%------------------------------------------------------------------------------
% 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    : Youngs_Inequality/0000_Youngs/prob_00061_002402__12834620_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1719 ( 504 unt; 439 typ;   0 def)
%            Number of atoms       : 4188 (1227 equ;   0 cnn)
%            Maximal formula atoms :   12 (   3 avg)
%            Number of connectives : 10584 ( 468   ~; 121   |; 330   &;7720   @)
%                                         (   0 <=>;1945  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   6 avg)
%            Number of types       :   71 (  70 usr)
%            Number of type conns  : 1183 (1183   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  372 ( 369 usr;  68 con; 0-4 aty)
%            Number of variables   : 3732 ( 592   ^;2976   !; 164   ?;3732   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 16:24:15.066
%------------------------------------------------------------------------------
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    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    collec4521028332464157409od_b_b: ( produc1536031394801701132od_b_b > $o ) > set_Pr5139338970096277698od_b_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    collec7702298003248673273_nat_b: ( product_prod_nat_b > $o ) > set_Pr4264375888882495962_nat_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    collec5699578864178162003_b_nat: ( product_prod_b_nat > $o ) > set_Pr1307281990691478580_b_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    collec3841725734524963642od_b_b: ( produc2840042325109449167od_b_b > $o ) > set_Pr4323519195528460463od_b_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    collec548942219715005266od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    collec4858231573021281987nnreal: ( set_Ex3793607809372303086nnreal > $o ) > set_se4580700918925141924nnreal ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
    collec3025963938859124489_b_nat: ( set_Pr1307281990691478580_b_nat > $o ) > set_se3342737189298382570_b_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    collec1108733003560609586od_b_b: ( set_Product_prod_b_b > $o ) > set_se5865286889322728103od_b_b ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
    collect_set_b: ( set_b > $o ) > set_set_b ).

thf(sy_c_Set_OCollect_001tf__b,type,
    collect_b: ( b > $o ) > set_b ).

thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
    filter_nat: ( nat > $o ) > set_nat > set_nat ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    filter5640266504077782706at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
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thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    filter4863506568596807603_nat_b: ( product_prod_nat_b > $o ) > set_Pr4264375888882495962_nat_b > set_Pr4264375888882495962_nat_b ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    filter2860787429526296333_b_nat: ( product_prod_b_nat > $o ) > set_Pr1307281990691478580_b_nat > set_Pr1307281990691478580_b_nat ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    filter6336438293541724672od_b_b: ( produc2840042325109449167od_b_b > $o ) > set_Pr4323519195528460463od_b_b > set_Pr4323519195528460463od_b_b ).

thf(sy_c_Set_Ofilter_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    filter1593123213581277208od_b_b: ( product_prod_b_b > $o ) > set_Product_prod_b_b > set_Product_prod_b_b ).

thf(sy_c_Set_Ofilter_001tf__b,type,
    filter_b: ( b > $o ) > set_b > set_b ).

thf(sy_c_Set_Oimage_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    image_8394674774369097847nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Set_Oimage_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Real__Oereal,type,
    image_6393943237584228047_ereal: ( extend8495563244428889912nnreal > extended_ereal ) > set_Ex3793607809372303086nnreal > set_Extended_ereal ).

thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__Extended____Nonnegative____Real__Oennreal,type,
    image_8614087454967683265nnreal: ( extended_ereal > extend8495563244428889912nnreal ) > set_Extended_ereal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__Extended____Real__Oereal,type,
    image_6042159593519690757_ereal: ( extended_ereal > extended_ereal ) > set_Extended_ereal > set_Extended_ereal ).

thf(sy_c_Set_Oimage_001t__Extended____Real__Oereal_001t__Nat__Onat,type,
    image_7659842161140344153al_nat: ( extended_ereal > nat ) > set_Extended_ereal > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
    image_8459861568512453903nnreal: ( nat > extend8495563244428889912nnreal ) > set_nat > set_Ex3793607809372303086nnreal ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Extended____Real__Oereal,type,
    image_4309273772856505399_ereal: ( nat > extended_ereal ) > set_nat > set_Extended_ereal ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    image_5000473625437735139od_b_b: ( nat > produc1536031394801701132od_b_b ) > set_nat > set_Pr5139338970096277698od_b_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    image_8668673924000913915_nat_b: ( nat > product_prod_nat_b ) > set_nat > set_Pr4264375888882495962_nat_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    image_6665954784930402645_b_nat: ( nat > product_prod_b_nat ) > set_nat > set_Pr1307281990691478580_b_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    image_2386253535795868088od_b_b: ( nat > produc2840042325109449167od_b_b ) > set_nat > set_Pr4323519195528460463od_b_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_6808858347418066896od_b_b: ( nat > product_prod_b_b ) > set_nat > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__b,type,
    image_nat_b: ( nat > b ) > set_nat > set_b ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001t__Nat__Onat,type,
    image_6770982514055950706_b_nat: ( product_prod_b_b > nat ) > set_Product_prod_b_b > set_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_Itf__b_Mtf__b_J_001tf__b,type,
    image_8398514867482601949_b_b_b: ( product_prod_b_b > b ) > set_Product_prod_b_b > set_b ).

thf(sy_c_Set_Oimage_001tf__b_001t__Nat__Onat,type,
    image_b_nat: ( b > nat ) > set_b > set_nat ).

thf(sy_c_Set_Oimage_001tf__b_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    image_3973729904588732333od_b_b: ( b > product_prod_b_b ) > set_b > set_Product_prod_b_b ).

thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
    image_b_b: ( b > b ) > set_b > set_b ).

thf(sy_c_Set_Ois__empty_001t__Extended____Nonnegative____Real__Oennreal,type,
    is_emp182806100662350310nnreal: set_Ex3793607809372303086nnreal > $o ).

thf(sy_c_Set_Ois__empty_001t__Extended____Real__Oereal,type,
    is_emp6845480677552319904_ereal: set_Extended_ereal > $o ).

thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
    member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).

