_606_UNIV__eq__I,axiom,
    ! [A: set_Pr3917002265660371076l_real] :
      ( ! [X4: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X4 @ A )
     => ( top_to5286755836160161588l_real = A ) ) ).

% UNIV_eq_I
thf(fact_607_UNIV__eq__I,axiom,
    ! [A: set_set_real] :
      ( ! [X4: set_real] : ( member_set_real @ X4 @ A )
     => ( top_top_set_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_608_UNIV__eq__I,axiom,
    ! [A: set_real] :
      ( ! [X4: real] : ( member_real @ X4 @ A )
     => ( top_top_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_609_UNIV__eq__I,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b] :
      ( ! [X4: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X4 @ A )
     => ( top_to8883559732078315298_a_b_b = A ) ) ).

% UNIV_eq_I
thf(fact_610_UNIV__eq__I,axiom,
    ! [A: set_real_real] :
      ( ! [X4: real > real] : ( member_real_real @ X4 @ A )
     => ( top_to2071711978144146653l_real = A ) ) ).

% UNIV_eq_I
thf(fact_611_UNIV__eq__I,axiom,
    ! [A: set_Pr1027596265373204187b_real] :
      ( ! [X4: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X4 @ A )
     => ( top_to1564948194967201163b_real = A ) ) ).

% UNIV_eq_I
thf(fact_612_subset__iff,axiom,
    ( ord_le8308227983669428452l_real
    = ( ^ [A2: set_Pr3917002265660371076l_real,B4: set_Pr3917002265660371076l_real] :
        ! [T2: produc7929879615859924995b_real > real] :
          ( ( member5467075257862553189l_real @ T2 @ A2 )
         => ( member5467075257862553189l_real @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_613_subset__iff,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A2: set_set_real,B4: set_set_real] :
        ! [T2: set_real] :
          ( ( member_set_real @ T2 @ A2 )
         => ( member_set_real @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_614_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A2: set_real,B4: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A2 )
         => ( member_real @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_615_subset__iff,axiom,
    ( ord_le8439411419596101842_a_b_b
    = ( ^ [A2: set_Pr3674455588107860082_a_b_b,B4: set_Pr3674455588107860082_a_b_b] :
        ! [T2: product_prod_a_b > b] :
          ( ( member1080921170058542931_a_b_b @ T2 @ A2 )
         => ( member1080921170058542931_a_b_b @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_616_subset__iff,axiom,
    ( ord_le4198349162570665613l_real
    = ( ^ [A2: set_real_real,B4: set_real_real] :
        ! [T2: real > real] :
          ( ( member_real_real @ T2 @ A2 )
         => ( member_real_real @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_617_subset__iff,axiom,
    ( ord_le2582152453551361851b_real
    = ( ^ [A2: set_Pr1027596265373204187b_real,B4: set_Pr1027596265373204187b_real] :
        ! [T2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
          ( ( member1118978200981565500b_real @ T2 @ A2 )
         => ( member1118978200981565500b_real @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_618_subset__eq,axiom,
    ( ord_le8308227983669428452l_real
    = ( ^ [A2: set_Pr3917002265660371076l_real,B4: set_Pr3917002265660371076l_real] :
        ! [X: produc7929879615859924995b_real > real] :
          ( ( member5467075257862553189l_real @ X @ A2 )
         => ( member5467075257862553189l_real @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_619_subset__eq,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A2: set_set_real,B4: set_set_real] :
        ! [X: set_real] :
          ( ( member_set_real @ X @ A2 )
         => ( member_set_real @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_620_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A2: set_real,B4: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A2 )
         => ( member_real @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_621_subset__eq,axiom,
    ( ord_le8439411419596101842_a_b_b
    = ( ^ [A2: set_Pr3674455588107860082_a_b_b,B4: set_Pr3674455588107860082_a_b_b] :
        ! [X: product_prod_a_b > b] :
          ( ( member1080921170058542931_a_b_b @ X @ A2 )
         => ( member1080921170058542931_a_b_b @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_622_subset__eq,axiom,
    ( ord_le4198349162570665613l_real
    = ( ^ [A2: set_real_real,B4: set_real_real] :
        ! [X: real > real] :
          ( ( member_real_real @ X @ A2 )
         => ( member_real_real @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_623_subset__eq,axiom,
    ( ord_le2582152453551361851b_real
    = ( ^ [A2: set_Pr1027596265373204187b_real,B4: set_Pr1027596265373204187b_real] :
        ! [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
          ( ( member1118978200981565500b_real @ X @ A2 )
         => ( member1118978200981565500b_real @ X @ B4 ) ) ) ) ).

% subset_eq
thf(fact_624_subsetD,axiom,
    ! [A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real,C: produc7929879615859924995b_real > real] :
      ( ( ord_le8308227983669428452l_real @ A @ B2 )
     => ( ( member5467075257862553189l_real @ C @ A )
       => ( member5467075257862553189l_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_625_subsetD,axiom,
    ! [A: set_set_real,B2: set_set_real,C: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B2 )
     => ( ( member_set_real @ C @ A )
       => ( member_set_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_626_subsetD,axiom,
    ! [A: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A @ B2 )
     => ( ( member_real @ C @ A )
       => ( member_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_627_subsetD,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b,C: product_prod_a_b > b] :
      ( ( ord_le8439411419596101842_a_b_b @ A @ B2 )
     => ( ( member1080921170058542931_a_b_b @ C @ A )
       => ( member1080921170058542931_a_b_b @ C @ B2 ) ) ) ).

% subsetD
thf(fact_628_subsetD,axiom,
    ! [A: set_real_real,B2: set_real_real,C: real > real] :
      ( ( ord_le4198349162570665613l_real @ A @ B2 )
     => ( ( member_real_real @ C @ A )
       => ( member_real_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_629_subsetD,axiom,
    ! [A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real,C: produc7929879615859924995b_real > produc7929879615859924995b_real] :
      ( ( ord_le2582152453551361851b_real @ A @ B2 )
     => ( ( member1118978200981565500b_real @ C @ A )
       => ( member1118978200981565500b_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_630_in__mono,axiom,
    ! [A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real,X2: produc7929879615859924995b_real > real] :
      ( ( ord_le8308227983669428452l_real @ A @ B2 )
     => ( ( member5467075257862553189l_real @ X2 @ A )
       => ( member5467075257862553189l_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_631_in__mono,axiom,
    ! [A: set_set_real,B2: set_set_real,X2: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B2 )
     => ( ( member_set_real @ X2 @ A )
       => ( member_set_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_632_in__mono,axiom,
    ! [A: set_real,B2: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A @ B2 )
     => ( ( member_real @ X2 @ A )
       => ( member_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_633_in__mono,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b,X2: product_prod_a_b > b] :
      ( ( ord_le8439411419596101842_a_b_b @ A @ B2 )
     => ( ( member1080921170058542931_a_b_b @ X2 @ A )
       => ( member1080921170058542931_a_b_b @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_634_in__mono,axiom,
    ! [A: set_real_real,B2: set_real_real,X2: real > real] :
      ( ( ord_le4198349162570665613l_real @ A @ B2 )
     => ( ( member_real_real @ X2 @ A )
       => ( member_real_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_635_in__mono,axiom,
    ! [A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real,X2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
      ( ( ord_le2582152453551361851b_real @ A @ B2 )
     => ( ( member1118978200981565500b_real @ X2 @ A )
       => ( member1118978200981565500b_real @ X2 @ B2 ) ) ) ).

% in_mono
thf(fact_636_fst__diag__snd,axiom,
    ( ( comp_P7068956567367034951b_real @ produc5828954698716094813l_real
      @ ( comp_r6448350170807567469b_real
        @ ^ [X: real] : ( produc4511245868158468465l_real @ X @ X )
        @ produc7037151529651111887b_real ) )
    = produc7037151529651111887b_real ) ).

% fst_diag_snd
thf(fact_637_fst__diag__snd,axiom,
    ( ( comp_P5715990135234274861od_a_b @ product_fst_b_b
      @ ( comp_b6573281471806798941od_a_b
        @ ^ [X: b] : ( product_Pair_b_b @ X @ X )
        @ product_snd_a_b ) )
    = product_snd_a_b ) ).

% fst_diag_snd
thf(fact_638_IntD2,axiom,
    ! [C: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ( member5467075257862553189l_real @ C @ ( inf_in3822636991831349782l_real @ A @ B2 ) )
     => ( member5467075257862553189l_real @ C @ B2 ) ) ).

% IntD2
thf(fact_639_IntD2,axiom,
    ! [C: set_real,A: set_set_real,B2: set_set_real] :
      ( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
     => ( member_set_real @ C @ B2 ) ) ).

% IntD2
thf(fact_640_IntD2,axiom,
    ! [C: real,A: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A @ B2 ) )
     => ( member_real @ C @ B2 ) ) ).

% IntD2
thf(fact_641_IntD2,axiom,
    ! [C: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ( member1080921170058542931_a_b_b @ C @ ( inf_in1278836612478006532_a_b_b @ A @ B2 ) )
     => ( member1080921170058542931_a_b_b @ C @ B2 ) ) ).

% IntD2
thf(fact_642_IntD2,axiom,
    ! [C: real > real,A: set_real_real,B2: set_real_real] :
      ( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B2 ) )
     => ( member_real_real @ C @ B2 ) ) ).

% IntD2
thf(fact_643_IntD2,axiom,
    ! [C: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ( member1118978200981565500b_real @ C @ ( inf_in7065115160788051053b_real @ A @ B2 ) )
     => ( member1118978200981565500b_real @ C @ B2 ) ) ).

% IntD2
thf(fact_644_IntD1,axiom,
    ! [C: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ( member5467075257862553189l_real @ C @ ( inf_in3822636991831349782l_real @ A @ B2 ) )
     => ( member5467075257862553189l_real @ C @ A ) ) ).

% IntD1
thf(fact_645_IntD1,axiom,
    ! [C: set_real,A: set_set_real,B2: set_set_real] :
      ( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
     => ( member_set_real @ C @ A ) ) ).

% IntD1
thf(fact_646_IntD1,axiom,
    ! [C: real,A: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A @ B2 ) )
     => ( member_real @ C @ A ) ) ).

% IntD1
thf(fact_647_IntD1,axiom,
    ! [C: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ( member1080921170058542931_a_b_b @ C @ ( inf_in1278836612478006532_a_b_b @ A @ B2 ) )
     => ( member1080921170058542931_a_b_b @ C @ A ) ) ).

% IntD1
thf(fact_648_IntD1,axiom,
    ! [C: real > real,A: set_real_real,B2: set_real_real] :
      ( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B2 ) )
     => ( member_real_real @ C @ A ) ) ).

% IntD1
thf(fact_649_IntD1,axiom,
    ! [C: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ( member1118978200981565500b_real @ C @ ( inf_in7065115160788051053b_real @ A @ B2 ) )
     => ( member1118978200981565500b_real @ C @ A ) ) ).

% IntD1
thf(fact_650_IntE,axiom,
    ! [C: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real] :
      ( ( member5467075257862553189l_real @ C @ ( inf_in3822636991831349782l_real @ A @ B2 ) )
     => ~ ( ( member5467075257862553189l_real @ C @ A )
         => ~ ( member5467075257862553189l_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_651_IntE,axiom,
    ! [C: set_real,A: set_set_real,B2: set_set_real] :
      ( ( member_set_real @ C @ ( inf_inf_set_set_real @ A @ B2 ) )
     => ~ ( ( member_set_real @ C @ A )
         => ~ ( member_set_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_652_IntE,axiom,
    ! [C: real,A: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A @ B2 ) )
     => ~ ( ( member_real @ C @ A )
         => ~ ( member_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_653_IntE,axiom,
    ! [C: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b] :
      ( ( member1080921170058542931_a_b_b @ C @ ( inf_in1278836612478006532_a_b_b @ A @ B2 ) )
     => ~ ( ( member1080921170058542931_a_b_b @ C @ A )
         => ~ ( member1080921170058542931_a_b_b @ C @ B2 ) ) ) ).

% IntE
thf(fact_654_IntE,axiom,
    ! [C: real > real,A: set_real_real,B2: set_real_real] :
      ( ( member_real_real @ C @ ( inf_in5166753670444608447l_real @ A @ B2 ) )
     => ~ ( ( member_real_real @ C @ A )
         => ~ ( member_real_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_655_IntE,axiom,
    ! [C: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real] :
      ( ( member1118978200981565500b_real @ C @ ( inf_in7065115160788051053b_real @ A @ B2 ) )
     => ~ ( ( member1118978200981565500b_real @ C @ A )
         => ~ ( member1118978200981565500b_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_656_Euclidean__dist__upper,axiom,
    ! [I: real,X2: real,Y: real] :
      ( ( member_real @ I @ euclid1305858884100475807s_real )
     => ( ord_less_eq_real @ ( real_V975177566351809787t_real @ ( inner_5569486634795291694r_real @ X2 @ I ) @ ( inner_5569486634795291694r_real @ Y @ I ) ) @ ( real_V975177566351809787t_real @ X2 @ Y ) ) ) ).

% Euclidean_dist_upper
thf(fact_657_hafspace__greater__eq__borel,axiom,
    ! [I: real,A3: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] : ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ X @ I ) @ A3 ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% hafspace_greater_eq_borel
thf(fact_658_hafspace__greater__eq__borel,axiom,
    ! [I: produc2422161461964618553l_real,A3: real] :
      ( member6767226683953100098l_real
      @ ( collec3799799289383736868l_real
        @ ^ [X: produc2422161461964618553l_real] : ( ord_less_eq_real @ ( inner_2340953980025574455l_real @ X @ I ) @ A3 ) )
      @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) ) ).

% hafspace_greater_eq_borel
thf(fact_659_hafspace__less__eq__borel,axiom,
    ! [A3: real,I: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X: real] : ( ord_less_eq_real @ A3 @ ( inner_5569486634795291694r_real @ X @ I ) ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% hafspace_less_eq_borel
thf(fact_660_hafspace__less__eq__borel,axiom,
    ! [A3: real,I: produc2422161461964618553l_real] :
      ( member6767226683953100098l_real
      @ ( collec3799799289383736868l_real
        @ ^ [X: produc2422161461964618553l_real] : ( ord_less_eq_real @ A3 @ ( inner_2340953980025574455l_real @ X @ I ) ) )
      @ ( sigma_1543982426361076760l_real @ borel_9155112475215227991l_real ) ) ).

% hafspace_less_eq_borel
thf(fact_661_Collect__subset,axiom,
    ! [A: set_Pr3917002265660371076l_real,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ord_le8308227983669428452l_real
      @ ( collec869437573433150115l_real
        @ ^ [X: produc7929879615859924995b_real > real] :
            ( ( member5467075257862553189l_real @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_662_Collect__subset,axiom,
    ! [A: set_set_real,P: set_real > $o] :
      ( ord_le3558479182127378552t_real
      @ ( collect_set_real
        @ ^ [X: set_real] :
            ( ( member_set_real @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_663_Collect__subset,axiom,
    ! [A: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_664_Collect__subset,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b,P: ( product_prod_a_b > b ) > $o] :
      ( ord_le8439411419596101842_a_b_b
      @ ( collec2250730345169186961_a_b_b
        @ ^ [X: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_665_Collect__subset,axiom,
    ! [A: set_real_real,P: ( real > real ) > $o] :
      ( ord_le4198349162570665613l_real
      @ ( collect_real_real
        @ ^ [X: real > real] :
            ( ( member_real_real @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_666_Collect__subset,axiom,
    ! [A: set_Pr1027596265373204187b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ord_le2582152453551361851b_real
      @ ( collec5735648035525910650b_real
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
            ( ( member1118978200981565500b_real @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_667_Int__def,axiom,
    ( inf_in3822636991831349782l_real
    = ( ^ [A2: set_Pr3917002265660371076l_real,B4: set_Pr3917002265660371076l_real] :
          ( collec869437573433150115l_real
          @ ^ [X: produc7929879615859924995b_real > real] :
              ( ( member5467075257862553189l_real @ X @ A2 )
              & ( member5467075257862553189l_real @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_668_Int__def,axiom,
    ( inf_inf_set_set_real
    = ( ^ [A2: set_set_real,B4: set_set_real] :
          ( collect_set_real
          @ ^ [X: set_real] :
              ( ( member_set_real @ X @ A2 )
              & ( member_set_real @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_669_Int__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A2: set_real,B4: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A2 )
              & ( member_real @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_670_Int__def,axiom,
    ( inf_in1278836612478006532_a_b_b
    = ( ^ [A2: set_Pr3674455588107860082_a_b_b,B4: set_Pr3674455588107860082_a_b_b] :
          ( collec2250730345169186961_a_b_b
          @ ^ [X: product_prod_a_b > b] :
              ( ( member1080921170058542931_a_b_b @ X @ A2 )
              & ( member1080921170058542931_a_b_b @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_671_Int__def,axiom,
    ( inf_in5166753670444608447l_real
    = ( ^ [A2: set_real_real,B4: set_real_real] :
          ( collect_real_real
          @ ^ [X: real > real] :
              ( ( member_real_real @ X @ A2 )
              & ( member_real_real @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_672_Int__def,axiom,
    ( inf_in7065115160788051053b_real
    = ( ^ [A2: set_Pr1027596265373204187b_real,B4: set_Pr1027596265373204187b_real] :
          ( collec5735648035525910650b_real
          @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] :
              ( ( member1118978200981565500b_real @ X @ A2 )
              & ( member1118978200981565500b_real @ X @ B4 ) ) ) ) ) ).