thf(sy_c_member_001t__Extended____Real__Oereal,type,
    member2350847679896131959_ereal: extended_ereal > set_Extended_ereal > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member5954771037092721571od_b_b: produc1536031394801701132od_b_b > set_Pr5139338970096277698od_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J,type,
    member8962352056413324475_nat_b: product_prod_nat_b > set_Pr4264375888882495962_nat_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    member6959632917342813205_b_nat: product_prod_b_nat > set_Pr1307281990691478580_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J,type,
    member1867066792959145720od_b_b: produc2840042325109449167od_b_b > set_Pr4323519195528460463od_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mtf__b_J,type,
    member7862447936710763792od_b_b: product_prod_b_b > set_Product_prod_b_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).

thf(sy_c_member_001t__Set__Oset_It__Extended____Real__Oereal_J,type,
    member5519481007471526743_ereal: set_Extended_ereal > set_se6634062954251873166_ereal > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
    member3392960547284361049od_b_b: set_Pr5139338970096277698od_b_b > set_se6506758776816089464od_b_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__b_J_J,type,
    member7364522358610929521_nat_b: set_Pr4264375888882495962_nat_b > set_se5362224065136839056_nat_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
    member4407428460419912139_b_nat: set_Pr1307281990691478580_b_nat > set_se3342737189298382570_b_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__b_Mtf__b_J_J_J,type,
    member3004383007682734552od_b_b: set_Pr4323519195528460463od_b_b > set_se7830547602675927951od_b_b > $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_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

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

thf(sy_v_I,type,
    i: set_b ).

% Relevant facts (1274)
thf(fact_0_assms_I1_J,axiom,
    finite_finite_b @ i ).

% assms(1)
thf(fact_1_finite__SigmaI,axiom,
    ! [A: set_b,B: b > set_nat] :
      ( ( finite_finite_b @ A )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite7880342692102525205_b_nat @ ( product_Sigma_b_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_2_finite__SigmaI,axiom,
    ! [A: set_nat,B: nat > set_b] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite659689794318260667_nat_b @ ( product_Sigma_nat_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_3_finite__SigmaI,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_4_finite__SigmaI,axiom,
    ! [A: set_b,B: b > set_b] :
      ( ( finite_finite_b @ A )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite3757003017338540048od_b_b @ ( product_Sigma_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_5_finite__SigmaI,axiom,
    ! [A: set_Product_prod_b_b,B: product_prod_b_b > set_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ! [A2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite1344950077393947688_b_b_b @ ( produc7340210106073890848_b_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_6_finite__SigmaI,axiom,
    ! [A: set_Product_prod_b_b,B: product_prod_b_b > set_nat] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ! [A2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite800119761285742333_b_nat @ ( produc7989754787496632431_b_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_7_finite__SigmaI,axiom,
    ! [A: set_b,B: b > set_Product_prod_b_b] :
      ( ( finite_finite_b @ A )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite3757003017338540048od_b_b @ ( B @ A2 ) ) )
       => ( finite4644902770518909432od_b_b @ ( produc2915425143180021232od_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_8_finite__SigmaI,axiom,
    ! [A: set_nat,B: nat > set_Product_prod_b_b] :
      ( ( finite_finite_nat @ A )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite3757003017338540048od_b_b @ ( B @ A2 ) ) )
       => ( finite7768965217515309219od_b_b @ ( produc8027630620858748621od_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_9_finite__SigmaI,axiom,
    ! [A: set_Product_prod_b_b,B: product_prod_b_b > set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ! [A2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ A2 @ A )
           => ( finite3757003017338540048od_b_b @ ( B @ A2 ) ) )
       => ( finite2059881226400990736od_b_b @ ( produc5716436088342041352od_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_10_finite__SigmaI,axiom,
    ! [A: set_b,B: b > set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_b @ A )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite6177210948735845034at_nat @ ( B @ A2 ) ) )
       => ( finite6415899795318129170at_nat @ ( produc4667218688476393610at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI
thf(fact_11_finite__cartesian__product,axiom,
    ! [A: set_b,B: set_nat] :
      ( ( finite_finite_b @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite7880342692102525205_b_nat
          @ ( product_Sigma_b_nat @ A
            @ ^ [Uu: b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_12_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_b] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_b @ B )
       => ( finite659689794318260667_nat_b
          @ ( product_Sigma_nat_b @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_13_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_14_finite__cartesian__product,axiom,
    ! [A: set_b,B: set_b] :
      ( ( finite_finite_b @ A )
     => ( ( finite_finite_b @ B )
       => ( finite3757003017338540048od_b_b
          @ ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_15_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ( finite_finite_b @ B )
       => ( finite1344950077393947688_b_b_b
          @ ( produc7340210106073890848_b_b_b @ A
            @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_16_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_nat] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ( finite_finite_nat @ B )
       => ( finite800119761285742333_b_nat
          @ ( produc7989754787496632431_b_nat @ A
            @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_17_finite__cartesian__product,axiom,
    ! [A: set_b,B: set_Product_prod_b_b] :
      ( ( finite_finite_b @ A )
     => ( ( finite3757003017338540048od_b_b @ B )
       => ( finite4644902770518909432od_b_b
          @ ( produc2915425143180021232od_b_b @ A
            @ ^ [Uu: b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_18_finite__cartesian__product,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b] :
      ( ( finite_finite_nat @ A )
     => ( ( finite3757003017338540048od_b_b @ B )
       => ( finite7768965217515309219od_b_b
          @ ( produc8027630620858748621od_b_b @ A
            @ ^ [Uu: nat] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_19_finite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ( finite3757003017338540048od_b_b @ B )
       => ( finite2059881226400990736od_b_b
          @ ( produc5716436088342041352od_b_b @ A
            @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_20_finite__cartesian__product,axiom,
    ! [A: set_b,B: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_b @ A )
     => ( ( finite6177210948735845034at_nat @ B )
       => ( finite6415899795318129170at_nat
          @ ( produc4667218688476393610at_nat @ A
            @ ^ [Uu: b] : B ) ) ) ) ).