% Int_def
thf(fact_673_Int__Collect,axiom,
    ! [X2: produc7929879615859924995b_real > real,A: set_Pr3917002265660371076l_real,P: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( member5467075257862553189l_real @ X2 @ ( inf_in3822636991831349782l_real @ A @ ( collec869437573433150115l_real @ P ) ) )
      = ( ( member5467075257862553189l_real @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_674_Int__Collect,axiom,
    ! [X2: set_real,A: set_set_real,P: set_real > $o] :
      ( ( member_set_real @ X2 @ ( inf_inf_set_set_real @ A @ ( collect_set_real @ P ) ) )
      = ( ( member_set_real @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_675_Int__Collect,axiom,
    ! [X2: real,A: set_real,P: real > $o] :
      ( ( member_real @ X2 @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) )
      = ( ( member_real @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_676_Int__Collect,axiom,
    ! [X2: product_prod_a_b > b,A: set_Pr3674455588107860082_a_b_b,P: ( product_prod_a_b > b ) > $o] :
      ( ( member1080921170058542931_a_b_b @ X2 @ ( inf_in1278836612478006532_a_b_b @ A @ ( collec2250730345169186961_a_b_b @ P ) ) )
      = ( ( member1080921170058542931_a_b_b @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_677_Int__Collect,axiom,
    ! [X2: real > real,A: set_real_real,P: ( real > real ) > $o] :
      ( ( member_real_real @ X2 @ ( inf_in5166753670444608447l_real @ A @ ( collect_real_real @ P ) ) )
      = ( ( member_real_real @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_678_Int__Collect,axiom,
    ! [X2: produc7929879615859924995b_real > produc7929879615859924995b_real,A: set_Pr1027596265373204187b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( member1118978200981565500b_real @ X2 @ ( inf_in7065115160788051053b_real @ A @ ( collec5735648035525910650b_real @ P ) ) )
      = ( ( member1118978200981565500b_real @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_679_borel__measurable__iff__ge,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member3057847817020379547l_real
            @ ( collec869437573433150115l_real
              @ ^ [W2: produc7929879615859924995b_real > real] :
                  ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_680_borel__measurable__iff__ge,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_set_real
            @ ( collect_set_real
              @ ^ [W2: set_real] :
                  ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_681_borel__measurable__iff__ge,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member4198857394546769929_a_b_b
            @ ( collec2250730345169186961_a_b_b
              @ ^ [W2: product_prod_a_b > b] :
                  ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_682_borel__measurable__iff__ge,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_real_real2
            @ ( collect_real_real
              @ ^ [W2: real > real] :
                  ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_683_borel__measurable__iff__ge,axiom,
    ! [F: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real,M: sigma_7931231546270528766b_real] :
      ( ( member5779234595813729927l_real @ F @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member566263741820371442b_real
            @ ( collec5735648035525910650b_real
              @ ^ [W2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                  ( ( member1118978200981565500b_real @ W2 @ ( sigma_6346777283556531613b_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_684_borel__measurable__iff__ge,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member2587080422514930128b_real
            @ ( collec4671965459240937304b_real
              @ ^ [W2: produc7929879615859924995b_real] :
                  ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_685_borel__measurable__iff__ge,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_real
            @ ( collect_real
              @ ^ [W2: real] :
                  ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                  & ( ord_less_eq_real @ A4 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_686_borel__measurable__iff__le,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member3057847817020379547l_real
            @ ( collec869437573433150115l_real
              @ ^ [W2: produc7929879615859924995b_real > real] :
                  ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_687_borel__measurable__iff__le,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_set_real
            @ ( collect_set_real
              @ ^ [W2: set_real] :
                  ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_sets_set_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_688_borel__measurable__iff__le,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member4198857394546769929_a_b_b
            @ ( collec2250730345169186961_a_b_b
              @ ^ [W2: product_prod_a_b > b] :
                  ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_689_borel__measurable__iff__le,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_real_real2
            @ ( collect_real_real
              @ ^ [W2: real > real] :
                  ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_sets_real_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_690_borel__measurable__iff__le,axiom,
    ! [F: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > real,M: sigma_7931231546270528766b_real] :
      ( ( member5779234595813729927l_real @ F @ ( sigma_3417798853871732131l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member566263741820371442b_real
            @ ( collec5735648035525910650b_real
              @ ^ [W2: produc7929879615859924995b_real > produc7929879615859924995b_real] :
                  ( ( member1118978200981565500b_real @ W2 @ ( sigma_6346777283556531613b_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_1595522527309588870b_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_691_borel__measurable__iff__le,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member2587080422514930128b_real
            @ ( collec4671965459240937304b_real
              @ ^ [W2: produc7929879615859924995b_real] :
                  ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_692_borel__measurable__iff__le,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A4: real] :
            ( member_set_real
            @ ( collect_real
              @ ^ [W2: real] :
                  ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A4 ) ) )
            @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_693_Pair__mono,axiom,
    ! [X2: nat,X5: nat,Y: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X2 @ X5 )
     => ( ( ord_less_eq_nat @ Y @ Y5 )
       => ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( product_Pair_nat_nat @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_694_Pair__mono,axiom,
    ! [X2: nat,X5: nat,Y: real,Y5: real] :
      ( ( ord_less_eq_nat @ X2 @ X5 )
     => ( ( ord_less_eq_real @ Y @ Y5 )
       => ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ X2 @ Y ) @ ( produc7837566107596912789t_real @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_695_Pair__mono,axiom,
    ! [X2: nat,X5: nat,Y: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ X2 @ X5 )
     => ( ( ord_le3935885782089961368nnreal @ Y @ Y5 )
       => ( ord_le8638621733772292105nnreal @ ( produc5075389201112886689nnreal @ X2 @ Y ) @ ( produc5075389201112886689nnreal @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_696_Pair__mono,axiom,
    ! [X2: nat,X5: nat,Y: extended_ereal,Y5: extended_ereal] :
      ( ( ord_less_eq_nat @ X2 @ X5 )
     => ( ( ord_le1083603963089353582_ereal @ Y @ Y5 )
       => ( ord_le4672251659283004883_ereal @ ( produc2502934844456986405_ereal @ X2 @ Y ) @ ( produc2502934844456986405_ereal @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_697_Pair__mono,axiom,
    ! [X2: real,X5: real,Y: nat,Y5: nat] :
      ( ( ord_less_eq_real @ X2 @ X5 )
     => ( ( ord_less_eq_nat @ Y @ Y5 )
       => ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ X2 @ Y ) @ ( produc3181502643871035669al_nat @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_698_Pair__mono,axiom,
    ! [X2: real,X5: real,Y: real,Y5: real] :
      ( ( ord_less_eq_real @ X2 @ X5 )
     => ( ( ord_less_eq_real @ Y @ Y5 )
       => ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X2 @ Y ) @ ( produc4511245868158468465l_real @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_699_Pair__mono,axiom,
    ! [X2: real,X5: real,Y: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ X2 @ X5 )
     => ( ( ord_le3935885782089961368nnreal @ Y @ Y5 )
       => ( ord_le4096773168995780197nnreal @ ( produc4778015194254607485nnreal @ X2 @ Y ) @ ( produc4778015194254607485nnreal @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_700_Pair__mono,axiom,
    ! [X2: real,X5: real,Y: extended_ereal,Y5: extended_ereal] :
      ( ( ord_less_eq_real @ X2 @ X5 )
     => ( ( ord_le1083603963089353582_ereal @ Y @ Y5 )
       => ( ord_le6751365068476080247_ereal @ ( produc6400082213244922185_ereal @ X2 @ Y ) @ ( produc6400082213244922185_ereal @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_701_Pair__mono,axiom,
    ! [X2: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal,Y: nat,Y5: nat] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ X5 )
     => ( ( ord_less_eq_nat @ Y @ Y5 )
       => ( ord_le6792163879939894281al_nat @ ( produc625717604924970401al_nat @ X2 @ Y ) @ ( produc625717604924970401al_nat @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_702_Pair__mono,axiom,
    ! [X2: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal,Y: real,Y5: real] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ X5 )
     => ( ( ord_less_eq_real @ Y @ Y5 )
       => ( ord_le4051224869651757541l_real @ ( produc2810268924804063229l_real @ X2 @ Y ) @ ( produc2810268924804063229l_real @ X5 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_703_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_704_inf_OcoboundedI2,axiom,
    ! [B: real,C: real,A3: real] :
      ( ( ord_less_eq_real @ B @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_705_inf_OcoboundedI2,axiom,
    ! [B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ C )
     => ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_706_inf_OcoboundedI2,axiom,
    ! [B: extended_ereal,C: extended_ereal,A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ C )
     => ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_707_inf_OcoboundedI1,axiom,
    ! [A3: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_708_inf_OcoboundedI1,axiom,
    ! [A3: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ C )
     => ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_709_inf_OcoboundedI1,axiom,
    ! [A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ C )
     => ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_710_inf_OcoboundedI1,axiom,
    ! [A3: extended_ereal,C: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ C )
     => ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_711_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A4: nat] :
          ( ( inf_inf_nat @ A4 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_712_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_real
    = ( ^ [B5: real,A4: real] :
          ( ( inf_inf_real @ A4 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_713_inf_Oabsorb__iff2,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [B5: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
          ( ( inf_in7439215052339218890nnreal @ A4 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_714_inf_Oabsorb__iff2,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [B5: extended_ereal,A4: extended_ereal] :
          ( ( inf_in2794916579150040252_ereal @ A4 @ B5 )
          = B5 ) ) ) ).

% inf.absorb_iff2
thf(fact_715_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B5: nat] :
          ( ( inf_inf_nat @ A4 @ B5 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_716_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B5: real] :
          ( ( inf_inf_real @ A4 @ B5 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_717_inf_Oabsorb__iff1,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( inf_in7439215052339218890nnreal @ A4 @ B5 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_718_inf_Oabsorb__iff1,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( inf_in2794916579150040252_ereal @ A4 @ B5 )
          = A4 ) ) ) ).

% inf.absorb_iff1
thf(fact_719_inf_Ocobounded2,axiom,
    ! [A3: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_720_inf_Ocobounded2,axiom,
    ! [A3: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_721_inf_Ocobounded2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_722_inf_Ocobounded2,axiom,
    ! [A3: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ B ) ).

% inf.cobounded2
thf(fact_723_inf_Ocobounded1,axiom,
    ! [A3: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ A3 ) ).

% inf.cobounded1
thf(fact_724_inf_Ocobounded1,axiom,
    ! [A3: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ A3 ) ).

% inf.cobounded1
thf(fact_725_inf_Ocobounded1,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ A3 ) ).

% inf.cobounded1
thf(fact_726_inf_Ocobounded1,axiom,
    ! [A3: extended_ereal,B: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ A3 ) ).

% inf.cobounded1
thf(fact_727_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B5: nat] :
          ( A4
          = ( inf_inf_nat @ A4 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_728_inf_Oorder__iff,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B5: real] :
          ( A4
          = ( inf_inf_real @ A4 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_729_inf_Oorder__iff,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( A4
          = ( inf_in7439215052339218890nnreal @ A4 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_730_inf_Oorder__iff,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( A4
          = ( inf_in2794916579150040252_ereal @ A4 @ B5 ) ) ) ) ).

% inf.order_iff
thf(fact_731_inf__greatest,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ X2 @ Z2 )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_732_inf__greatest,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ X2 @ Z2 )
       => ( ord_less_eq_real @ X2 @ ( inf_inf_real @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_733_inf__greatest,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ X2 @ Z2 )
       => ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_734_inf__greatest,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( ord_le1083603963089353582_ereal @ X2 @ Z2 )
       => ( ord_le1083603963089353582_ereal @ X2 @ ( inf_in2794916579150040252_ereal @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_735_inf_OboundedI,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_eq_nat @ A3 @ C )
       => ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_736_inf_OboundedI,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_eq_real @ A3 @ C )
       => ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_737_inf_OboundedI,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ A3 @ C )
       => ( ord_le3935885782089961368nnreal @ A3 @ ( inf_in7439215052339218890nnreal @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_738_inf_OboundedI,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ A3 @ C )
       => ( ord_le1083603963089353582_ereal @ A3 @ ( inf_in2794916579150040252_ereal @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_739_inf_OboundedE,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A3 @ B )
         => ~ ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_740_inf_OboundedE,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( inf_inf_real @ B @ C ) )
     => ~ ( ( ord_less_eq_real @ A3 @ B )
         => ~ ( ord_less_eq_real @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_741_inf_OboundedE,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( inf_in7439215052339218890nnreal @ B @ C ) )
     => ~ ( ( ord_le3935885782089961368nnreal @ A3 @ B )
         => ~ ( ord_le3935885782089961368nnreal @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_742_inf_OboundedE,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ ( inf_in2794916579150040252_ereal @ B @ C ) )
     => ~ ( ( ord_le1083603963089353582_ereal @ A3 @ B )
         => ~ ( ord_le1083603963089353582_ereal @ A3 @ C ) ) ) ).

% inf.boundedE
thf(fact_743_inf__absorb2,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( inf_inf_nat @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_744_inf__absorb2,axiom,
    ! [Y: real,X2: real] :
      ( ( ord_less_eq_real @ Y @ X2 )
     => ( ( inf_inf_real @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_745_inf__absorb2,axiom,
    ! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
     => ( ( inf_in7439215052339218890nnreal @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_746_inf__absorb2,axiom,
    ! [Y: extended_ereal,X2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ Y @ X2 )
     => ( ( inf_in2794916579150040252_ereal @ X2 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_747_inf__absorb1,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( inf_inf_nat @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_748_inf__absorb1,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( inf_inf_real @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_749_inf__absorb1,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( inf_in7439215052339218890nnreal @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_750_inf__absorb1,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( inf_in2794916579150040252_ereal @ X2 @ Y )
        = X2 ) ) ).

% inf_absorb1
thf(fact_751_inf_Oabsorb2,axiom,
    ! [B: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B @ A3 )
     => ( ( inf_inf_nat @ A3 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_752_inf_Oabsorb2,axiom,
    ! [B: real,A3: real] :
      ( ( ord_less_eq_real @ B @ A3 )
     => ( ( inf_inf_real @ A3 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_753_inf_Oabsorb2,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A3 )
     => ( ( inf_in7439215052339218890nnreal @ A3 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_754_inf_Oabsorb2,axiom,
    ! [B: extended_ereal,A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ A3 )
     => ( ( inf_in2794916579150040252_ereal @ A3 @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_755_inf_Oabsorb1,axiom,
    ! [A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( inf_inf_nat @ A3 @ B )
        = A3 ) ) ).

% inf.absorb1
thf(fact_756_inf_Oabsorb1,axiom,
    ! [A3: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( inf_inf_real @ A3 @ B )
        = A3 ) ) ).

% inf.absorb1
thf(fact_757_inf_Oabsorb1,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( inf_in7439215052339218890nnreal @ A3 @ B )
        = A3 ) ) ).

% inf.absorb1
thf(fact_758_inf_Oabsorb1,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( inf_in2794916579150040252_ereal @ A3 @ B )
        = A3 ) ) ).

% inf.absorb1
thf(fact_759_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( inf_inf_nat @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_760_le__iff__inf,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y2: real] :
          ( ( inf_inf_real @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_761_le__iff__inf,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
          ( ( inf_in7439215052339218890nnreal @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_762_le__iff__inf,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( inf_in2794916579150040252_ereal @ X @ Y2 )
          = X ) ) ) ).