% finite_cartesian_product
thf(fact_21_infinite__cartesian__product,axiom,
    ! [A: set_b,B: set_nat] :
      ( ~ ( finite_finite_b @ A )
     => ( ~ ( finite_finite_nat @ B )
       => ~ ( finite7880342692102525205_b_nat
            @ ( product_Sigma_b_nat @ A
              @ ^ [Uu: b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_22_infinite__cartesian__product,axiom,
    ! [A: set_nat,B: set_b] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_b @ B )
       => ~ ( finite659689794318260667_nat_b
            @ ( product_Sigma_nat_b @ A
              @ ^ [Uu: nat] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_23_infinite__cartesian__product,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_nat @ B )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A
              @ ^ [Uu: nat] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_24_infinite__cartesian__product,axiom,
    ! [A: set_b,B: set_b] :
      ( ~ ( finite_finite_b @ A )
     => ( ~ ( finite_finite_b @ B )
       => ~ ( finite3757003017338540048od_b_b
            @ ( product_Sigma_b_b @ A
              @ ^ [Uu: b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_25_infinite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_b] :
      ( ~ ( finite3757003017338540048od_b_b @ A )
     => ( ~ ( finite_finite_b @ B )
       => ~ ( finite1344950077393947688_b_b_b
            @ ( produc7340210106073890848_b_b_b @ A
              @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_26_infinite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_nat] :
      ( ~ ( finite3757003017338540048od_b_b @ A )
     => ( ~ ( finite_finite_nat @ B )
       => ~ ( finite800119761285742333_b_nat
            @ ( produc7989754787496632431_b_nat @ A
              @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_27_infinite__cartesian__product,axiom,
    ! [A: set_b,B: set_Product_prod_b_b] :
      ( ~ ( finite_finite_b @ A )
     => ( ~ ( finite3757003017338540048od_b_b @ B )
       => ~ ( finite4644902770518909432od_b_b
            @ ( produc2915425143180021232od_b_b @ A
              @ ^ [Uu: b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_28_infinite__cartesian__product,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite3757003017338540048od_b_b @ B )
       => ~ ( finite7768965217515309219od_b_b
            @ ( produc8027630620858748621od_b_b @ A
              @ ^ [Uu: nat] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_29_infinite__cartesian__product,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
      ( ~ ( finite3757003017338540048od_b_b @ A )
     => ( ~ ( finite3757003017338540048od_b_b @ B )
       => ~ ( finite2059881226400990736od_b_b
            @ ( produc5716436088342041352od_b_b @ A
              @ ^ [Uu: product_prod_b_b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_30_infinite__cartesian__product,axiom,
    ! [A: set_b,B: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_b @ A )
     => ( ~ ( finite6177210948735845034at_nat @ B )
       => ~ ( finite6415899795318129170at_nat
            @ ( produc4667218688476393610at_nat @ A
              @ ^ [Uu: b] : B ) ) ) ) ).

% infinite_cartesian_product
thf(fact_31_finite__Collect__conjI,axiom,
    ! [P: produc1536031394801701132od_b_b > $o,Q: produc1536031394801701132od_b_b > $o] :
      ( ( ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ P ) )
        | ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ Q ) ) )
     => ( finite7768965217515309219od_b_b
        @ ( collec4521028332464157409od_b_b
          @ ^ [X: produc1536031394801701132od_b_b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_32_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_33_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_b > $o,Q: product_prod_nat_b > $o] :
      ( ( ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ P ) )
        | ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ Q ) ) )
     => ( finite659689794318260667_nat_b
        @ ( collec7702298003248673273_nat_b
          @ ^ [X: product_prod_nat_b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_34_finite__Collect__conjI,axiom,
    ! [P: produc2840042325109449167od_b_b > $o,Q: produc2840042325109449167od_b_b > $o] :
      ( ( ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ P ) )
        | ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ Q ) ) )
     => ( finite4644902770518909432od_b_b
        @ ( collec3841725734524963642od_b_b
          @ ^ [X: produc2840042325109449167od_b_b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_35_finite__Collect__conjI,axiom,
    ! [P: product_prod_b_nat > $o,Q: product_prod_b_nat > $o] :
      ( ( ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ P ) )
        | ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ Q ) ) )
     => ( finite7880342692102525205_b_nat
        @ ( collec5699578864178162003_b_nat
          @ ^ [X: product_prod_b_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_36_finite__Collect__conjI,axiom,
    ! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ( ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ P ) )
        | ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ Q ) ) )
     => ( finite3757003017338540048od_b_b
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_37_finite__Collect__conjI,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( ( finite_finite_b @ ( collect_b @ P ) )
        | ( finite_finite_b @ ( collect_b @ Q ) ) )
     => ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_38_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_39_finite__Collect__disjI,axiom,
    ! [P: produc1536031394801701132od_b_b > $o,Q: produc1536031394801701132od_b_b > $o] :
      ( ( finite7768965217515309219od_b_b
        @ ( collec4521028332464157409od_b_b
          @ ^ [X: produc1536031394801701132od_b_b] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ P ) )
        & ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_40_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_41_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_b > $o,Q: product_prod_nat_b > $o] :
      ( ( finite659689794318260667_nat_b
        @ ( collec7702298003248673273_nat_b
          @ ^ [X: product_prod_nat_b] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ P ) )
        & ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_42_finite__Collect__disjI,axiom,
    ! [P: produc2840042325109449167od_b_b > $o,Q: produc2840042325109449167od_b_b > $o] :
      ( ( finite4644902770518909432od_b_b
        @ ( collec3841725734524963642od_b_b
          @ ^ [X: produc2840042325109449167od_b_b] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ P ) )
        & ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_43_finite__Collect__disjI,axiom,
    ! [P: product_prod_b_nat > $o,Q: product_prod_b_nat > $o] :
      ( ( finite7880342692102525205_b_nat
        @ ( collec5699578864178162003_b_nat
          @ ^ [X: product_prod_b_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ P ) )
        & ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_44_finite__Collect__disjI,axiom,
    ! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ( finite3757003017338540048od_b_b
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ P ) )
        & ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_45_finite__Collect__disjI,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_b @ ( collect_b @ P ) )
        & ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_46_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_47_Sigma__cong,axiom,
    ! [A: set_nat,B: set_nat,C: nat > set_Product_prod_b_b,D: nat > set_Product_prod_b_b] :
      ( ( A = B )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( produc8027630620858748621od_b_b @ A @ C )
          = ( produc8027630620858748621od_b_b @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_48_Sigma__cong,axiom,
    ! [A: set_nat,B: set_nat,C: nat > set_nat,D: nat > set_nat] :
      ( ( A = B )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( produc457027306803732586at_nat @ A @ C )
          = ( produc457027306803732586at_nat @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_49_Sigma__cong,axiom,
    ! [A: set_nat,B: set_nat,C: nat > set_b,D: nat > set_b] :
      ( ( A = B )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( product_Sigma_nat_b @ A @ C )
          = ( product_Sigma_nat_b @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_50_Sigma__cong,axiom,
    ! [A: set_b,B: set_b,C: b > set_Product_prod_b_b,D: b > set_Product_prod_b_b] :
      ( ( A = B )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( produc2915425143180021232od_b_b @ A @ C )
          = ( produc2915425143180021232od_b_b @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_51_Sigma__cong,axiom,
    ! [A: set_b,B: set_b,C: b > set_nat,D: b > set_nat] :
      ( ( A = B )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( product_Sigma_b_nat @ A @ C )
          = ( product_Sigma_b_nat @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_52_Sigma__cong,axiom,
    ! [A: set_b,B: set_b,C: b > set_b,D: b > set_b] :
      ( ( A = B )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ B )
           => ( ( C @ X2 )
              = ( D @ X2 ) ) )
       => ( ( product_Sigma_b_b @ A @ C )
          = ( product_Sigma_b_b @ B @ D ) ) ) ) ).