% le_iff_inf
thf(fact_763_inf__unique,axiom,
    ! [F: nat > nat > nat,X2: nat,Y: nat] :
      ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ( ord_less_eq_nat @ X4 @ Z3 )
               => ( ord_less_eq_nat @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_764_inf__unique,axiom,
    ! [F: real > real > real,X2: real,Y: real] :
      ( ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: real,Y3: real] : ( ord_less_eq_real @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: real,Y3: real,Z3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ( ord_less_eq_real @ X4 @ Z3 )
               => ( ord_less_eq_real @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_inf_real @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_765_inf__unique,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal > extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ( ord_le3935885782089961368nnreal @ X4 @ Z3 )
               => ( ord_le3935885782089961368nnreal @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in7439215052339218890nnreal @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_766_inf__unique,axiom,
    ! [F: extended_ereal > extended_ereal > extended_ereal,X2: extended_ereal,Y: extended_ereal] :
      ( ! [X4: extended_ereal,Y3: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F @ X4 @ Y3 ) @ X4 )
     => ( ! [X4: extended_ereal,Y3: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( F @ X4 @ Y3 ) @ Y3 )
       => ( ! [X4: extended_ereal,Y3: extended_ereal,Z3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ( ord_le1083603963089353582_ereal @ X4 @ Z3 )
               => ( ord_le1083603963089353582_ereal @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
         => ( ( inf_in2794916579150040252_ereal @ X2 @ Y )
            = ( F @ X2 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_767_inf_OorderI,axiom,
    ! [A3: nat,B: nat] :
      ( ( A3
        = ( inf_inf_nat @ A3 @ B ) )
     => ( ord_less_eq_nat @ A3 @ B ) ) ).

% inf.orderI
thf(fact_768_inf_OorderI,axiom,
    ! [A3: real,B: real] :
      ( ( A3
        = ( inf_inf_real @ A3 @ B ) )
     => ( ord_less_eq_real @ A3 @ B ) ) ).

% inf.orderI
thf(fact_769_inf_OorderI,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( A3
        = ( inf_in7439215052339218890nnreal @ A3 @ B ) )
     => ( ord_le3935885782089961368nnreal @ A3 @ B ) ) ).

% inf.orderI
thf(fact_770_inf_OorderI,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( A3
        = ( inf_in2794916579150040252_ereal @ A3 @ B ) )
     => ( ord_le1083603963089353582_ereal @ A3 @ B ) ) ).

% inf.orderI
thf(fact_771_inf_OorderE,axiom,
    ! [A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( A3
        = ( inf_inf_nat @ A3 @ B ) ) ) ).

% inf.orderE
thf(fact_772_inf_OorderE,axiom,
    ! [A3: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( A3
        = ( inf_inf_real @ A3 @ B ) ) ) ).

% inf.orderE
thf(fact_773_inf_OorderE,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( A3
        = ( inf_in7439215052339218890nnreal @ A3 @ B ) ) ) ).

% inf.orderE
thf(fact_774_inf_OorderE,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( A3
        = ( inf_in2794916579150040252_ereal @ A3 @ B ) ) ) ).

% inf.orderE
thf(fact_775_le__infI2,axiom,
    ! [B: nat,X2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ X2 ) ) ).

% le_infI2
thf(fact_776_le__infI2,axiom,
    ! [B: real,X2: real,A3: real] :
      ( ( ord_less_eq_real @ B @ X2 )
     => ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ X2 ) ) ).

% le_infI2
thf(fact_777_le__infI2,axiom,
    ! [B: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ X2 )
     => ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ X2 ) ) ).

% le_infI2
thf(fact_778_le__infI2,axiom,
    ! [B: extended_ereal,X2: extended_ereal,A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ X2 )
     => ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ X2 ) ) ).

% le_infI2
thf(fact_779_le__infI1,axiom,
    ! [A3: nat,X2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ X2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ X2 ) ) ).

% le_infI1
thf(fact_780_le__infI1,axiom,
    ! [A3: real,X2: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ X2 )
     => ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ X2 ) ) ).

% le_infI1
thf(fact_781_le__infI1,axiom,
    ! [A3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ X2 )
     => ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ X2 ) ) ).

% le_infI1
thf(fact_782_le__infI1,axiom,
    ! [A3: extended_ereal,X2: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ X2 )
     => ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ X2 ) ) ).

% le_infI1
thf(fact_783_inf__mono,axiom,
    ! [A3: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A3 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_784_inf__mono,axiom,
    ! [A3: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ C )
     => ( ( ord_less_eq_real @ B @ D )
       => ( ord_less_eq_real @ ( inf_inf_real @ A3 @ B ) @ ( inf_inf_real @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_785_inf__mono,axiom,
    ! [A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ C )
     => ( ( ord_le3935885782089961368nnreal @ B @ D )
       => ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ A3 @ B ) @ ( inf_in7439215052339218890nnreal @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_786_inf__mono,axiom,
    ! [A3: extended_ereal,C: extended_ereal,B: extended_ereal,D: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ C )
     => ( ( ord_le1083603963089353582_ereal @ B @ D )
       => ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ A3 @ B ) @ ( inf_in2794916579150040252_ereal @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_787_le__infI,axiom,
    ! [X2: nat,A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ X2 @ A3 )
     => ( ( ord_less_eq_nat @ X2 @ B )
       => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A3 @ B ) ) ) ) ).

% le_infI
thf(fact_788_le__infI,axiom,
    ! [X2: real,A3: real,B: real] :
      ( ( ord_less_eq_real @ X2 @ A3 )
     => ( ( ord_less_eq_real @ X2 @ B )
       => ( ord_less_eq_real @ X2 @ ( inf_inf_real @ A3 @ B ) ) ) ) ).

% le_infI
thf(fact_789_le__infI,axiom,
    ! [X2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ A3 )
     => ( ( ord_le3935885782089961368nnreal @ X2 @ B )
       => ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ A3 @ B ) ) ) ) ).

% le_infI
thf(fact_790_le__infI,axiom,
    ! [X2: extended_ereal,A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ X2 @ B )
       => ( ord_le1083603963089353582_ereal @ X2 @ ( inf_in2794916579150040252_ereal @ A3 @ B ) ) ) ) ).

% le_infI
thf(fact_791_le__infE,axiom,
    ! [X2: nat,A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A3 @ B ) )
     => ~ ( ( ord_less_eq_nat @ X2 @ A3 )
         => ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).

% le_infE
thf(fact_792_le__infE,axiom,
    ! [X2: real,A3: real,B: real] :
      ( ( ord_less_eq_real @ X2 @ ( inf_inf_real @ A3 @ B ) )
     => ~ ( ( ord_less_eq_real @ X2 @ A3 )
         => ~ ( ord_less_eq_real @ X2 @ B ) ) ) ).

% le_infE
thf(fact_793_le__infE,axiom,
    ! [X2: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ ( inf_in7439215052339218890nnreal @ A3 @ B ) )
     => ~ ( ( ord_le3935885782089961368nnreal @ X2 @ A3 )
         => ~ ( ord_le3935885782089961368nnreal @ X2 @ B ) ) ) ).

% le_infE
thf(fact_794_le__infE,axiom,
    ! [X2: extended_ereal,A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ ( inf_in2794916579150040252_ereal @ A3 @ B ) )
     => ~ ( ( ord_le1083603963089353582_ereal @ X2 @ A3 )
         => ~ ( ord_le1083603963089353582_ereal @ X2 @ B ) ) ) ).

% le_infE
thf(fact_795_inf__le2,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_796_inf__le2,axiom,
    ! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_797_inf__le2,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_798_inf__le2,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X2 @ Y ) @ Y ) ).

% inf_le2
thf(fact_799_inf__le1,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_800_inf__le1,axiom,
    ! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_801_inf__le1,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_802_inf__le1,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X2 @ Y ) @ X2 ) ).

% inf_le1
thf(fact_803_inf__sup__ord_I1_J,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_804_inf__sup__ord_I1_J,axiom,
    ! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_805_inf__sup__ord_I1_J,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_806_inf__sup__ord_I1_J,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X2 @ Y ) @ X2 ) ).

% inf_sup_ord(1)
thf(fact_807_inf__sup__ord_I2_J,axiom,
    ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_808_inf__sup__ord_I2_J,axiom,
    ! [X2: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_809_inf__sup__ord_I2_J,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( inf_in7439215052339218890nnreal @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_810_inf__sup__ord_I2_J,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] : ( ord_le1083603963089353582_ereal @ ( inf_in2794916579150040252_ereal @ X2 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_811_top__prod__def,axiom,
    ( top_to1606914339385671969nnreal
    = ( produc344325839068023049nnreal @ top_to1496364449551166952nnreal @ top_to1496364449551166952nnreal ) ) ).

% top_prod_def
thf(fact_812_Int__Collect__mono,axiom,
    ! [A: set_Pr3917002265660371076l_real,B2: set_Pr3917002265660371076l_real,P: ( produc7929879615859924995b_real > real ) > $o,Q: ( produc7929879615859924995b_real > real ) > $o] :
      ( ( ord_le8308227983669428452l_real @ A @ B2 )
     => ( ! [X4: produc7929879615859924995b_real > real] :
            ( ( member5467075257862553189l_real @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le8308227983669428452l_real @ ( inf_in3822636991831349782l_real @ A @ ( collec869437573433150115l_real @ P ) ) @ ( inf_in3822636991831349782l_real @ B2 @ ( collec869437573433150115l_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_813_Int__Collect__mono,axiom,
    ! [A: set_set_real,B2: set_set_real,P: set_real > $o,Q: set_real > $o] :
      ( ( ord_le3558479182127378552t_real @ A @ B2 )
     => ( ! [X4: set_real] :
            ( ( member_set_real @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le3558479182127378552t_real @ ( inf_inf_set_set_real @ A @ ( collect_set_real @ P ) ) @ ( inf_inf_set_set_real @ B2 @ ( collect_set_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_814_Int__Collect__mono,axiom,
    ! [A: set_real,B2: set_real,P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ A @ B2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_815_Int__Collect__mono,axiom,
    ! [A: set_Pr3674455588107860082_a_b_b,B2: set_Pr3674455588107860082_a_b_b,P: ( product_prod_a_b > b ) > $o,Q: ( product_prod_a_b > b ) > $o] :
      ( ( ord_le8439411419596101842_a_b_b @ A @ B2 )
     => ( ! [X4: product_prod_a_b > b] :
            ( ( member1080921170058542931_a_b_b @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le8439411419596101842_a_b_b @ ( inf_in1278836612478006532_a_b_b @ A @ ( collec2250730345169186961_a_b_b @ P ) ) @ ( inf_in1278836612478006532_a_b_b @ B2 @ ( collec2250730345169186961_a_b_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_816_Int__Collect__mono,axiom,
    ! [A: set_real_real,B2: set_real_real,P: ( real > real ) > $o,Q: ( real > real ) > $o] :
      ( ( ord_le4198349162570665613l_real @ A @ B2 )
     => ( ! [X4: real > real] :
            ( ( member_real_real @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le4198349162570665613l_real @ ( inf_in5166753670444608447l_real @ A @ ( collect_real_real @ P ) ) @ ( inf_in5166753670444608447l_real @ B2 @ ( collect_real_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_817_Int__Collect__mono,axiom,
    ! [A: set_Pr1027596265373204187b_real,B2: set_Pr1027596265373204187b_real,P: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o,Q: ( produc7929879615859924995b_real > produc7929879615859924995b_real ) > $o] :
      ( ( ord_le2582152453551361851b_real @ A @ B2 )
     => ( ! [X4: produc7929879615859924995b_real > produc7929879615859924995b_real] :
            ( ( member1118978200981565500b_real @ X4 @ A )
           => ( ( P @ X4 )
             => ( Q @ X4 ) ) )
       => ( ord_le2582152453551361851b_real @ ( inf_in7065115160788051053b_real @ A @ ( collec5735648035525910650b_real @ P ) ) @ ( inf_in7065115160788051053b_real @ B2 @ ( collec5735648035525910650b_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_818_borel__measurable__iff__halfspace__le,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_819_borel__measurable__iff__halfspace__le,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_820_borel__measurable__iff__halfspace__le,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_821_borel__measurable__iff__halfspace__le,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_822_borel__measurable__iff__halfspace__le,axiom,
    ! [F: set_real > produc2422161461964618553l_real,M: sigma_3733394171116455995t_real] :
      ( ( member1965154327494239031l_real @ F @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_823_borel__measurable__iff__halfspace__le,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_824_borel__measurable__iff__halfspace__le,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member2587080422514930128b_real
                @ ( collec4671965459240937304b_real
                  @ ^ [W2: produc7929879615859924995b_real] :
                      ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_825_borel__measurable__iff__halfspace__le,axiom,
    ! [F: ( real > real ) > produc2422161461964618553l_real,M: sigma_4258434043392614480l_real] :
      ( ( member4094776897846995810l_real @ F @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_826_borel__measurable__iff__halfspace__le,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member3057847817020379547l_real
                @ ( collec869437573433150115l_real
                  @ ^ [W2: produc7929879615859924995b_real > real] :
                      ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                      & ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_827_borel__measurable__iff__halfspace__le,axiom,
    ! [F: ( product_prod_a_b > b ) > produc2422161461964618553l_real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member861154087952452317l_real @ F @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_eq_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_le
thf(fact_828_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_829_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_830_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_831_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_832_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: set_real > produc2422161461964618553l_real,M: sigma_3733394171116455995t_real] :
      ( ( member1965154327494239031l_real @ F @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_833_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_834_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member2587080422514930128b_real
                @ ( collec4671965459240937304b_real
                  @ ^ [W2: produc7929879615859924995b_real] :
                      ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_835_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: ( real > real ) > produc2422161461964618553l_real,M: sigma_4258434043392614480l_real] :
      ( ( member4094776897846995810l_real @ F @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_836_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member3057847817020379547l_real
                @ ( collec869437573433150115l_real
                  @ ^ [W2: produc7929879615859924995b_real > real] :
                      ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_837_borel__measurable__iff__halfspace__ge,axiom,
    ! [F: ( product_prod_a_b > b ) > produc2422161461964618553l_real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member861154087952452317l_real @ F @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_eq_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_ge
thf(fact_838_less__eq__prod__def,axiom,
    ( ord_le8460144461188290721at_nat
    = ( ^ [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
          ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X ) @ ( product_fst_nat_nat @ Y2 ) )
          & ( ord_less_eq_nat @ ( product_snd_nat_nat @ X ) @ ( product_snd_nat_nat @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_839_less__eq__prod__def,axiom,
    ( ord_le8710666929947597437t_real
    = ( ^ [X: produc7716430852924023517t_real,Y2: produc7716430852924023517t_real] :
          ( ( ord_less_eq_nat @ ( product_fst_nat_real @ X ) @ ( product_fst_nat_real @ Y2 ) )
          & ( ord_less_eq_real @ ( product_snd_nat_real @ X ) @ ( product_snd_nat_real @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_840_less__eq__prod__def,axiom,
    ( ord_le8638621733772292105nnreal
    = ( ^ [X: produc5192943231052834921nnreal,Y2: produc5192943231052834921nnreal] :
          ( ( ord_less_eq_nat @ ( produc3274940128995334541nnreal @ X ) @ ( produc3274940128995334541nnreal @ Y2 ) )
          & ( ord_le3935885782089961368nnreal @ ( produc940077783810965967nnreal @ X ) @ ( produc940077783810965967nnreal @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_841_less__eq__prod__def,axiom,
    ( ord_le4672251659283004883_ereal
    = ( ^ [X: produc7896515489273245043_ereal,Y2: produc7896515489273245043_ereal] :
          ( ( ord_less_eq_nat @ ( produc405286064046379065_ereal @ X ) @ ( produc405286064046379065_ereal @ Y2 ) )
          & ( ord_le1083603963089353582_ereal @ ( produc2020276525179383671_ereal @ X ) @ ( produc2020276525179383671_ereal @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_842_less__eq__prod__def,axiom,
    ( ord_le4735619238470717181al_nat
    = ( ^ [X: produc3741383161447143261al_nat,Y2: produc3741383161447143261al_nat] :
          ( ( ord_less_eq_real @ ( product_fst_real_nat @ X ) @ ( product_fst_real_nat @ Y2 ) )
          & ( ord_less_eq_nat @ ( product_snd_real_nat @ X ) @ ( product_snd_real_nat @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_843_less__eq__prod__def,axiom,
    ( ord_le1075799226346578649l_real
    = ( ^ [X: produc2422161461964618553l_real,Y2: produc2422161461964618553l_real] :
          ( ( ord_less_eq_real @ ( produc5828954698716094813l_real @ X ) @ ( produc5828954698716094813l_real @ Y2 ) )
          & ( ord_less_eq_real @ ( produc3484788084999411615l_real @ X ) @ ( produc3484788084999411615l_real @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_844_less__eq__prod__def,axiom,
    ( ord_le4096773168995780197nnreal
    = ( ^ [X: produc1565745902094061253nnreal,Y2: produc1565745902094061253nnreal] :
          ( ( ord_less_eq_real @ ( produc1135137091964385897nnreal @ X ) @ ( produc1135137091964385897nnreal @ Y2 ) )
          & ( ord_le3935885782089961368nnreal @ ( produc4591475017210782123nnreal @ X ) @ ( produc4591475017210782123nnreal @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_845_less__eq__prod__def,axiom,
    ( ord_le6751365068476080247_ereal
    = ( ^ [X: produc1986409236987838487_ereal,Y2: produc1986409236987838487_ereal] :
          ( ( ord_less_eq_real @ ( produc5152095971134359645_ereal @ X ) @ ( produc5152095971134359645_ereal @ Y2 ) )
          & ( ord_le1083603963089353582_ereal @ ( produc4749583790187325595_ereal @ X ) @ ( produc4749583790187325595_ereal @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_846_less__eq__prod__def,axiom,
    ( ord_le6792163879939894281al_nat
    = ( ^ [X: produc3346485377220437097al_nat,Y2: produc3346485377220437097al_nat] :
          ( ( ord_le3935885782089961368nnreal @ ( produc8048640569662194061al_nat @ X ) @ ( produc8048640569662194061al_nat @ Y2 ) )
          & ( ord_less_eq_nat @ ( produc5713778224477825487al_nat @ X ) @ ( produc5713778224477825487al_nat @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_847_less__eq__prod__def,axiom,
    ( ord_le4051224869651757541l_real
    = ( ^ [X: produc1520197602750038597l_real,Y2: produc1520197602750038597l_real] :
          ( ( ord_le3935885782089961368nnreal @ ( produc8390762859368617449l_real @ X ) @ ( produc8390762859368617449l_real @ Y2 ) )
          & ( ord_less_eq_real @ ( produc2623728747760237867l_real @ X ) @ ( produc2623728747760237867l_real @ Y2 ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_848_measurable__mono,axiom,
    ! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ N2 ) )
       => ( ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ M ) @ ( sigma_5719578828308484196b_real @ M4 ) )
         => ( ( ( sigma_709238742955535483b_real @ M )
              = ( sigma_709238742955535483b_real @ M4 ) )
           => ( ord_le8308227983669428452l_real @ ( sigma_2402725360379274625l_real @ M @ N ) @ ( sigma_2402725360379274625l_real @ M4 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_849_measurable__mono,axiom,
    ! [N2: sigma_measure_b,N: sigma_measure_b,M: sigma_5782782616773303333od_a_b,M4: sigma_5782782616773303333od_a_b] :
      ( ( ord_le3795704787696855135_set_b @ ( sigma_sets_b @ N2 ) @ ( sigma_sets_b @ N ) )
     => ( ( ( sigma_space_b @ N )
          = ( sigma_space_b @ N2 ) )
       => ( ( ord_le8697249284497989384od_a_b @ ( sigma_377313353287227655od_a_b @ M ) @ ( sigma_377313353287227655od_a_b @ M4 ) )
         => ( ( ( sigma_1564438922714704432od_a_b @ M )
              = ( sigma_1564438922714704432od_a_b @ M4 ) )
           => ( ord_le8439411419596101842_a_b_b @ ( sigma_7225552367035416487_a_b_b @ M @ N ) @ ( sigma_7225552367035416487_a_b_b @ M4 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_850_measurable__mono,axiom,
    ! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_measure_real,M4: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ N2 ) )
       => ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M4 ) )
         => ( ( ( sigma_space_real @ M )
              = ( sigma_space_real @ M4 ) )
           => ( ord_le4198349162570665613l_real @ ( sigma_5267869275261027754l_real @ M @ N ) @ ( sigma_5267869275261027754l_real @ M4 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_851_measurable__mono,axiom,
    ! [N2: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,M: sigma_4174442854635832540b_real,M4: sigma_4174442854635832540b_real] :
      ( ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ N2 ) @ ( sigma_5719578828308484196b_real @ N ) )
     => ( ( ( sigma_709238742955535483b_real @ N )
          = ( sigma_709238742955535483b_real @ N2 ) )
       => ( ( ord_le2707404280535930575b_real @ ( sigma_5719578828308484196b_real @ M ) @ ( sigma_5719578828308484196b_real @ M4 ) )
         => ( ( ( sigma_709238742955535483b_real @ M )
              = ( sigma_709238742955535483b_real @ M4 ) )
           => ( ord_le2582152453551361851b_real @ ( sigma_1108951568718238936b_real @ M @ N ) @ ( sigma_1108951568718238936b_real @ M4 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_852_inner__Basis__inf__left,axiom,
    ! [I: real,X2: real,Y: real] :
      ( ( member_real @ I @ euclid1305858884100475807s_real )
     => ( ( inner_5569486634795291694r_real @ ( inf_inf_real @ X2 @ Y ) @ I )
        = ( inf_inf_real @ ( inner_5569486634795291694r_real @ X2 @ I ) @ ( inner_5569486634795291694r_real @ Y @ I ) ) ) ) ).

% inner_Basis_inf_left
thf(fact_853_le__measureD2,axiom,
    ! [A: sigma_measure_real,B2: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B2 )
     => ( ( ( sigma_space_real @ A )
          = ( sigma_space_real @ B2 ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B2 ) ) ) ) ).

% le_measureD2
thf(fact_854_eucl__le,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y2: real] :
        ! [Z4: real] :
          ( ( member_real @ Z4 @ euclid1305858884100475807s_real )
         => ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ X @ Z4 ) @ ( inner_5569486634795291694r_real @ Y2 @ Z4 ) ) ) ) ) ).

% eucl_le
thf(fact_855_inner__Basis__mono,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( member_real @ C @ euclid1305858884100475807s_real )
       => ( ord_less_eq_real @ ( inner_5569486634795291694r_real @ A3 @ C ) @ ( inner_5569486634795291694r_real @ B @ C ) ) ) ) ).

% inner_Basis_mono
thf(fact_856_iso__tuple__UNIV__I,axiom,
    ! [X2: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X2 @ top_to5286755836160161588l_real ) ).

% iso_tuple_UNIV_I
thf(fact_857_iso__tuple__UNIV__I,axiom,
    ! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_858_iso__tuple__UNIV__I,axiom,
    ! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_859_iso__tuple__UNIV__I,axiom,
    ! [X2: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X2 @ top_to8883559732078315298_a_b_b ) ).

% iso_tuple_UNIV_I
thf(fact_860_iso__tuple__UNIV__I,axiom,
    ! [X2: real > real] : ( member_real_real @ X2 @ top_to2071711978144146653l_real ) ).

% iso_tuple_UNIV_I
thf(fact_861_iso__tuple__UNIV__I,axiom,
    ! [X2: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X2 @ top_to1564948194967201163b_real ) ).

% iso_tuple_UNIV_I
thf(fact_862_top__empty__eq,axiom,
    ( top_to7996754997647394897real_o
    = ( ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ top_to5286755836160161588l_real ) ) ) ).

% top_empty_eq
thf(fact_863_top__empty__eq,axiom,
    ( top_top_set_real_o
    = ( ^ [X: set_real] : ( member_set_real @ X @ top_top_set_set_real ) ) ) ).

% top_empty_eq
thf(fact_864_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X: real] : ( member_real @ X @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_865_top__empty__eq,axiom,
    ( top_to1923726438858641379_b_b_o
    = ( ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ top_to8883559732078315298_a_b_b ) ) ) ).

% top_empty_eq
thf(fact_866_top__empty__eq,axiom,
    ( top_top_real_real_o
    = ( ^ [X: real > real] : ( member_real_real @ X @ top_to2071711978144146653l_real ) ) ) ).

% top_empty_eq
thf(fact_867_top__empty__eq,axiom,
    ( top_to7812595598144044730real_o
    = ( ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ top_to1564948194967201163b_real ) ) ) ).

% top_empty_eq
thf(fact_868_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_real,B2: sigma_measure_real,C2: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B2 )
     => ( ( ord_le487379304121309861e_real @ B2 @ C2 )
       => ( ( ( sigma_sets_real @ A )
            = ( sigma_sets_real @ C2 ) )
         => ( ( sigma_sets_real @ B2 )
            = ( sigma_sets_real @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_869_pred__subset__eq,axiom,
    ! [R: set_Pr3917002265660371076l_real,S2: set_Pr3917002265660371076l_real] :
      ( ( ord_le8040096449130281633real_o
        @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ R )
        @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ S2 ) )
      = ( ord_le8308227983669428452l_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_870_pred__subset__eq,axiom,
    ! [R: set_set_real,S2: set_set_real] :
      ( ( ord_le2392157289819280397real_o
        @ ^ [X: set_real] : ( member_set_real @ X @ R )
        @ ^ [X: set_real] : ( member_set_real @ X @ S2 ) )
      = ( ord_le3558479182127378552t_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_871_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S2 ) )
      = ( ord_less_eq_set_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_872_pred__subset__eq,axiom,
    ! [R: set_Pr3674455588107860082_a_b_b,S2: set_Pr3674455588107860082_a_b_b] :
      ( ( ord_le3099313929993222707_b_b_o
        @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ R )
        @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ S2 ) )
      = ( ord_le8439411419596101842_a_b_b @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_873_pred__subset__eq,axiom,
    ! [R: set_real_real,S2: set_real_real] :
      ( ( ord_le5273791883478943800real_o
        @ ^ [X: real > real] : ( member_real_real @ X @ R )
        @ ^ [X: real > real] : ( member_real_real @ X @ S2 ) )
      = ( ord_le4198349162570665613l_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_874_pred__subset__eq,axiom,
    ! [R: set_Pr1027596265373204187b_real,S2: set_Pr1027596265373204187b_real] :
      ( ( ord_le5987956920600383754real_o
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ R )
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ S2 ) )
      = ( ord_le2582152453551361851b_real @ R @ S2 ) ) ).

% pred_subset_eq
thf(fact_875_inf__Int__eq,axiom,
    ! [R: set_Pr3917002265660371076l_real,S2: set_Pr3917002265660371076l_real] :
      ( ( inf_in7897423780479368175real_o
        @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ R )
        @ ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ S2 ) )
      = ( ^ [X: produc7929879615859924995b_real > real] : ( member5467075257862553189l_real @ X @ ( inf_in3822636991831349782l_real @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_876_inf__Int__eq,axiom,
    ! [R: set_set_real,S2: set_set_real] :
      ( ( inf_inf_set_real_o
        @ ^ [X: set_real] : ( member_set_real @ X @ R )
        @ ^ [X: set_real] : ( member_set_real @ X @ S2 ) )
      = ( ^ [X: set_real] : ( member_set_real @ X @ ( inf_inf_set_set_real @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_877_inf__Int__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ( inf_inf_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S2 ) )
      = ( ^ [X: real] : ( member_real @ X @ ( inf_inf_set_real @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_878_inf__Int__eq,axiom,
    ! [R: set_Pr3674455588107860082_a_b_b,S2: set_Pr3674455588107860082_a_b_b] :
      ( ( inf_in9192623926311598209_b_b_o
        @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ R )
        @ ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ S2 ) )
      = ( ^ [X: product_prod_a_b > b] : ( member1080921170058542931_a_b_b @ X @ ( inf_in1278836612478006532_a_b_b @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_879_inf__Int__eq,axiom,
    ! [R: set_real_real,S2: set_real_real] :
      ( ( inf_inf_real_real_o
        @ ^ [X: real > real] : ( member_real_real @ X @ R )
        @ ^ [X: real > real] : ( member_real_real @ X @ S2 ) )
      = ( ^ [X: real > real] : ( member_real_real @ X @ ( inf_in5166753670444608447l_real @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_880_inf__Int__eq,axiom,
    ! [R: set_Pr1027596265373204187b_real,S2: set_Pr1027596265373204187b_real] :
      ( ( inf_in2390107153597142616real_o
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ R )
        @ ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ S2 ) )
      = ( ^ [X: produc7929879615859924995b_real > produc7929879615859924995b_real] : ( member1118978200981565500b_real @ X @ ( inf_in7065115160788051053b_real @ R @ S2 ) ) ) ) ).

% inf_Int_eq
thf(fact_881_dual__order_Orefl,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_882_dual__order_Orefl,axiom,
    ! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_883_dual__order_Orefl,axiom,
    ! [A3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_884_dual__order_Orefl,axiom,
    ! [A3: extended_ereal] : ( ord_le1083603963089353582_ereal @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_885_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_886_order__refl,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).

% order_refl
thf(fact_887_order__refl,axiom,
    ! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ X2 ) ).

% order_refl
thf(fact_888_order__refl,axiom,
    ! [X2: extended_ereal] : ( ord_le1083603963089353582_ereal @ X2 @ X2 ) ).

% order_refl
thf(fact_889_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_890_order__antisym__conv,axiom,
    ! [Y: real,X2: real] :
      ( ( ord_less_eq_real @ Y @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_891_order__antisym__conv,axiom,
    ! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
     => ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_892_order__antisym__conv,axiom,
    ! [Y: extended_ereal,X2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ Y @ X2 )
     => ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_893_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_894_linorder__le__cases,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_eq_real @ X2 @ Y )
     => ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_895_linorder__le__cases,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_896_linorder__le__cases,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ~ ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ord_le1083603963089353582_ereal @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_897_ord__le__eq__subst,axiom,
    ! [A3: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_898_ord__le__eq__subst,axiom,
    ! [A3: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_899_ord__le__eq__subst,axiom,
    ! [A3: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_900_ord__le__eq__subst,axiom,
    ! [A3: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_901_ord__le__eq__subst,axiom,
    ! [A3: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_902_ord__le__eq__subst,axiom,
    ! [A3: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_903_ord__le__eq__subst,axiom,
    ! [A3: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_904_ord__le__eq__subst,axiom,
    ! [A3: real,B: real,F: real > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_905_ord__le__eq__subst,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_906_ord__le__eq__subst,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_907_ord__eq__le__subst,axiom,
    ! [A3: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_908_ord__eq__le__subst,axiom,
    ! [A3: real,F: nat > real,B: nat,C: nat] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_909_ord__eq__le__subst,axiom,
    ! [A3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_910_ord__eq__le__subst,axiom,
    ! [A3: extended_ereal,F: nat > extended_ereal,B: nat,C: nat] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_911_ord__eq__le__subst,axiom,
    ! [A3: nat,F: real > nat,B: real,C: real] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_912_ord__eq__le__subst,axiom,
    ! [A3: real,F: real > real,B: real,C: real] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_913_ord__eq__le__subst,axiom,
    ! [A3: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_914_ord__eq__le__subst,axiom,
    ! [A3: extended_ereal,F: real > extended_ereal,B: real,C: real] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_915_ord__eq__le__subst,axiom,
    ! [A3: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_916_ord__eq__le__subst,axiom,
    ! [A3: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A3
        = ( F @ B ) )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_917_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_918_linorder__linear,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
      | ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_919_linorder__linear,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
      | ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_920_linorder__linear,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
      | ( ord_le1083603963089353582_ereal @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_921_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_922_order__eq__refl,axiom,
    ! [X2: real,Y: real] :
      ( ( X2 = Y )
     => ( ord_less_eq_real @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_923_order__eq__refl,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( X2 = Y )
     => ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_924_order__eq__refl,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( X2 = Y )
     => ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_925_order__subst2,axiom,
    ! [A3: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_926_order__subst2,axiom,
    ! [A3: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_927_order__subst2,axiom,
    ! [A3: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_928_order__subst2,axiom,
    ! [A3: nat,B: nat,F: nat > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_929_order__subst2,axiom,
    ! [A3: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_930_order__subst2,axiom,
    ! [A3: real,B: real,F: real > real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_931_order__subst2,axiom,
    ! [A3: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_932_order__subst2,axiom,
    ! [A3: real,B: real,F: real > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_933_order__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_934_order__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_935_order__subst1,axiom,
    ! [A3: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_936_order__subst1,axiom,
    ! [A3: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_937_order__subst1,axiom,
    ! [A3: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_938_order__subst1,axiom,
    ! [A3: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_le1083603963089353582_ereal @ B @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_939_order__subst1,axiom,
    ! [A3: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_940_order__subst1,axiom,
    ! [A3: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_941_order__subst1,axiom,
    ! [A3: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_942_order__subst1,axiom,
    ! [A3: real,F: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_le1083603963089353582_ereal @ B @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_943_order__subst1,axiom,
    ! [A3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_944_order__subst1,axiom,
    ! [A3: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( F @ B ) )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le3935885782089961368nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_945_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [A4: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A4 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_946_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
    = ( ^ [A4: real,B5: real] :
          ( ( ord_less_eq_real @ A4 @ B5 )
          & ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_947_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A4 @ B5 )
          & ( ord_le3935885782089961368nnreal @ B5 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_948_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z ) )
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A4 @ B5 )
          & ( ord_le1083603963089353582_ereal @ B5 @ A4 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_949_antisym,axiom,
    ! [A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_eq_nat @ B @ A3 )
       => ( A3 = B ) ) ) ).