% Sigma_cong
thf(fact_53_Times__eq__cancel2,axiom,
    ! [X3: product_prod_b_b,C: set_Product_prod_b_b,A: set_nat,B: set_nat] :
      ( ( member7862447936710763792od_b_b @ X3 @ C )
     => ( ( ( produc8027630620858748621od_b_b @ A
            @ ^ [Uu: nat] : C )
          = ( produc8027630620858748621od_b_b @ B
            @ ^ [Uu: nat] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_54_Times__eq__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ X3 @ C )
     => ( ( ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : C )
          = ( produc457027306803732586at_nat @ B
            @ ^ [Uu: nat] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_55_Times__eq__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_nat,B: set_nat] :
      ( ( member_b @ X3 @ C )
     => ( ( ( product_Sigma_nat_b @ A
            @ ^ [Uu: nat] : C )
          = ( product_Sigma_nat_b @ B
            @ ^ [Uu: nat] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_56_Times__eq__cancel2,axiom,
    ! [X3: product_prod_b_b,C: set_Product_prod_b_b,A: set_b,B: set_b] :
      ( ( member7862447936710763792od_b_b @ X3 @ C )
     => ( ( ( produc2915425143180021232od_b_b @ A
            @ ^ [Uu: b] : C )
          = ( produc2915425143180021232od_b_b @ B
            @ ^ [Uu: b] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_57_Times__eq__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_b,B: set_b] :
      ( ( member_nat @ X3 @ C )
     => ( ( ( product_Sigma_b_nat @ A
            @ ^ [Uu: b] : C )
          = ( product_Sigma_b_nat @ B
            @ ^ [Uu: b] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_58_Times__eq__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_b,B: set_b] :
      ( ( member_b @ X3 @ C )
     => ( ( ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : C )
          = ( product_Sigma_b_b @ B
            @ ^ [Uu: b] : C ) )
        = ( A = B ) ) ) ).

% Times_eq_cancel2
thf(fact_59_not__finite__existsD,axiom,
    ! [P: produc1536031394801701132od_b_b > $o] :
      ( ~ ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ P ) )
     => ? [X_1: produc1536031394801701132od_b_b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_60_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_61_not__finite__existsD,axiom,
    ! [P: product_prod_nat_b > $o] :
      ( ~ ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ P ) )
     => ? [X_1: product_prod_nat_b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_62_not__finite__existsD,axiom,
    ! [P: produc2840042325109449167od_b_b > $o] :
      ( ~ ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ P ) )
     => ? [X_1: produc2840042325109449167od_b_b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_63_not__finite__existsD,axiom,
    ! [P: product_prod_b_nat > $o] :
      ( ~ ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ P ) )
     => ? [X_1: product_prod_b_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_64_not__finite__existsD,axiom,
    ! [P: product_prod_b_b > $o] :
      ( ~ ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ P ) )
     => ? [X_1: product_prod_b_b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_65_not__finite__existsD,axiom,
    ! [P: b > $o] :
      ( ~ ( finite_finite_b @ ( collect_b @ P ) )
     => ? [X_1: b] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_66_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_67_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_b,R: b > b > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: b] :
              ( ( member_b @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_68_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_nat,R: b > nat > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_69_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_b,R: nat > b > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: b] :
              ( ( member_b @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_70_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_71_pigeonhole__infinite__rel,axiom,
    ! [A: set_Product_prod_b_b,B: set_b,R: product_prod_b_b > b > $o] :
      ( ~ ( finite3757003017338540048od_b_b @ A )
     => ( ( finite_finite_b @ B )
       => ( ! [X2: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X2 @ A )
             => ? [Xa: b] :
                  ( ( member_b @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: b] :
              ( ( member_b @ X2 @ B )
              & ~ ( finite3757003017338540048od_b_b
                  @ ( collec548942219715005266od_b_b
                    @ ^ [A3: product_prod_b_b] :
                        ( ( member7862447936710763792od_b_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_72_pigeonhole__infinite__rel,axiom,
    ! [A: set_Product_prod_b_b,B: set_nat,R: product_prod_b_b > nat > $o] :
      ( ~ ( finite3757003017338540048od_b_b @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite3757003017338540048od_b_b
                  @ ( collec548942219715005266od_b_b
                    @ ^ [A3: product_prod_b_b] :
                        ( ( member7862447936710763792od_b_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_73_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_Product_prod_b_b,R: b > product_prod_b_b > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite3757003017338540048od_b_b @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: product_prod_b_b] :
                  ( ( member7862447936710763792od_b_b @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_74_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_Pr1261947904930325089at_nat,R: b > product_prod_nat_nat > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite6177210948735845034at_nat @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_75_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_Pr4264375888882495962_nat_b,R: b > product_prod_nat_b > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite659689794318260667_nat_b @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: product_prod_nat_b] :
                  ( ( member8962352056413324475_nat_b @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: product_prod_nat_b] :
              ( ( member8962352056413324475_nat_b @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_76_pigeonhole__infinite__rel,axiom,
    ! [A: set_b,B: set_Pr1307281990691478580_b_nat,R: b > product_prod_b_nat > $o] :
      ( ~ ( finite_finite_b @ A )
     => ( ( finite7880342692102525205_b_nat @ B )
       => ( ! [X2: b] :
              ( ( member_b @ X2 @ A )
             => ? [Xa: product_prod_b_nat] :
                  ( ( member6959632917342813205_b_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: product_prod_b_nat] :
              ( ( member6959632917342813205_b_nat @ X2 @ B )
              & ~ ( finite_finite_b
                  @ ( collect_b
                    @ ^ [A3: b] :
                        ( ( member_b @ A3 @ A )
                        & ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_77_member__product,axiom,
    ! [X3: product_prod_b_b,A: set_b,B: set_b] :
      ( ( member7862447936710763792od_b_b @ X3 @ ( product_product_b_b @ A @ B ) )
      = ( member7862447936710763792od_b_b @ X3
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B ) ) ) ).