% antisym
thf(fact_950_antisym,axiom,
    ! [A3: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_eq_real @ B @ A3 )
       => ( A3 = B ) ) ) ).

% antisym
thf(fact_951_antisym,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ B @ A3 )
       => ( A3 = B ) ) ) ).

% antisym
thf(fact_952_antisym,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ B @ A3 )
       => ( A3 = B ) ) ) ).

% antisym
thf(fact_953_dual__order_Otrans,axiom,
    ! [B: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A3 )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_954_dual__order_Otrans,axiom,
    ! [B: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B @ A3 )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_955_dual__order_Otrans,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A3 )
     => ( ( ord_le3935885782089961368nnreal @ C @ B )
       => ( ord_le3935885782089961368nnreal @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_956_dual__order_Otrans,axiom,
    ! [B: extended_ereal,A3: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ C @ B )
       => ( ord_le1083603963089353582_ereal @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_957_dual__order_Oantisym,axiom,
    ! [B: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ B )
       => ( A3 = B ) ) ) ).

% dual_order.antisym
thf(fact_958_dual__order_Oantisym,axiom,
    ! [B: real,A3: real] :
      ( ( ord_less_eq_real @ B @ A3 )
     => ( ( ord_less_eq_real @ A3 @ B )
       => ( A3 = B ) ) ) ).

% dual_order.antisym
thf(fact_959_dual__order_Oantisym,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A3 )
     => ( ( ord_le3935885782089961368nnreal @ A3 @ B )
       => ( A3 = B ) ) ) ).

% dual_order.antisym
thf(fact_960_dual__order_Oantisym,axiom,
    ! [B: extended_ereal,A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ A3 @ B )
       => ( A3 = B ) ) ) ).

% dual_order.antisym
thf(fact_961_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [A4: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A4 )
          & ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_962_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
    = ( ^ [A4: real,B5: real] :
          ( ( ord_less_eq_real @ B5 @ A4 )
          & ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_963_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ B5 @ A4 )
          & ( ord_le3935885782089961368nnreal @ A4 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_964_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z ) )
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ B5 @ A4 )
          & ( ord_le1083603963089353582_ereal @ A4 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_965_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A3: nat,B: nat] :
      ( ! [A5: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: nat,B3: nat] :
            ( ( P @ B3 @ A5 )
           => ( P @ A5 @ B3 ) )
       => ( P @ A3 @ B ) ) ) ).

% linorder_wlog
thf(fact_966_linorder__wlog,axiom,
    ! [P: real > real > $o,A3: real,B: real] :
      ( ! [A5: real,B3: real] :
          ( ( ord_less_eq_real @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: real,B3: real] :
            ( ( P @ B3 @ A5 )
           => ( P @ A5 @ B3 ) )
       => ( P @ A3 @ B ) ) ) ).

% linorder_wlog
thf(fact_967_linorder__wlog,axiom,
    ! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
            ( ( P @ B3 @ A5 )
           => ( P @ A5 @ B3 ) )
       => ( P @ A3 @ B ) ) ) ).

% linorder_wlog
thf(fact_968_linorder__wlog,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A3: extended_ereal,B: extended_ereal] :
      ( ! [A5: extended_ereal,B3: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
         => ( P @ A5 @ B3 ) )
     => ( ! [A5: extended_ereal,B3: extended_ereal] :
            ( ( P @ B3 @ A5 )
           => ( P @ A5 @ B3 ) )
       => ( P @ A3 @ B ) ) ) ).

% linorder_wlog
thf(fact_969_order__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_970_order__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_971_order__trans,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
       => ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_972_order__trans,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( ord_le1083603963089353582_ereal @ Y @ Z2 )
       => ( ord_le1083603963089353582_ereal @ X2 @ Z2 ) ) ) ).

% order_trans
thf(fact_973_order_Otrans,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% order.trans
thf(fact_974_order_Otrans,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% order.trans
thf(fact_975_order_Otrans,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ord_le3935885782089961368nnreal @ A3 @ C ) ) ) ).

% order.trans
thf(fact_976_order_Otrans,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ B @ C )
       => ( ord_le1083603963089353582_ereal @ A3 @ C ) ) ) ).

% order.trans
thf(fact_977_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_978_order__antisym,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_979_order__antisym,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_980_order__antisym,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( ord_le1083603963089353582_ereal @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_981_ord__le__eq__trans,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_982_ord__le__eq__trans,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( B = C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_983_ord__le__eq__trans,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( B = C )
       => ( ord_le3935885782089961368nnreal @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_984_ord__le__eq__trans,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( B = C )
       => ( ord_le1083603963089353582_ereal @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_985_ord__eq__le__trans,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( A3 = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_986_ord__eq__le__trans,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( A3 = B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_987_ord__eq__le__trans,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( A3 = B )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ord_le3935885782089961368nnreal @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_988_ord__eq__le__trans,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( A3 = B )
     => ( ( ord_le1083603963089353582_ereal @ B @ C )
       => ( ord_le1083603963089353582_ereal @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_989_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_990_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
    = ( ^ [X: real,Y2: real] :
          ( ( ord_less_eq_real @ X @ Y2 )
          & ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_991_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
    = ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ X @ Y2 )
          & ( ord_le3935885782089961368nnreal @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_992_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: extended_ereal,Z: extended_ereal] : ( Y4 = Z ) )
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ X @ Y2 )
          & ( ord_le1083603963089353582_ereal @ Y2 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_993_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_994_le__cases3,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ( ord_less_eq_real @ X2 @ Y )
       => ~ ( ord_less_eq_real @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_real @ Y @ X2 )
         => ~ ( ord_less_eq_real @ X2 @ Z2 ) )
       => ( ( ( ord_less_eq_real @ X2 @ Z2 )
           => ~ ( ord_less_eq_real @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_real @ Z2 @ Y )
             => ~ ( ord_less_eq_real @ Y @ X2 ) )
           => ( ( ( ord_less_eq_real @ Y @ Z2 )
               => ~ ( ord_less_eq_real @ Z2 @ X2 ) )
             => ~ ( ( ord_less_eq_real @ Z2 @ X2 )
                 => ~ ( ord_less_eq_real @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_995_le__cases3,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
       => ~ ( ord_le3935885782089961368nnreal @ Y @ Z2 ) )
     => ( ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
         => ~ ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) )
       => ( ( ( ord_le3935885782089961368nnreal @ X2 @ Z2 )
           => ~ ( ord_le3935885782089961368nnreal @ Z2 @ Y ) )
         => ( ( ( ord_le3935885782089961368nnreal @ Z2 @ Y )
             => ~ ( ord_le3935885782089961368nnreal @ Y @ X2 ) )
           => ( ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
               => ~ ( ord_le3935885782089961368nnreal @ Z2 @ X2 ) )
             => ~ ( ( ord_le3935885782089961368nnreal @ Z2 @ X2 )
                 => ~ ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_996_le__cases3,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
       => ~ ( ord_le1083603963089353582_ereal @ Y @ Z2 ) )
     => ( ( ( ord_le1083603963089353582_ereal @ Y @ X2 )
         => ~ ( ord_le1083603963089353582_ereal @ X2 @ Z2 ) )
       => ( ( ( ord_le1083603963089353582_ereal @ X2 @ Z2 )
           => ~ ( ord_le1083603963089353582_ereal @ Z2 @ Y ) )
         => ( ( ( ord_le1083603963089353582_ereal @ Z2 @ Y )
             => ~ ( ord_le1083603963089353582_ereal @ Y @ X2 ) )
           => ( ( ( ord_le1083603963089353582_ereal @ Y @ Z2 )
               => ~ ( ord_le1083603963089353582_ereal @ Z2 @ X2 ) )
             => ~ ( ( ord_le1083603963089353582_ereal @ Z2 @ X2 )
                 => ~ ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_997_nle__le,axiom,
    ! [A3: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A3 @ B ) )
      = ( ( ord_less_eq_nat @ B @ A3 )
        & ( B != A3 ) ) ) ).

% nle_le
thf(fact_998_nle__le,axiom,
    ! [A3: real,B: real] :
      ( ( ~ ( ord_less_eq_real @ A3 @ B ) )
      = ( ( ord_less_eq_real @ B @ A3 )
        & ( B != A3 ) ) ) ).

% nle_le
thf(fact_999_nle__le,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le3935885782089961368nnreal @ A3 @ B ) )
      = ( ( ord_le3935885782089961368nnreal @ B @ A3 )
        & ( B != A3 ) ) ) ).

% nle_le
thf(fact_1000_nle__le,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ~ ( ord_le1083603963089353582_ereal @ A3 @ B ) )
      = ( ( ord_le1083603963089353582_ereal @ B @ A3 )
        & ( B != A3 ) ) ) ).

% nle_le
thf(fact_1001_top_Oextremum__uniqueI,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A3 )
     => ( A3 = top_to1496364449551166952nnreal ) ) ).

% top.extremum_uniqueI
thf(fact_1002_top_Oextremum__uniqueI,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ top_to6662034908053899550_ereal @ A3 )
     => ( A3 = top_to6662034908053899550_ereal ) ) ).

% top.extremum_uniqueI
thf(fact_1003_top_Oextremum__unique,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A3 )
      = ( A3 = top_to1496364449551166952nnreal ) ) ).

% top.extremum_unique
thf(fact_1004_top_Oextremum__unique,axiom,
    ! [A3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ top_to6662034908053899550_ereal @ A3 )
      = ( A3 = top_to6662034908053899550_ereal ) ) ).

% top.extremum_unique
thf(fact_1005_top__greatest,axiom,
    ! [A3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A3 @ top_to1496364449551166952nnreal ) ).

% top_greatest
thf(fact_1006_top__greatest,axiom,
    ! [A3: extended_ereal] : ( ord_le1083603963089353582_ereal @ A3 @ top_to6662034908053899550_ereal ) ).

% top_greatest
thf(fact_1007_increasingD,axiom,
    ! [M: set_set_real,F: set_real > nat,X2: set_real,Y: set_real] :
      ( ( measur1249124728302858545al_nat @ M @ F )
     => ( ( ord_less_eq_set_real @ X2 @ Y )
       => ( ( member_set_real @ X2 @ M )
         => ( ( member_set_real @ Y @ M )
           => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_1008_increasingD,axiom,
    ! [M: set_set_real,F: set_real > real,X2: set_real,Y: set_real] :
      ( ( measur4480787322886042509l_real @ M @ F )
     => ( ( ord_less_eq_set_real @ X2 @ Y )
       => ( ( member_set_real @ X2 @ M )
         => ( ( member_set_real @ Y @ M )
           => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_1009_increasingD,axiom,
    ! [M: set_set_real,F: set_real > extend8495563244428889912nnreal,X2: set_real,Y: set_real] :
      ( ( measur4858252777561384089nnreal @ M @ F )
     => ( ( ord_less_eq_set_real @ X2 @ Y )
       => ( ( member_set_real @ X2 @ M )
         => ( ( member_set_real @ Y @ M )
           => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_1010_increasingD,axiom,
    ! [M: set_set_real,F: set_real > extended_ereal,X2: set_real,Y: set_real] :
      ( ( measur5991237585687085_ereal @ M @ F )
     => ( ( ord_less_eq_set_real @ X2 @ Y )
       => ( ( member_set_real @ X2 @ M )
         => ( ( member_set_real @ Y @ M )
           => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_1011_euclidean__eqI,axiom,
    ! [X2: real,Y: real] :
      ( ! [B3: real] :
          ( ( member_real @ B3 @ euclid1305858884100475807s_real )
         => ( ( inner_5569486634795291694r_real @ X2 @ B3 )
            = ( inner_5569486634795291694r_real @ Y @ B3 ) ) )
     => ( X2 = Y ) ) ).

% euclidean_eqI
thf(fact_1012_borel__measurable__restrict__space__iff,axiom,
    ! [Omega: set_Pr6157794097399793337b_real,M: sigma_4174442854635832540b_real,F: produc7929879615859924995b_real > real] :
      ( ( member2587080422514930128b_real @ ( inf_in1453021003782806603b_real @ Omega @ ( sigma_709238742955535483b_real @ M ) ) @ ( sigma_5719578828308484196b_real @ M ) )
     => ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ ( sigma_5676250515055399871b_real @ M @ Omega ) @ borel_5078946678739801102l_real ) )
        = ( member5467075257862553189l_real
          @ ^ [X: produc7929879615859924995b_real] : ( real_V1485227260804924795R_real @ ( indica5149351975077090117l_real @ Omega @ X ) @ ( F @ X ) )
          @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_restrict_space_iff
thf(fact_1013_borel__measurable__restrict__space__iff,axiom,
    ! [Omega: set_real,M: sigma_measure_real,F: real > real] :
      ( ( member_set_real @ ( inf_inf_set_real @ Omega @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ borel_5078946678739801102l_real ) )
        = ( member_real_real
          @ ^ [X: real] : ( real_V1485227260804924795R_real @ ( indica3952125700972739438l_real @ Omega @ X ) @ ( F @ X ) )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).

% borel_measurable_restrict_space_iff
thf(fact_1014_borel__measurable__restrict__space__iff,axiom,
    ! [Omega: set_real,M: sigma_measure_real,F: real > produc2422161461964618553l_real] :
      ( ( member_set_real @ ( inf_inf_set_real @ Omega @ ( sigma_space_real @ M ) ) @ ( sigma_sets_real @ M ) )
     => ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ borel_9155112475215227991l_real ) )
        = ( member9086635009091248365l_real
          @ ^ [X: real] : ( real_V8334778624115123498l_real @ ( indica3952125700972739438l_real @ Omega @ X ) @ ( F @ X ) )
          @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_restrict_space_iff
thf(fact_1015_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: real,B2: set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_less_eq_nat @ ( indica8086626469483108754al_nat @ A @ X2 ) @ ( indica8086626469483108754al_nat @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1016_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: real,B2: set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_le3935885782089961368nnreal @ ( indica5945566441542501626nnreal @ A @ X2 ) @ ( indica5945566441542501626nnreal @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1017_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: real,B2: set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_less_eq_real @ ( indica3952125700972739438l_real @ A @ X2 ) @ ( indica3952125700972739438l_real @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1018_indicator__leI,axiom,
    ! [X2: set_real,A: set_set_real,Y: real,B2: set_real] :
      ( ( ( member_set_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_less_eq_nat @ ( indica1436675305277707720al_nat @ A @ X2 ) @ ( indica8086626469483108754al_nat @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1019_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: set_real,B2: set_set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_set_real @ Y @ B2 ) )
     => ( ord_less_eq_nat @ ( indica8086626469483108754al_nat @ A @ X2 ) @ ( indica1436675305277707720al_nat @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1020_indicator__leI,axiom,
    ! [X2: set_real,A: set_set_real,Y: real,B2: set_real] :
      ( ( ( member_set_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_le3935885782089961368nnreal @ ( indica5251532090073125168nnreal @ A @ X2 ) @ ( indica5945566441542501626nnreal @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1021_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: set_real,B2: set_set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_set_real @ Y @ B2 ) )
     => ( ord_le3935885782089961368nnreal @ ( indica5945566441542501626nnreal @ A @ X2 ) @ ( indica5251532090073125168nnreal @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1022_indicator__leI,axiom,
    ! [X2: set_real,A: set_set_real,Y: real,B2: set_real] :
      ( ( ( member_set_real @ X2 @ A )
       => ( member_real @ Y @ B2 ) )
     => ( ord_less_eq_real @ ( indica2804201007932554404l_real @ A @ X2 ) @ ( indica3952125700972739438l_real @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1023_indicator__leI,axiom,
    ! [X2: real,A: set_real,Y: set_real,B2: set_set_real] :
      ( ( ( member_real @ X2 @ A )
       => ( member_set_real @ Y @ B2 ) )
     => ( ord_less_eq_real @ ( indica3952125700972739438l_real @ A @ X2 ) @ ( indica2804201007932554404l_real @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1024_indicator__leI,axiom,
    ! [X2: set_real,A: set_set_real,Y: set_real,B2: set_set_real] :
      ( ( ( member_set_real @ X2 @ A )
       => ( member_set_real @ Y @ B2 ) )
     => ( ord_less_eq_nat @ ( indica1436675305277707720al_nat @ A @ X2 ) @ ( indica1436675305277707720al_nat @ B2 @ Y ) ) ) ).

% indicator_leI
thf(fact_1025_borel__measurable__iff__halfspace__less,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1026_borel__measurable__iff__halfspace__less,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1027_borel__measurable__iff__halfspace__less,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1028_borel__measurable__iff__halfspace__less,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1029_borel__measurable__iff__halfspace__less,axiom,
    ! [F: set_real > produc2422161461964618553l_real,M: sigma_3733394171116455995t_real] :
      ( ( member1965154327494239031l_real @ F @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1030_borel__measurable__iff__halfspace__less,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1031_borel__measurable__iff__halfspace__less,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member2587080422514930128b_real
                @ ( collec4671965459240937304b_real
                  @ ^ [W2: produc7929879615859924995b_real] :
                      ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1032_borel__measurable__iff__halfspace__less,axiom,
    ! [F: ( real > real ) > produc2422161461964618553l_real,M: sigma_4258434043392614480l_real] :
      ( ( member4094776897846995810l_real @ F @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1033_borel__measurable__iff__halfspace__less,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member3057847817020379547l_real
                @ ( collec869437573433150115l_real
                  @ ^ [W2: produc7929879615859924995b_real > real] :
                      ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                      & ( ord_less_real @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1034_borel__measurable__iff__halfspace__less,axiom,
    ! [F: ( product_prod_a_b > b ) > produc2422161461964618553l_real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member861154087952452317l_real @ F @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_real @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) @ A4 ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_less
thf(fact_1035_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1036_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: set_real > real,M: sigma_3733394171116455995t_real] :
      ( ( member_set_real_real @ F @ ( sigma_397049400287467232l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1037_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: ( real > real ) > real,M: sigma_4258434043392614480l_real] :
      ( ( member5749659578190367193l_real @ F @ ( sigma_314226851082786037l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1038_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: real > produc2422161461964618553l_real,M: sigma_measure_real] :
      ( ( member9086635009091248365l_real @ F @ ( sigma_7998147297565726139l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real
                @ ( collect_real
                  @ ^ [W2: real] :
                      ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1039_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: set_real > produc2422161461964618553l_real,M: sigma_3733394171116455995t_real] :
      ( ( member1965154327494239031l_real @ F @ ( sigma_4780502115161236485l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_set_real
                @ ( collect_set_real
                  @ ^ [W2: set_real] :
                      ( ( member_set_real @ W2 @ ( sigma_space_set_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_set_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1040_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: ( product_prod_a_b > b ) > real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member5852679122225541790b_real @ F @ ( sigma_1604199761323110458b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1041_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member2587080422514930128b_real
                @ ( collec4671965459240937304b_real
                  @ ^ [W2: produc7929879615859924995b_real] :
                      ( ( member6125958737618052890b_real @ W2 @ ( sigma_709238742955535483b_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_5719578828308484196b_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1042_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: ( real > real ) > produc2422161461964618553l_real,M: sigma_4258434043392614480l_real] :
      ( ( member4094776897846995810l_real @ F @ ( sigma_7266237356533690928l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member_set_real_real2
                @ ( collect_real_real
                  @ ^ [W2: real > real] :
                      ( ( member_real_real @ W2 @ ( sigma_3619470280215722479l_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_sets_real_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1043_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: ( produc7929879615859924995b_real > real ) > real,M: sigma_1368263781377809575l_real] :
      ( ( member497696658880670256l_real @ F @ ( sigma_7830024550720126796l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [X: real] :
            ( ( member_real @ X @ euclid1305858884100475807s_real )
           => ! [A4: real] :
                ( member3057847817020379547l_real
                @ ( collec869437573433150115l_real
                  @ ^ [W2: produc7929879615859924995b_real > real] :
                      ( ( member5467075257862553189l_real @ W2 @ ( sigma_1202137500299886022l_real @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_5569486634795291694r_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7475115973638204335l_real @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1044_borel__measurable__iff__halfspace__greater,axiom,
    ! [F: ( product_prod_a_b > b ) > produc2422161461964618553l_real,M: sigma_7535330959625612565_a_b_b] :
      ( ( member861154087952452317l_real @ F @ ( sigma_6089308910023032619l_real @ M @ borel_9155112475215227991l_real ) )
      = ( ! [X: produc2422161461964618553l_real] :
            ( ( member7849222048561428706l_real @ X @ euclid5187495680784233478l_real )
           => ! [A4: real] :
                ( member4198857394546769929_a_b_b
                @ ( collec2250730345169186961_a_b_b
                  @ ^ [W2: product_prod_a_b > b] :
                      ( ( member1080921170058542931_a_b_b @ W2 @ ( sigma_8376800615221029940_a_b_b @ M ) )
                      & ( ord_less_real @ A4 @ ( inner_2340953980025574455l_real @ ( F @ W2 ) @ X ) ) ) )
                @ ( sigma_7736500445245453213_a_b_b @ M ) ) ) ) ) ).

% borel_measurable_iff_halfspace_greater
thf(fact_1045_space__sup__measure_H,axiom,
    ! [B2: sigma_measure_real,A: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B2 )
        = ( sigma_sets_real @ A ) )
     => ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A @ B2 ) )
        = ( sigma_space_real @ A ) ) ) ).

% space_sup_measure'
thf(fact_1046_sets__sup__measure_H,axiom,
    ! [B2: sigma_measure_real,A: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B2 )
        = ( sigma_sets_real @ A ) )
     => ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A @ B2 ) )
        = ( sigma_sets_real @ A ) ) ) ).

% sets_sup_measure'
thf(fact_1047_sets__restrict__UNIV,axiom,
    ! [M: sigma_measure_real] :
      ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ top_top_set_real ) )
      = ( sigma_sets_real @ M ) ) ).

% sets_restrict_UNIV
thf(fact_1048_space__restrict__space2,axiom,
    ! [Omega: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ Omega @ ( sigma_sets_real @ M ) )
     => ( ( sigma_space_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
        = Omega ) ) ).

% space_restrict_space2
thf(fact_1049_restrict__space__sets__cong,axiom,
    ! [A: set_real,B2: set_real,M: sigma_measure_real,N: sigma_measure_real] :
      ( ( A = B2 )
     => ( ( ( sigma_sets_real @ M )
          = ( sigma_sets_real @ N ) )
       => ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ A ) )
          = ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ B2 ) ) ) ) ) ).

% restrict_space_sets_cong
thf(fact_1050_sets__restrict__space__cong,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,Omega: set_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ N ) )
     => ( ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) )
        = ( sigma_sets_real @ ( sigma_5414646170262037096e_real @ N @ Omega ) ) ) ) ).

% sets_restrict_space_cong
thf(fact_1051_measurable__restrict__space1,axiom,
    ! [F: produc7929879615859924995b_real > real,M: sigma_4174442854635832540b_real,N: sigma_measure_real,Omega: set_Pr6157794097399793337b_real] :
      ( ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ M @ N ) )
     => ( member5467075257862553189l_real @ F @ ( sigma_2402725360379274625l_real @ ( sigma_5676250515055399871b_real @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1052_measurable__restrict__space1,axiom,
    ! [F: product_prod_a_b > b,M: sigma_5782782616773303333od_a_b,N: sigma_measure_b,Omega: set_Product_prod_a_b] :
      ( ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ M @ N ) )
     => ( member1080921170058542931_a_b_b @ F @ ( sigma_7225552367035416487_a_b_b @ ( sigma_8382023653677975020od_a_b @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1053_measurable__restrict__space1,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,Omega: set_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ ( sigma_5414646170262037096e_real @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1054_measurable__restrict__space1,axiom,
    ! [F: produc7929879615859924995b_real > produc7929879615859924995b_real,M: sigma_4174442854635832540b_real,N: sigma_4174442854635832540b_real,Omega: set_Pr6157794097399793337b_real] :
      ( ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ M @ N ) )
     => ( member1118978200981565500b_real @ F @ ( sigma_1108951568718238936b_real @ ( sigma_5676250515055399871b_real @ M @ Omega ) @ N ) ) ) ).

% measurable_restrict_space1
thf(fact_1055_leD,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ~ ( ord_less_nat @ X2 @ Y ) ) ).

% leD
thf(fact_1056_leD,axiom,
    ! [Y: real,X2: real] :
      ( ( ord_less_eq_real @ Y @ X2 )
     => ~ ( ord_less_real @ X2 @ Y ) ) ).

% leD
thf(fact_1057_leD,axiom,
    ! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Y @ X2 )
     => ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) ) ).

% leD
thf(fact_1058_leD,axiom,
    ! [Y: extended_ereal,X2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ Y @ X2 )
     => ~ ( ord_le1188267648640031866_ereal @ X2 @ Y ) ) ).

% leD
thf(fact_1059_leI,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% leI
thf(fact_1060_leI,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_real @ X2 @ Y )
     => ( ord_less_eq_real @ Y @ X2 ) ) ).

% leI
thf(fact_1061_leI,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
     => ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).

% leI
thf(fact_1062_leI,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ~ ( ord_le1188267648640031866_ereal @ X2 @ Y )
     => ( ord_le1083603963089353582_ereal @ Y @ X2 ) ) ).

% leI
thf(fact_1063_nless__le,axiom,
    ! [A3: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A3 @ B ) )
      = ( ~ ( ord_less_eq_nat @ A3 @ B )
        | ( A3 = B ) ) ) ).

% nless_le
thf(fact_1064_nless__le,axiom,
    ! [A3: real,B: real] :
      ( ( ~ ( ord_less_real @ A3 @ B ) )
      = ( ~ ( ord_less_eq_real @ A3 @ B )
        | ( A3 = B ) ) ) ).

% nless_le
thf(fact_1065_nless__le,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le7381754540660121996nnreal @ A3 @ B ) )
      = ( ~ ( ord_le3935885782089961368nnreal @ A3 @ B )
        | ( A3 = B ) ) ) ).

% nless_le
thf(fact_1066_nless__le,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ~ ( ord_le1188267648640031866_ereal @ A3 @ B ) )
      = ( ~ ( ord_le1083603963089353582_ereal @ A3 @ B )
        | ( A3 = B ) ) ) ).

% nless_le
thf(fact_1067_antisym__conv1,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1068_antisym__conv1,axiom,
    ! [X2: real,Y: real] :
      ( ~ ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1069_antisym__conv1,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1070_antisym__conv1,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ~ ( ord_le1188267648640031866_ereal @ X2 @ Y )
     => ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% antisym_conv1
thf(fact_1071_antisym__conv2,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ~ ( ord_less_nat @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1072_antisym__conv2,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ~ ( ord_less_real @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1073_antisym__conv2,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1074_antisym__conv2,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( ~ ( ord_le1188267648640031866_ereal @ X2 @ Y ) )
        = ( X2 = Y ) ) ) ).

% antisym_conv2
thf(fact_1075_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ Z2 @ X4 )
         => ( ord_less_eq_real @ Y @ X4 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_1076_dense__ge,axiom,
    ! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ! [X4: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ Z2 @ X4 )
         => ( ord_le3935885782089961368nnreal @ Y @ X4 ) )
     => ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_1077_dense__ge,axiom,
    ! [Z2: extended_ereal,Y: extended_ereal] :
      ( ! [X4: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ Z2 @ X4 )
         => ( ord_le1083603963089353582_ereal @ Y @ X4 ) )
     => ( ord_le1083603963089353582_ereal @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_1078_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_eq_real @ X4 @ Z2 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_le
thf(fact_1079_dense__le,axiom,
    ! [Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ! [X4: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ X4 @ Y )
         => ( ord_le3935885782089961368nnreal @ X4 @ Z2 ) )
     => ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ).

% dense_le
thf(fact_1080_dense__le,axiom,
    ! [Y: extended_ereal,Z2: extended_ereal] :
      ( ! [X4: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ X4 @ Y )
         => ( ord_le1083603963089353582_ereal @ X4 @ Z2 ) )
     => ( ord_le1083603963089353582_ereal @ Y @ Z2 ) ) ).

% dense_le
thf(fact_1081_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1082_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y2: real] :
          ( ( ord_less_eq_real @ X @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1083_less__le__not__le,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ X @ Y2 )
          & ~ ( ord_le3935885782089961368nnreal @ Y2 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1084_less__le__not__le,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ X @ Y2 )
          & ~ ( ord_le1083603963089353582_ereal @ Y2 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1085_not__le__imp__less,axiom,
    ! [Y: nat,X2: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X2 )
     => ( ord_less_nat @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_1086_not__le__imp__less,axiom,
    ! [Y: real,X2: real] :
      ( ~ ( ord_less_eq_real @ Y @ X2 )
     => ( ord_less_real @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_1087_not__le__imp__less,axiom,
    ! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ~ ( ord_le3935885782089961368nnreal @ Y @ X2 )
     => ( ord_le7381754540660121996nnreal @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_1088_not__le__imp__less,axiom,
    ! [Y: extended_ereal,X2: extended_ereal] :
      ( ~ ( ord_le1083603963089353582_ereal @ Y @ X2 )
     => ( ord_le1188267648640031866_ereal @ X2 @ Y ) ) ).

% not_le_imp_less
thf(fact_1089_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A4: nat,B5: nat] :
          ( ( ord_less_nat @ A4 @ B5 )
          | ( A4 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1090_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A4: real,B5: real] :
          ( ( ord_less_real @ A4 @ B5 )
          | ( A4 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1091_order_Oorder__iff__strict,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ A4 @ B5 )
          | ( A4 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1092_order_Oorder__iff__strict,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ A4 @ B5 )
          | ( A4 = B5 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1093_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A4 @ B5 )
          & ( A4 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1094_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B5: real] :
          ( ( ord_less_eq_real @ A4 @ B5 )
          & ( A4 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1095_order_Ostrict__iff__order,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A4 @ B5 )
          & ( A4 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1096_order_Ostrict__iff__order,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A4 @ B5 )
          & ( A4 != B5 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1097_order_Ostrict__trans1,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1098_order_Ostrict__trans1,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1099_order_Ostrict__trans1,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le7381754540660121996nnreal @ B @ C )
       => ( ord_le7381754540660121996nnreal @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1100_order_Ostrict__trans1,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1188267648640031866_ereal @ B @ C )
       => ( ord_le1188267648640031866_ereal @ A3 @ C ) ) ) ).

% order.strict_trans1
thf(fact_1101_order_Ostrict__trans2,axiom,
    ! [A3: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1102_order_Ostrict__trans2,axiom,
    ! [A3: real,B: real,C: real] :
      ( ( ord_less_real @ A3 @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1103_order_Ostrict__trans2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A3 @ B )
     => ( ( ord_le3935885782089961368nnreal @ B @ C )
       => ( ord_le7381754540660121996nnreal @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1104_order_Ostrict__trans2,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ B @ C )
       => ( ord_le1188267648640031866_ereal @ A3 @ C ) ) ) ).