% member_product
thf(fact_78_member__product,axiom,
    ! [X3: produc1536031394801701132od_b_b,A: set_nat,B: set_Product_prod_b_b] :
      ( ( member5954771037092721571od_b_b @ X3 @ ( produc4251392375023618883od_b_b @ A @ B ) )
      = ( member5954771037092721571od_b_b @ X3
        @ ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% member_product
thf(fact_79_member__product,axiom,
    ! [X3: product_prod_nat_nat,A: set_nat,B: set_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( produc929334515565554804at_nat @ A @ B ) )
      = ( member8440522571783428010at_nat @ X3
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% member_product
thf(fact_80_member__product,axiom,
    ! [X3: product_prod_nat_b,A: set_nat,B: set_b] :
      ( ( member8962352056413324475_nat_b @ X3 @ ( produc840569836383677275_nat_b @ A @ B ) )
      = ( member8962352056413324475_nat_b @ X3
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% member_product
thf(fact_81_member__product,axiom,
    ! [X3: produc2840042325109449167od_b_b,A: set_b,B: set_Product_prod_b_b] :
      ( ( member1867066792959145720od_b_b @ X3 @ ( produc8433888527511044858od_b_b @ A @ B ) )
      = ( member1867066792959145720od_b_b @ X3
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : B ) ) ) ).

% member_product
thf(fact_82_member__product,axiom,
    ! [X3: product_prod_b_nat,A: set_b,B: set_nat] :
      ( ( member6959632917342813205_b_nat @ X3 @ ( produc7267454701016561149_b_nat @ A @ B ) )
      = ( member6959632917342813205_b_nat @ X3
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B ) ) ) ).

% member_product
thf(fact_83_Product__Type_Oproduct__def,axiom,
    ( product_product_b_b
    = ( ^ [A4: set_b,B2: set_b] :
          ( product_Sigma_b_b @ A4
          @ ^ [Uu: b] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_84_Product__Type_Oproduct__def,axiom,
    ( produc4251392375023618883od_b_b
    = ( ^ [A4: set_nat,B2: set_Product_prod_b_b] :
          ( produc8027630620858748621od_b_b @ A4
          @ ^ [Uu: nat] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_85_Product__Type_Oproduct__def,axiom,
    ( produc929334515565554804at_nat
    = ( ^ [A4: set_nat,B2: set_nat] :
          ( produc457027306803732586at_nat @ A4
          @ ^ [Uu: nat] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_86_Product__Type_Oproduct__def,axiom,
    ( produc840569836383677275_nat_b
    = ( ^ [A4: set_nat,B2: set_b] :
          ( product_Sigma_nat_b @ A4
          @ ^ [Uu: nat] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_87_Product__Type_Oproduct__def,axiom,
    ( produc8433888527511044858od_b_b
    = ( ^ [A4: set_b,B2: set_Product_prod_b_b] :
          ( produc2915425143180021232od_b_b @ A4
          @ ^ [Uu: b] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_88_Product__Type_Oproduct__def,axiom,
    ( produc7267454701016561149_b_nat
    = ( ^ [A4: set_b,B2: set_nat] :
          ( product_Sigma_b_nat @ A4
          @ ^ [Uu: b] : B2 ) ) ) ).

% Product_Type.product_def
thf(fact_89_finite__cartesian__product__iff,axiom,
    ! [A: set_b,B: set_b] :
      ( ( finite3757003017338540048od_b_b
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B ) )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bot_set_b )
        | ( ( finite_finite_b @ A )
          & ( finite_finite_b @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_90_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_91_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_b] :
      ( ( finite659689794318260667_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_b )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_b @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_92_finite__cartesian__product__iff,axiom,
    ! [A: set_b,B: set_nat] :
      ( ( finite7880342692102525205_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B ) )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bot_set_nat )
        | ( ( finite_finite_b @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_93_finite__cartesian__product__iff,axiom,
    ! [A: set_b,B: set_Ex3793607809372303086nnreal] :
      ( ( finite6788980017087891581nnreal
        @ ( produc151940477230798575nnreal @ A
          @ ^ [Uu: b] : B ) )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bo4854962954004695426nnreal )
        | ( ( finite_finite_b @ A )
          & ( finite3782138982310603983nnreal @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_94_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_Ex3793607809372303086nnreal] :
      ( ( finite7647684622998704658nnreal
        @ ( produc648375469580864466nnreal @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bo4854962954004695426nnreal )
        | ( ( finite_finite_nat @ A )
          & ( finite3782138982310603983nnreal @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_95_finite__cartesian__product__iff,axiom,
    ! [A: set_b,B: set_Extended_ereal] :
      ( ( finite6405620804434884703_ereal
        @ ( produc4178612860395885143_ereal @ A
          @ ^ [Uu: b] : B ) )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bo8367695208629047834_ereal )
        | ( ( finite_finite_b @ A )
          & ( finite7198162374296863863_ereal @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_96_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B: set_Extended_ereal] :
      ( ( finite2240905580840527242_ereal
        @ ( produc870331913724930228_ereal @ A
          @ ^ [Uu: nat] : B ) )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bo8367695208629047834_ereal )
        | ( ( finite_finite_nat @ A )
          & ( finite7198162374296863863_ereal @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_97_finite__cartesian__product__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_b] :
      ( ( finite5295106166121109587real_b
        @ ( produc6071992953914654589real_b @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B ) )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( B = bot_bot_set_b )
        | ( ( finite3782138982310603983nnreal @ A )
          & ( finite_finite_b @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_98_finite__cartesian__product__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_nat] :
      ( ( finite5801226769166306834al_nat
        @ ( produc5422075910247723986al_nat @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B ) )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( B = bot_bot_set_nat )
        | ( ( finite3782138982310603983nnreal @ A )
          & ( finite_finite_nat @ B ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_99_finite__cartesian__productD2,axiom,
    ! [A: set_b,B: set_b] :
      ( ( finite3757003017338540048od_b_b
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( A != bot_bot_set_b )
       => ( finite_finite_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_100_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_101_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_b] :
      ( ( finite659689794318260667_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite_finite_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_102_finite__cartesian__productD2,axiom,
    ! [A: set_b,B: set_nat] :
      ( ( finite7880342692102525205_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B ) )
     => ( ( A != bot_bot_set_b )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_103_finite__cartesian__productD2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_b] :
      ( ( finite5295106166121109587real_b
        @ ( produc6071992953914654589real_b @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B ) )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( finite_finite_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_104_finite__cartesian__productD2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_nat] :
      ( ( finite5801226769166306834al_nat
        @ ( produc5422075910247723986al_nat @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B ) )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_105_finite__cartesian__productD2,axiom,
    ! [A: set_Extended_ereal,B: set_b] :
      ( ( finite7540987213771331073real_b
        @ ( produc2583502849437520505real_b @ A
          @ ^ [Uu: extended_ereal] : B ) )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( finite_finite_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_106_finite__cartesian__productD2,axiom,
    ! [A: set_Extended_ereal,B: set_nat] :
      ( ( finite9115718422570557668al_nat
        @ ( produc4220900302008768982al_nat @ A
          @ ^ [Uu: extended_ereal] : B ) )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_107_finite__cartesian__productD2,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b] :
      ( ( finite7768965217515309219od_b_b
        @ ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( A != bot_bot_set_nat )
       => ( finite3757003017338540048od_b_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_108_finite__cartesian__productD2,axiom,
    ! [A: set_b,B: set_Product_prod_b_b] :
      ( ( finite4644902770518909432od_b_b
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( A != bot_bot_set_b )
       => ( finite3757003017338540048od_b_b @ B ) ) ) ).

% finite_cartesian_productD2
thf(fact_109_finite__cartesian__productD1,axiom,
    ! [A: set_b,B: set_b] :
      ( ( finite3757003017338540048od_b_b
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( B != bot_bot_set_b )
       => ( finite_finite_b @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_110_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bot_set_nat )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_111_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_b] :
      ( ( finite659689794318260667_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bot_set_b )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_112_finite__cartesian__productD1,axiom,
    ! [A: set_b,B: set_nat] :
      ( ( finite7880342692102525205_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B ) )
     => ( ( B != bot_bot_set_nat )
       => ( finite_finite_b @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_113_finite__cartesian__productD1,axiom,
    ! [A: set_b,B: set_Ex3793607809372303086nnreal] :
      ( ( finite6788980017087891581nnreal
        @ ( produc151940477230798575nnreal @ A
          @ ^ [Uu: b] : B ) )
     => ( ( B != bot_bo4854962954004695426nnreal )
       => ( finite_finite_b @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_114_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_Ex3793607809372303086nnreal] :
      ( ( finite7647684622998704658nnreal
        @ ( produc648375469580864466nnreal @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bo4854962954004695426nnreal )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_115_finite__cartesian__productD1,axiom,
    ! [A: set_b,B: set_Extended_ereal] :
      ( ( finite6405620804434884703_ereal
        @ ( produc4178612860395885143_ereal @ A
          @ ^ [Uu: b] : B ) )
     => ( ( B != bot_bo8367695208629047834_ereal )
       => ( finite_finite_b @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_116_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_Extended_ereal] :
      ( ( finite2240905580840527242_ereal
        @ ( produc870331913724930228_ereal @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bo8367695208629047834_ereal )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_117_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b] :
      ( ( finite7768965217515309219od_b_b
        @ ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( B != bot_bo2792761326896053555od_b_b )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_118_finite__cartesian__productD1,axiom,
    ! [A: set_b,B: set_Product_prod_b_b] :
      ( ( finite4644902770518909432od_b_b
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( B != bot_bo2792761326896053555od_b_b )
       => ( finite_finite_b @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_119_finite__SigmaI2,axiom,
    ! [A: set_b,B: b > set_b] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_b ) ) ) )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite3757003017338540048od_b_b @ ( product_Sigma_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_120_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_121_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_b] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_b ) ) ) )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite659689794318260667_nat_b @ ( product_Sigma_nat_b @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_122_finite__SigmaI2,axiom,
    ! [A: set_b,B: b > set_nat] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite7880342692102525205_b_nat @ ( product_Sigma_b_nat @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_123_finite__SigmaI2,axiom,
    ! [A: set_b,B: b > set_Ex3793607809372303086nnreal] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( ( B @ X )
               != bot_bo4854962954004695426nnreal ) ) ) )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite3782138982310603983nnreal @ ( B @ A2 ) ) )
       => ( finite6788980017087891581nnreal @ ( produc151940477230798575nnreal @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_124_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_Ex3793607809372303086nnreal] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B @ X )
               != bot_bo4854962954004695426nnreal ) ) ) )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite3782138982310603983nnreal @ ( B @ A2 ) ) )
       => ( finite7647684622998704658nnreal @ ( produc648375469580864466nnreal @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_125_finite__SigmaI2,axiom,
    ! [A: set_b,B: b > set_Extended_ereal] :
      ( ( finite_finite_b
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( ( B @ X )
               != bot_bo8367695208629047834_ereal ) ) ) )
     => ( ! [A2: b] :
            ( ( member_b @ A2 @ A )
           => ( finite7198162374296863863_ereal @ ( B @ A2 ) ) )
       => ( finite6405620804434884703_ereal @ ( produc4178612860395885143_ereal @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_126_finite__SigmaI2,axiom,
    ! [A: set_nat,B: nat > set_Extended_ereal] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B @ X )
               != bot_bo8367695208629047834_ereal ) ) ) )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A )
           => ( finite7198162374296863863_ereal @ ( B @ A2 ) ) )
       => ( finite2240905580840527242_ereal @ ( produc870331913724930228_ereal @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_127_finite__SigmaI2,axiom,
    ! [A: set_Product_prod_b_b,B: product_prod_b_b > set_b] :
      ( ( finite3757003017338540048od_b_b
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_b ) ) ) )
     => ( ! [A2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ A2 @ A )
           => ( finite_finite_b @ ( B @ A2 ) ) )
       => ( finite1344950077393947688_b_b_b @ ( produc7340210106073890848_b_b_b @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_128_finite__SigmaI2,axiom,
    ! [A: set_Product_prod_b_b,B: product_prod_b_b > set_nat] :
      ( ( finite3757003017338540048od_b_b
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ A )
              & ( ( B @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ A2 @ A )
           => ( finite_finite_nat @ ( B @ A2 ) ) )
       => ( finite800119761285742333_b_nat @ ( produc7989754787496632431_b_nat @ A @ B ) ) ) ) ).