% order.strict_trans2
thf(fact_1105_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A4: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A4 @ B5 )
          & ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1106_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A4: real,B5: real] :
          ( ( ord_less_eq_real @ A4 @ B5 )
          & ~ ( ord_less_eq_real @ B5 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1107_order_Ostrict__iff__not,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [A4: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A4 @ B5 )
          & ~ ( ord_le3935885782089961368nnreal @ B5 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1108_order_Ostrict__iff__not,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [A4: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A4 @ B5 )
          & ~ ( ord_le1083603963089353582_ereal @ B5 @ A4 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1109_dense__ge__bounded,axiom,
    ! [Z2: real,X2: real,Y: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z2 @ W )
           => ( ( ord_less_real @ W @ X2 )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1110_dense__ge__bounded,axiom,
    ! [Z2: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ Z2 @ X2 )
     => ( ! [W: extend8495563244428889912nnreal] :
            ( ( ord_le7381754540660121996nnreal @ Z2 @ W )
           => ( ( ord_le7381754540660121996nnreal @ W @ X2 )
             => ( ord_le3935885782089961368nnreal @ Y @ W ) ) )
       => ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1111_dense__ge__bounded,axiom,
    ! [Z2: extended_ereal,X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ Z2 @ X2 )
     => ( ! [W: extended_ereal] :
            ( ( ord_le1188267648640031866_ereal @ Z2 @ W )
           => ( ( ord_le1188267648640031866_ereal @ W @ X2 )
             => ( ord_le1083603963089353582_ereal @ Y @ W ) ) )
       => ( ord_le1083603963089353582_ereal @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1112_dense__le__bounded,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X2 @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1113_dense__le__bounded,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ X2 @ Y )
     => ( ! [W: extend8495563244428889912nnreal] :
            ( ( ord_le7381754540660121996nnreal @ X2 @ W )
           => ( ( ord_le7381754540660121996nnreal @ W @ Y )
             => ( ord_le3935885782089961368nnreal @ W @ Z2 ) ) )
       => ( ord_le3935885782089961368nnreal @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1114_dense__le__bounded,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X2 @ Y )
     => ( ! [W: extended_ereal] :
            ( ( ord_le1188267648640031866_ereal @ X2 @ W )
           => ( ( ord_le1188267648640031866_ereal @ W @ Y )
             => ( ord_le1083603963089353582_ereal @ W @ Z2 ) ) )
       => ( ord_le1083603963089353582_ereal @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1115_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B5: nat,A4: nat] :
          ( ( ord_less_nat @ B5 @ A4 )
          | ( A4 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1116_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B5: real,A4: real] :
          ( ( ord_less_real @ B5 @ A4 )
          | ( A4 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1117_dual__order_Oorder__iff__strict,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [B5: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ B5 @ A4 )
          | ( A4 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1118_dual__order_Oorder__iff__strict,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [B5: extended_ereal,A4: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ B5 @ A4 )
          | ( A4 = B5 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1119_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B5 @ A4 )
          & ( A4 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1120_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B5: real,A4: real] :
          ( ( ord_less_eq_real @ B5 @ A4 )
          & ( A4 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1121_dual__order_Ostrict__iff__order,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [B5: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ B5 @ A4 )
          & ( A4 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1122_dual__order_Ostrict__iff__order,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [B5: extended_ereal,A4: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ B5 @ A4 )
          & ( A4 != B5 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1123_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A3 )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1124_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B @ A3 )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1125_dual__order_Ostrict__trans1,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B @ A3 )
     => ( ( ord_le7381754540660121996nnreal @ C @ B )
       => ( ord_le7381754540660121996nnreal @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1126_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_ereal,A3: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B @ A3 )
     => ( ( ord_le1188267648640031866_ereal @ C @ B )
       => ( ord_le1188267648640031866_ereal @ C @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1127_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A3: nat,C: nat] :
      ( ( ord_less_nat @ B @ A3 )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1128_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A3: real,C: real] :
      ( ( ord_less_real @ B @ A3 )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1129_dual__order_Ostrict__trans2,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ B @ A3 )
     => ( ( ord_le3935885782089961368nnreal @ C @ B )
       => ( ord_le7381754540660121996nnreal @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1130_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_ereal,A3: extended_ereal,C: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ B @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ C @ B )
       => ( ord_le1188267648640031866_ereal @ C @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1131_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B5: nat,A4: nat] :
          ( ( ord_less_eq_nat @ B5 @ A4 )
          & ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1132_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B5: real,A4: real] :
          ( ( ord_less_eq_real @ B5 @ A4 )
          & ~ ( ord_less_eq_real @ A4 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1133_dual__order_Ostrict__iff__not,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [B5: extend8495563244428889912nnreal,A4: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ B5 @ A4 )
          & ~ ( ord_le3935885782089961368nnreal @ A4 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1134_dual__order_Ostrict__iff__not,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [B5: extended_ereal,A4: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ B5 @ A4 )
          & ~ ( ord_le1083603963089353582_ereal @ A4 @ B5 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1135_order_Ostrict__implies__order,axiom,
    ! [A3: nat,B: nat] :
      ( ( ord_less_nat @ A3 @ B )
     => ( ord_less_eq_nat @ A3 @ B ) ) ).

% order.strict_implies_order
thf(fact_1136_order_Ostrict__implies__order,axiom,
    ! [A3: real,B: real] :
      ( ( ord_less_real @ A3 @ B )
     => ( ord_less_eq_real @ A3 @ B ) ) ).

% order.strict_implies_order
thf(fact_1137_order_Ostrict__implies__order,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A3 @ B )
     => ( ord_le3935885782089961368nnreal @ A3 @ B ) ) ).

% order.strict_implies_order
thf(fact_1138_order_Ostrict__implies__order,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A3 @ B )
     => ( ord_le1083603963089353582_ereal @ A3 @ B ) ) ).

% order.strict_implies_order
thf(fact_1139_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A3: nat] :
      ( ( ord_less_nat @ B @ A3 )
     => ( ord_less_eq_nat @ B @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1140_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A3: real] :
      ( ( ord_less_real @ B @ A3 )
     => ( ord_less_eq_real @ B @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1141_dual__order_Ostrict__implies__order,axiom,
    ! [B: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ B @ A3 )
     => ( ord_le3935885782089961368nnreal @ B @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1142_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_ereal,A3: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ B @ A3 )
     => ( ord_le1083603963089353582_ereal @ B @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1143_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_nat @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1144_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y2: real] :
          ( ( ord_less_real @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1145_order__le__less,axiom,
    ( ord_le3935885782089961368nnreal
    = ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1146_order__le__less,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% order_le_less
thf(fact_1147_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X @ Y2 )
          & ( X != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1148_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y2: real] :
          ( ( ord_less_eq_real @ X @ Y2 )
          & ( X != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1149_order__less__le,axiom,
    ( ord_le7381754540660121996nnreal
    = ( ^ [X: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ X @ Y2 )
          & ( X != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1150_order__less__le,axiom,
    ( ord_le1188267648640031866_ereal
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ X @ Y2 )
          & ( X != Y2 ) ) ) ) ).

% order_less_le
thf(fact_1151_linorder__not__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
      = ( ord_less_nat @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_1152_linorder__not__le,axiom,
    ! [X2: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X2 @ Y ) )
      = ( ord_less_real @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_1153_linorder__not__le,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y ) )
      = ( ord_le7381754540660121996nnreal @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_1154_linorder__not__le,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ~ ( ord_le1083603963089353582_ereal @ X2 @ Y ) )
      = ( ord_le1188267648640031866_ereal @ Y @ X2 ) ) ).

% linorder_not_le
thf(fact_1155_linorder__not__less,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_1156_linorder__not__less,axiom,
    ! [X2: real,Y: real] :
      ( ( ~ ( ord_less_real @ X2 @ Y ) )
      = ( ord_less_eq_real @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_1157_linorder__not__less,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le7381754540660121996nnreal @ X2 @ Y ) )
      = ( ord_le3935885782089961368nnreal @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_1158_linorder__not__less,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ~ ( ord_le1188267648640031866_ereal @ X2 @ Y ) )
      = ( ord_le1083603963089353582_ereal @ Y @ X2 ) ) ).

% linorder_not_less
thf(fact_1159_order__less__imp__le,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1160_order__less__imp__le,axiom,
    ! [X2: real,Y: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ord_less_eq_real @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1161_order__less__imp__le,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ X2 @ Y )
     => ( ord_le3935885782089961368nnreal @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1162_order__less__imp__le,axiom,
    ! [X2: extended_ereal,Y: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X2 @ Y )
     => ( ord_le1083603963089353582_ereal @ X2 @ Y ) ) ).

% order_less_imp_le
thf(fact_1163_order__le__neq__trans,axiom,
    ! [A3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A3 @ B )
     => ( ( A3 != B )
       => ( ord_less_nat @ A3 @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1164_order__le__neq__trans,axiom,
    ! [A3: real,B: real] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( A3 != B )
       => ( ord_less_real @ A3 @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1165_order__le__neq__trans,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( A3 != B )
       => ( ord_le7381754540660121996nnreal @ A3 @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1166_order__le__neq__trans,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( A3 != B )
       => ( ord_le1188267648640031866_ereal @ A3 @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1167_order__neq__le__trans,axiom,
    ! [A3: nat,B: nat] :
      ( ( A3 != B )
     => ( ( ord_less_eq_nat @ A3 @ B )
       => ( ord_less_nat @ A3 @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1168_order__neq__le__trans,axiom,
    ! [A3: real,B: real] :
      ( ( A3 != B )
     => ( ( ord_less_eq_real @ A3 @ B )
       => ( ord_less_real @ A3 @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1169_order__neq__le__trans,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( A3 != B )
     => ( ( ord_le3935885782089961368nnreal @ A3 @ B )
       => ( ord_le7381754540660121996nnreal @ A3 @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1170_order__neq__le__trans,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( A3 != B )
     => ( ( ord_le1083603963089353582_ereal @ A3 @ B )
       => ( ord_le1188267648640031866_ereal @ A3 @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1171_order__le__less__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1172_order__le__less__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X2 @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1173_order__le__less__trans,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
     => ( ( ord_le7381754540660121996nnreal @ Y @ Z2 )
       => ( ord_le7381754540660121996nnreal @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1174_order__le__less__trans,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X2 @ Y )
     => ( ( ord_le1188267648640031866_ereal @ Y @ Z2 )
       => ( ord_le1188267648640031866_ereal @ X2 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1175_order__less__le__trans,axiom,
    ! [X2: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1176_order__less__le__trans,axiom,
    ! [X2: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X2 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_real @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1177_order__less__le__trans,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ X2 @ Y )
     => ( ( ord_le3935885782089961368nnreal @ Y @ Z2 )
       => ( ord_le7381754540660121996nnreal @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1178_order__less__le__trans,axiom,
    ! [X2: extended_ereal,Y: extended_ereal,Z2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X2 @ Y )
     => ( ( ord_le1083603963089353582_ereal @ Y @ Z2 )
       => ( ord_le1188267648640031866_ereal @ X2 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1179_order__le__less__subst1,axiom,
    ! [A3: nat,F: real > nat,B: real,C: real] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1180_order__le__less__subst1,axiom,
    ! [A3: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_le7381754540660121996nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1181_order__le__less__subst1,axiom,
    ! [A3: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1182_order__le__less__subst1,axiom,
    ! [A3: nat,F: extended_ereal > nat,B: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B ) )
     => ( ( ord_le1188267648640031866_ereal @ B @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X4 @ Y3 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1183_order__le__less__subst1,axiom,
    ! [A3: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1184_order__le__less__subst1,axiom,
    ! [A3: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_le7381754540660121996nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1185_order__le__less__subst1,axiom,
    ! [A3: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1186_order__le__less__subst1,axiom,
    ! [A3: real,F: extended_ereal > real,B: extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ ( F @ B ) )
     => ( ( ord_le1188267648640031866_ereal @ B @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X4 @ Y3 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1187_order__le__less__subst1,axiom,
    ! [A3: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le7381754540660121996nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1188_order__le__less__subst1,axiom,
    ! [A3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ ( F @ B ) )
     => ( ( ord_le7381754540660121996nnreal @ B @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le7381754540660121996nnreal @ X4 @ Y3 )
             => ( ord_le7381754540660121996nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le7381754540660121996nnreal @ A3 @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1189_order__le__less__subst2,axiom,
    ! [A3: real,B: real,F: real > extended_ereal,C: extended_ereal] :
      ( ( ord_less_eq_real @ A3 @ B )
     => ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_eq_real @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1190_order__le__less__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1191_order__le__less__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1192_order__le__less__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le7381754540660121996nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1193_order__le__less__subst2,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal,C: extended_ereal] :
      ( ( ord_le3935885782089961368nnreal @ A3 @ B )
     => ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
       => ( ! [X4: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1194_order__le__less__subst2,axiom,
    ! [A3: extended_ereal,B: extended_ereal,F: extended_ereal > nat,C: nat] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1195_order__le__less__subst2,axiom,
    ! [A3: extended_ereal,B: extended_ereal,F: extended_ereal > real,C: real] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1196_order__le__less__subst2,axiom,
    ! [A3: extended_ereal,B: extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le7381754540660121996nnreal @ ( F @ B ) @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le7381754540660121996nnreal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1197_order__le__less__subst2,axiom,
    ! [A3: extended_ereal,B: extended_ereal,F: extended_ereal > extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1188267648640031866_ereal @ ( F @ B ) @ C )
       => ( ! [X4: extended_ereal,Y3: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X4 @ Y3 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A3 ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1198_seq__mono__lemma,axiom,
    ! [M5: nat,D: nat > real,E: nat > real] :
      ( ! [N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_real @ ( D @ N3 ) @ ( E @ N3 ) ) )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
           => ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M5 ) ) )
       => ! [N4: nat] :
            ( ( ord_less_eq_nat @ M5 @ N4 )
           => ( ord_less_real @ ( D @ N4 ) @ ( E @ M5 ) ) ) ) ) ).

% seq_mono_lemma
thf(fact_1199_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y2: real] :
          ( ( ord_less_real @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_1200_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X6: real] : ( member_real @ X6 @ S2 )
     => ( ? [Z5: real] :
          ! [X4: real] :
            ( ( member_real @ X4 @ S2 )
           => ( ord_less_eq_real @ X4 @ Z5 ) )
       => ? [Y3: real] :
            ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ord_less_eq_real @ X6 @ Y3 ) )
            & ! [Z5: real] :
                ( ! [X4: real] :
                    ( ( member_real @ X4 @ S2 )
                   => ( ord_less_eq_real @ X4 @ Z5 ) )
               => ( ord_less_eq_real @ Y3 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_1201_neq__top__trans,axiom,
    ! [Y: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
      ( ( Y != top_to1496364449551166952nnreal )
     => ( ( ord_le3935885782089961368nnreal @ X2 @ Y )
       => ( X2 != top_to1496364449551166952nnreal ) ) ) ).

% neq_top_trans
thf(fact_1202_ennreal__top__mult__right,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( X2 != zero_z7100319975126383169nnreal )
     => ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ X2 )
        = top_to1496364449551166952nnreal ) ) ).

% ennreal_top_mult_right
thf(fact_1203_ennreal__top__mult__left,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( X2 != zero_z7100319975126383169nnreal )
     => ( ( times_1893300245718287421nnreal @ X2 @ top_to1496364449551166952nnreal )
        = top_to1496364449551166952nnreal ) ) ).

% ennreal_top_mult_left
thf(fact_1204_ennreal__approx__unit,axiom,
    ! [Z2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
      ( ! [A5: extend8495563244428889912nnreal] :
          ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ A5 )
         => ( ( ord_le7381754540660121996nnreal @ A5 @ one_on2969667320475766781nnreal )
           => ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A5 @ Z2 ) @ Y ) ) )
     => ( ord_le3935885782089961368nnreal @ Z2 @ Y ) ) ).

% ennreal_approx_unit
thf(fact_1205_ennreal__zero__less__top,axiom,
    ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ top_to1496364449551166952nnreal ).

% ennreal_zero_less_top
thf(fact_1206_ennreal__one__less__top,axiom,
    ord_le7381754540660121996nnreal @ one_on2969667320475766781nnreal @ top_to1496364449551166952nnreal ).

% ennreal_one_less_top
thf(fact_1207_ennreal__zero__neq__top,axiom,
    zero_z7100319975126383169nnreal != top_to1496364449551166952nnreal ).

% ennreal_zero_neq_top
thf(fact_1208_ennreal__top__neq__one,axiom,
    top_to1496364449551166952nnreal != one_on2969667320475766781nnreal ).

% ennreal_top_neq_one
thf(fact_1209_ennreal__mult__strict__right__mono,axiom,
    ! [A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A3 @ C )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B )
       => ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
         => ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A3 @ B ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ) ).

% ennreal_mult_strict_right_mono
thf(fact_1210_ennreal__mult__strict__left__mono,axiom,
    ! [A3: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ A3 @ C )
     => ( ( ord_le7381754540660121996nnreal @ zero_z7100319975126383169nnreal @ B )
       => ( ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal )
         => ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ B @ A3 ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ) ) ).

% ennreal_mult_strict_left_mono
thf(fact_1211_ennreal__mult__less__top,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ ( times_1893300245718287421nnreal @ A3 @ B ) @ top_to1496364449551166952nnreal )
      = ( ( A3 = zero_z7100319975126383169nnreal )
        | ( B = zero_z7100319975126383169nnreal )
        | ( ( ord_le7381754540660121996nnreal @ A3 @ top_to1496364449551166952nnreal )
          & ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) ) ) ) ).