% finite_SigmaI2
thf(fact_129_finite__filter,axiom,
    ! [S: set_Product_prod_b_b,P: product_prod_b_b > $o] :
      ( ( finite3757003017338540048od_b_b @ S )
     => ( finite3757003017338540048od_b_b @ ( filter1593123213581277208od_b_b @ P @ S ) ) ) ).

% finite_filter
thf(fact_130_finite__filter,axiom,
    ! [S: set_b,P: b > $o] :
      ( ( finite_finite_b @ S )
     => ( finite_finite_b @ ( filter_b @ P @ S ) ) ) ).

% finite_filter
thf(fact_131_finite__filter,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( finite_finite_nat @ ( filter_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_132_finite__filter,axiom,
    ! [S: set_Pr5139338970096277698od_b_b,P: produc1536031394801701132od_b_b > $o] :
      ( ( finite7768965217515309219od_b_b @ S )
     => ( finite7768965217515309219od_b_b @ ( filter8572989054268757659od_b_b @ P @ S ) ) ) ).

% finite_filter
thf(fact_133_finite__filter,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( finite6177210948735845034at_nat @ ( filter5640266504077782706at_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_134_finite__filter,axiom,
    ! [S: set_Pr4264375888882495962_nat_b,P: product_prod_nat_b > $o] :
      ( ( finite659689794318260667_nat_b @ S )
     => ( finite659689794318260667_nat_b @ ( filter4863506568596807603_nat_b @ P @ S ) ) ) ).

% finite_filter
thf(fact_135_finite__filter,axiom,
    ! [S: set_Pr4323519195528460463od_b_b,P: produc2840042325109449167od_b_b > $o] :
      ( ( finite4644902770518909432od_b_b @ S )
     => ( finite4644902770518909432od_b_b @ ( filter6336438293541724672od_b_b @ P @ S ) ) ) ).

% finite_filter
thf(fact_136_finite__filter,axiom,
    ! [S: set_Pr1307281990691478580_b_nat,P: product_prod_b_nat > $o] :
      ( ( finite7880342692102525205_b_nat @ S )
     => ( finite7880342692102525205_b_nat @ ( filter2860787429526296333_b_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_137_finite__Collect__not,axiom,
    ! [P: product_prod_b_b > $o] :
      ( ( finite3757003017338540048od_b_b @ ( collec548942219715005266od_b_b @ P ) )
     => ( ( finite3757003017338540048od_b_b
          @ ( collec548942219715005266od_b_b
            @ ^ [X: product_prod_b_b] :
                ~ ( P @ X ) ) )
        = ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b ) ) ) ).

% finite_Collect_not
thf(fact_138_finite__Collect__not,axiom,
    ! [P: b > $o] :
      ( ( finite_finite_b @ ( collect_b @ P ) )
     => ( ( finite_finite_b
          @ ( collect_b
            @ ^ [X: b] :
                ~ ( P @ X ) ) )
        = ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_Collect_not
thf(fact_139_finite__Collect__not,axiom,
    ! [P: produc1536031394801701132od_b_b > $o] :
      ( ( finite7768965217515309219od_b_b @ ( collec4521028332464157409od_b_b @ P ) )
     => ( ( finite7768965217515309219od_b_b
          @ ( collec4521028332464157409od_b_b
            @ ^ [X: produc1536031394801701132od_b_b] :
                ~ ( P @ X ) ) )
        = ( finite7768965217515309219od_b_b @ top_to3802939291301058418od_b_b ) ) ) ).

% finite_Collect_not
thf(fact_140_finite__Collect__not,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ~ ( P @ X ) ) )
        = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Collect_not
thf(fact_141_finite__Collect__not,axiom,
    ! [P: product_prod_nat_b > $o] :
      ( ( finite659689794318260667_nat_b @ ( collec7702298003248673273_nat_b @ P ) )
     => ( ( finite659689794318260667_nat_b
          @ ( collec7702298003248673273_nat_b
            @ ^ [X: product_prod_nat_b] :
                ~ ( P @ X ) ) )
        = ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b ) ) ) ).

% finite_Collect_not
thf(fact_142_finite__Collect__not,axiom,
    ! [P: produc2840042325109449167od_b_b > $o] :
      ( ( finite4644902770518909432od_b_b @ ( collec3841725734524963642od_b_b @ P ) )
     => ( ( finite4644902770518909432od_b_b
          @ ( collec3841725734524963642od_b_b
            @ ^ [X: produc2840042325109449167od_b_b] :
                ~ ( P @ X ) ) )
        = ( finite4644902770518909432od_b_b @ top_to3066526200275256831od_b_b ) ) ) ).