% ennreal_mult_less_top
thf(fact_1212_ennreal__mult__cancel__left,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
      ( ( ( times_1893300245718287421nnreal @ A3 @ B )
        = ( times_1893300245718287421nnreal @ A3 @ C ) )
      = ( ( ( A3 = top_to1496364449551166952nnreal )
          & ( B != zero_z7100319975126383169nnreal )
          & ( C != zero_z7100319975126383169nnreal ) )
        | ( A3 = zero_z7100319975126383169nnreal )
        | ( B = C ) ) ) ).

% ennreal_mult_cancel_left
thf(fact_1213_ennreal__top__eq__mult__iff,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( top_to1496364449551166952nnreal
        = ( times_1893300245718287421nnreal @ A3 @ B ) )
      = ( ( ( A3 = top_to1496364449551166952nnreal )
          & ( B != zero_z7100319975126383169nnreal ) )
        | ( ( B = top_to1496364449551166952nnreal )
          & ( A3 != zero_z7100319975126383169nnreal ) ) ) ) ).

% ennreal_top_eq_mult_iff
thf(fact_1214_ennreal__mult__eq__top__iff,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( times_1893300245718287421nnreal @ A3 @ B )
        = top_to1496364449551166952nnreal )
      = ( ( ( A3 = top_to1496364449551166952nnreal )
          & ( B != zero_z7100319975126383169nnreal ) )
        | ( ( B = top_to1496364449551166952nnreal )
          & ( A3 != zero_z7100319975126383169nnreal ) ) ) ) ).

% ennreal_mult_eq_top_iff
thf(fact_1215_ennreal__top__mult,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( ( A3 = zero_z7100319975126383169nnreal )
       => ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ A3 )
          = zero_z7100319975126383169nnreal ) )
      & ( ( A3 != zero_z7100319975126383169nnreal )
       => ( ( times_1893300245718287421nnreal @ top_to1496364449551166952nnreal @ A3 )
          = top_to1496364449551166952nnreal ) ) ) ).

% ennreal_top_mult
thf(fact_1216_ennreal__mult__top,axiom,
    ! [A3: extend8495563244428889912nnreal] :
      ( ( ( A3 = zero_z7100319975126383169nnreal )
       => ( ( times_1893300245718287421nnreal @ A3 @ top_to1496364449551166952nnreal )
          = zero_z7100319975126383169nnreal ) )
      & ( ( A3 != zero_z7100319975126383169nnreal )
       => ( ( times_1893300245718287421nnreal @ A3 @ top_to1496364449551166952nnreal )
          = top_to1496364449551166952nnreal ) ) ) ).

% ennreal_mult_top
thf(fact_1217_ennreal__mult__le__mult__iff,axiom,
    ! [C: extend8495563244428889912nnreal,A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( C != zero_z7100319975126383169nnreal )
     => ( ( C != top_to1496364449551166952nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A3 ) @ ( times_1893300245718287421nnreal @ C @ B ) )
          = ( ord_le3935885782089961368nnreal @ A3 @ B ) ) ) ) ).

% ennreal_mult_le_mult_iff
thf(fact_1218_kuhn__labelling__lemma_H,axiom,
    ! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
      ( ! [X4: nat > real] :
          ( ( P @ X4 )
         => ( P @ ( F @ X4 ) ) )
     => ( ! [X4: nat > real] :
            ( ( P @ X4 )
           => ! [I2: nat] :
                ( ( Q @ I2 )
               => ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I2 ) )
                  & ( ord_less_eq_real @ ( X4 @ I2 ) @ one_one_real ) ) ) )
       => ? [L2: ( nat > real ) > nat > nat] :
            ( ! [X6: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X6 @ I3 ) @ one_one_nat )
            & ! [X6: nat > real,I3: nat] :
                ( ( ( P @ X6 )
                  & ( Q @ I3 )
                  & ( ( X6 @ I3 )
                    = zero_zero_real ) )
               => ( ( L2 @ X6 @ I3 )
                  = zero_zero_nat ) )
            & ! [X6: nat > real,I3: nat] :
                ( ( ( P @ X6 )
                  & ( Q @ I3 )
                  & ( ( X6 @ I3 )
                    = one_one_real ) )
               => ( ( L2 @ X6 @ I3 )
                  = one_one_nat ) )
            & ! [X6: nat > real,I3: nat] :
                ( ( ( P @ X6 )
                  & ( Q @ I3 )
                  & ( ( L2 @ X6 @ I3 )
                    = zero_zero_nat ) )
               => ( ord_less_eq_real @ ( X6 @ I3 ) @ ( F @ X6 @ I3 ) ) )
            & ! [X6: nat > real,I3: nat] :
                ( ( ( P @ X6 )
                  & ( Q @ I3 )
                  & ( ( L2 @ X6 @ I3 )
                    = one_one_nat ) )
               => ( ord_less_eq_real @ ( F @ X6 @ I3 ) @ ( X6 @ I3 ) ) ) ) ) ) ).

% kuhn_labelling_lemma'
thf(fact_1219_le0,axiom,
    ! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).

% le0
thf(fact_1220_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).

% bot_nat_0.extremum
thf(fact_1221_mult__le__cancel2,axiom,
    ! [M5: nat,K: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N5 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M5 @ N5 ) ) ) ).

% mult_le_cancel2
thf(fact_1222_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_1223_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_1224_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L3 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L3 ) ) ) ) ).

% mult_le_mono
thf(fact_1225_le__square,axiom,
    ! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ).

% le_square
thf(fact_1226_le__cube,axiom,
    ! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ) ).

% le_cube
thf(fact_1227_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N6: nat] :
          ( ( ord_less_eq_nat @ M6 @ N6 )
          & ( M6 != N6 ) ) ) ) ).

% nat_less_le
thf(fact_1228_less__imp__le__nat,axiom,
    ! [M5: nat,N5: nat] :
      ( ( ord_less_nat @ M5 @ N5 )
     => ( ord_less_eq_nat @ M5 @ N5 ) ) ).

% less_imp_le_nat
thf(fact_1229_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N6: nat] :
          ( ( ord_less_nat @ M6 @ N6 )
          | ( M6 = N6 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1230_less__or__eq__imp__le,axiom,
    ! [M5: nat,N5: nat] :
      ( ( ( ord_less_nat @ M5 @ N5 )
        | ( M5 = N5 ) )
     => ( ord_less_eq_nat @ M5 @ N5 ) ) ).

% less_or_eq_imp_le
thf(fact_1231_le__neq__implies__less,axiom,
    ! [M5: nat,N5: nat] :
      ( ( ord_less_eq_nat @ M5 @ N5 )
     => ( ( M5 != N5 )
       => ( ord_less_nat @ M5 @ N5 ) ) ) ).

% le_neq_implies_less
thf(fact_1232_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1233_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1234_nat__le__linear,axiom,
    ! [M5: nat,N5: nat] :
      ( ( ord_less_eq_nat @ M5 @ N5 )
      | ( ord_less_eq_nat @ N5 @ M5 ) ) ).

% nat_le_linear
thf(fact_1235_le__antisym,axiom,
    ! [M5: nat,N5: nat] :
      ( ( ord_less_eq_nat @ M5 @ N5 )
     => ( ( ord_less_eq_nat @ N5 @ M5 )
       => ( M5 = N5 ) ) ) ).

% le_antisym
thf(fact_1236_eq__imp__le,axiom,
    ! [M5: nat,N5: nat] :
      ( ( M5 = N5 )
     => ( ord_less_eq_nat @ M5 @ N5 ) ) ).

% eq_imp_le
thf(fact_1237_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_1238_le__refl,axiom,
    ! [N5: nat] : ( ord_less_eq_nat @ N5 @ N5 ) ).

% le_refl
thf(fact_1239_less__eq__nat_Osimps_I1_J,axiom,
    ! [N5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N5 ) ).

% less_eq_nat.simps(1)
thf(fact_1240_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1241_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1242_le__0__eq,axiom,
    ! [N5: nat] :
      ( ( ord_less_eq_nat @ N5 @ zero_zero_nat )
      = ( N5 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1243_ex__least__nat__le,axiom,
    ! [P: nat > $o,N5: nat] :
      ( ( P @ N5 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N5 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1244_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M5: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N5 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M5 @ N5 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1245_nat__mult__le__cancel1,axiom,
    ! [K: nat,M5: nat,N5: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N5 ) )
        = ( ord_less_eq_nat @ M5 @ N5 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1246_less__eq__ereal__def,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [X: extended_ereal,Y2: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ X @ Y2 )
          | ( X = Y2 ) ) ) ) ).

% less_eq_ereal_def
thf(fact_1247_ereal__complete__Inf,axiom,
    ! [S2: set_Extended_ereal] :
    ? [X4: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member2350847679896131959_ereal @ Xa @ S2 )
         => ( ord_le1083603963089353582_ereal @ X4 @ Xa ) )
      & ! [Z5: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ Xa2 @ S2 )
             => ( ord_le1083603963089353582_ereal @ Z5 @ Xa2 ) )
         => ( ord_le1083603963089353582_ereal @ Z5 @ X4 ) ) ) ).

% ereal_complete_Inf
thf(fact_1248_ereal__complete__Sup,axiom,
    ! [S2: set_Extended_ereal] :
    ? [X4: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member2350847679896131959_ereal @ Xa @ S2 )
         => ( ord_le1083603963089353582_ereal @ Xa @ X4 ) )
      & ! [Z5: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ Xa2 @ S2 )
             => ( ord_le1083603963089353582_ereal @ Xa2 @ Z5 ) )
         => ( ord_le1083603963089353582_ereal @ X4 @ Z5 ) ) ) ).

% ereal_complete_Sup
thf(fact_1249_zero__less__one__ereal,axiom,
    ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ one_on4623092294121504201_ereal ).

% zero_less_one_ereal
thf(fact_1250_one__not__le__zero__ereal,axiom,
    ~ ( ord_le1083603963089353582_ereal @ one_on4623092294121504201_ereal @ zero_z2744965634713055877_ereal ) ).

% one_not_le_zero_ereal
thf(fact_1251_ereal__0__le__mult,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
       => ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A3 @ B ) ) ) ) ).

% ereal_0_le_mult
thf(fact_1252_ereal__mult__mono,axiom,
    ! [B: extended_ereal,C: extended_ereal,A3: extended_ereal,D: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B )
     => ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
       => ( ( ord_le1083603963089353582_ereal @ A3 @ B )
         => ( ( ord_le1083603963089353582_ereal @ C @ D )
           => ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A3 @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).

% ereal_mult_mono
thf(fact_1253_ereal__mult__mono_H,axiom,
    ! [A3: extended_ereal,C: extended_ereal,B: extended_ereal,D: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A3 )
     => ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
       => ( ( ord_le1083603963089353582_ereal @ A3 @ B )
         => ( ( ord_le1083603963089353582_ereal @ C @ D )
           => ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A3 @ C ) @ ( times_7703590493115627913_ereal @ B @ D ) ) ) ) ) ) ).

% ereal_mult_mono'
thf(fact_1254_ereal__mult__le__0__iff,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A3 @ B ) @ zero_z2744965634713055877_ereal )
      = ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A3 )
          & ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) )
        | ( ( ord_le1083603963089353582_ereal @ A3 @ zero_z2744965634713055877_ereal )
          & ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) ) ) ) ).

% ereal_mult_le_0_iff
thf(fact_1255_ereal__zero__le__0__iff,axiom,
    ! [A3: extended_ereal,B: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ ( times_7703590493115627913_ereal @ A3 @ B ) )
      = ( ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ A3 )
          & ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ B ) )
        | ( ( ord_le1083603963089353582_ereal @ A3 @ zero_z2744965634713055877_ereal )
          & ( ord_le1083603963089353582_ereal @ B @ zero_z2744965634713055877_ereal ) ) ) ) ).

% ereal_zero_le_0_iff
thf(fact_1256_ereal__mult__left__mono,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
       => ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ C @ A3 ) @ ( times_7703590493115627913_ereal @ C @ B ) ) ) ) ).

% ereal_mult_left_mono
thf(fact_1257_ereal__mult__right__mono,axiom,
    ! [A3: extended_ereal,B: extended_ereal,C: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A3 @ B )
     => ( ( ord_le1083603963089353582_ereal @ zero_z2744965634713055877_ereal @ C )
       => ( ord_le1083603963089353582_ereal @ ( times_7703590493115627913_ereal @ A3 @ C ) @ ( times_7703590493115627913_ereal @ B @ C ) ) ) ) ).

% ereal_mult_right_mono
thf(fact_1258_nat__add__left__cancel__le,axiom,
    ! [K: nat,M5: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N5 ) )
      = ( ord_less_eq_nat @ M5 @ N5 ) ) ).

% nat_add_left_cancel_le
thf(fact_1259_add__top__right__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ X2 @ top_to1496364449551166952nnreal )
      = top_to1496364449551166952nnreal ) ).

% add_top_right_ennreal
thf(fact_1260_add__top__left__ennreal,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ top_to1496364449551166952nnreal @ X2 )
      = top_to1496364449551166952nnreal ) ).

% add_top_left_ennreal
thf(fact_1261_ennreal__add__eq__top,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( plus_p1859984266308609217nnreal @ A3 @ B )
        = top_to1496364449551166952nnreal )
      = ( ( A3 = top_to1496364449551166952nnreal )
        | ( B = top_to1496364449551166952nnreal ) ) ) ).

% ennreal_add_eq_top
thf(fact_1262_ennreal__add__less__top,axiom,
    ! [A3: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le7381754540660121996nnreal @ ( plus_p1859984266308609217nnreal @ A3 @ B ) @ top_to1496364449551166952nnreal )
      = ( ( ord_le7381754540660121996nnreal @ A3 @ top_to1496364449551166952nnreal )
        & ( ord_le7381754540660121996nnreal @ B @ top_to1496364449551166952nnreal ) ) ) ).

% ennreal_add_less_top
thf(fact_1263_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M5: nat,K: nat] :
      ( ! [M7: nat,N3: nat] :
          ( ( ord_less_nat @ M7 @ N3 )
         => ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M5 ) @ K ) @ ( F @ ( plus_plus_nat @ M5 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1264_add__leE,axiom,
    ! [M5: nat,K: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N5 )
     => ~ ( ( ord_less_eq_nat @ M5 @ N5 )
         => ~ ( ord_less_eq_nat @ K @ N5 ) ) ) ).

% add_leE
thf(fact_1265_le__add1,axiom,
    ! [N5: nat,M5: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ N5 @ M5 ) ) ).

% le_add1
thf(fact_1266_le__add2,axiom,
    ! [N5: nat,M5: nat] : ( ord_less_eq_nat @ N5 @ ( plus_plus_nat @ M5 @ N5 ) ) ).

% le_add2
thf(fact_1267_add__leD1,axiom,
    ! [M5: nat,K: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N5 )
     => ( ord_less_eq_nat @ M5 @ N5 ) ) ).

% add_leD1
thf(fact_1268_add__leD2,axiom,
    ! [M5: nat,K: nat,N5: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N5 )
     => ( ord_less_eq_nat @ K @ N5 ) ) ).

% add_leD2
thf(fact_1269_le__Suc__ex,axiom,
    ! [K: nat,L3: nat] :
      ( ( ord_less_eq_nat @ K @ L3 )
     => ? [N3: nat] :
          ( L3
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_1270_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L3: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ) ).

% add_le_mono
thf(fact_1271_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_1272_trans__le__add1,axiom,
    ! [I: nat,J: nat,M5: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).

% trans_le_add1
thf(fact_1273_trans__le__add2,axiom,
    ! [I: nat,J: nat,M5: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).

% trans_le_add2
thf(fact_1274_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N6: nat] :
        ? [K3: nat] :
          ( N6
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add

% Helper facts (7)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X2: b,Y: b] :
      ( ( if_b @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X2: b,Y: b] :
      ( ( if_b @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ( if_real @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y: real] :
      ( ( if_real @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_T,axiom,
    ! [X2: produc7929879615859924995b_real,Y: produc7929879615859924995b_real] :
      ( ( if_Pro2237973481812034749b_real @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Real__Oreal_J_T,axiom,
    ! [X2: produc7929879615859924995b_real,Y: produc7929879615859924995b_real] :
      ( ( if_Pro2237973481812034749b_real @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( member5467075257862553189l_real
    @ ^ [X: produc7929879615859924995b_real] : ( indica3952125700972739438l_real @ a2 @ ( produc7037151529651111887b_real @ X ) )
    @ ( sigma_2402725360379274625l_real @ ( binary496794562119935756b_real @ ( binary867438762418767561re_a_b @ borel_5459123734250506524orel_a @ borel_5459123734250506525orel_b ) @ borel_5078946678739801102l_real ) @ borel_5078946678739801102l_real ) ) ).

%------------------------------------------------------------------------------