% finite_Collect_not
thf(fact_143_finite__Collect__not,axiom,
    ! [P: product_prod_b_nat > $o] :
      ( ( finite7880342692102525205_b_nat @ ( collec5699578864178162003_b_nat @ P ) )
     => ( ( finite7880342692102525205_b_nat
          @ ( collec5699578864178162003_b_nat
            @ ^ [X: product_prod_b_nat] :
                ~ ( P @ X ) ) )
        = ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat ) ) ) ).

% finite_Collect_not
thf(fact_144_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_145_finite__Collect__not,axiom,
    ! [P: extended_ereal > $o] :
      ( ( finite7198162374296863863_ereal @ ( collec5835592288176408249_ereal @ P ) )
     => ( ( finite7198162374296863863_ereal
          @ ( collec5835592288176408249_ereal
            @ ^ [X: extended_ereal] :
                ~ ( P @ X ) ) )
        = ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_Collect_not
thf(fact_146_finite__Collect__subsets,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( finite8153119339416810480od_b_b
        @ ( collec1108733003560609586od_b_b
          @ ^ [B2: set_Product_prod_b_b] : ( ord_le182087997850975847od_b_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_147_finite__Collect__subsets,axiom,
    ! [A: set_b] :
      ( ( finite_finite_b @ A )
     => ( finite_finite_set_b
        @ ( collect_set_b
          @ ^ [B2: set_b] : ( ord_less_eq_set_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_148_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_149_finite__Collect__subsets,axiom,
    ! [A: set_Pr5139338970096277698od_b_b] :
      ( ( finite7768965217515309219od_b_b @ A )
     => ( finite8041041396987327577od_b_b
        @ ( collec1400520969827591831od_b_b
          @ ^ [B2: set_Pr5139338970096277698od_b_b] : ( ord_le2005546836280662306od_b_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_150_finite__Collect__subsets,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_151_finite__Collect__subsets,axiom,
    ! [A: set_Pr4264375888882495962_nat_b] :
      ( ( finite659689794318260667_nat_b @ A )
     => ( finite9121790342551665777_nat_b
        @ ( collec5983057837050141871_nat_b
          @ ^ [B2: set_Pr4264375888882495962_nat_b] : ( ord_le7995947752535495226_nat_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_152_finite__Collect__subsets,axiom,
    ! [A: set_Pr4323519195528460463od_b_b] :
      ( ( finite4644902770518909432od_b_b @ A )
     => ( finite8128242158466110680od_b_b
        @ ( collec4149886324997387802od_b_b
          @ ^ [B2: set_Pr4323519195528460463od_b_b] : ( ord_le8131691160565715023od_b_b @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_153_finite__Collect__subsets,axiom,
    ! [A: set_Pr1307281990691478580_b_nat] :
      ( ( finite7880342692102525205_b_nat @ A )
     => ( finite6164696444360648395_b_nat
        @ ( collec3025963938859124489_b_nat
          @ ^ [B2: set_Pr1307281990691478580_b_nat] : ( ord_le5038853854344477844_b_nat @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_154_finite__Collect__subsets,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( finite3583719589609615493nnreal
        @ ( collec4858231573021281987nnreal
          @ ^ [B2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_155_finite__Collect__subsets,axiom,
    ! [A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( finite2737741666826350167_ereal
        @ ( collec85322473871370393_ereal
          @ ^ [B2: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ B2 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_156_finite__Plus__UNIV__iff,axiom,
    ( ( finite6487621606449413500um_b_b @ top_to8284291040046899147um_b_b )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_157_finite__Plus__UNIV__iff,axiom,
    ( ( finite1737549346599821481_b_nat @ top_to6391837080160651824_b_nat )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_158_finite__Plus__UNIV__iff,axiom,
    ( ( finite8967218968993403851_ereal @ top_to7275872289100691610_ereal )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_159_finite__Plus__UNIV__iff,axiom,
    ( ( finite3740268485670332751_nat_b @ top_to125558941496893398_nat_b )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_160_finite__Plus__UNIV__iff,axiom,
    ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_161_finite__Plus__UNIV__iff,axiom,
    ( ( finite9026038361767888670_ereal @ top_to1312714130927029285_ereal )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_162_finite__Plus__UNIV__iff,axiom,
    ( ( finite879213341475074413real_b @ top_to2802908295386670396real_b )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_163_finite__Plus__UNIV__iff,axiom,
    ( ( finite6677479166643143288al_nat @ top_to2575454765182564991al_nat )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_164_finite__Plus__UNIV__iff,axiom,
    ( ( finite4926319553379158588_ereal @ top_to363421576304148619_ereal )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_165_finite__Plus__UNIV__iff,axiom,
    ( ( finite6530867283557821844_b_b_b @ top_to4177792431638822883_b_b_b )
    = ( ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_166_Sigma__empty1,axiom,
    ! [B: b > set_b] :
      ( ( product_Sigma_b_b @ bot_bot_set_b @ B )
      = bot_bo2792761326896053555od_b_b ) ).

% Sigma_empty1
thf(fact_167_Sigma__empty1,axiom,
    ! [B: nat > set_Product_prod_b_b] :
      ( ( produc8027630620858748621od_b_b @ bot_bot_set_nat @ B )
      = bot_bo4493299828277062486od_b_b ) ).

% Sigma_empty1
thf(fact_168_Sigma__empty1,axiom,
    ! [B: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ bot_bot_set_nat @ B )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty1
thf(fact_169_Sigma__empty1,axiom,
    ! [B: nat > set_b] :
      ( ( product_Sigma_nat_b @ bot_bot_set_nat @ B )
      = bot_bo8379049445785516142_nat_b ) ).

% Sigma_empty1
thf(fact_170_Sigma__empty1,axiom,
    ! [B: b > set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ bot_bot_set_b @ B )
      = bot_bo3230273597238993691od_b_b ) ).

% Sigma_empty1
thf(fact_171_Sigma__empty1,axiom,
    ! [B: b > set_nat] :
      ( ( product_Sigma_b_nat @ bot_bot_set_b @ B )
      = bot_bo5421955547594498760_b_nat ) ).

% Sigma_empty1
thf(fact_172_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_b_b @ top_top_set_b
      @ ^ [Uu: b] : top_top_set_b )
    = top_to7498756471699006487od_b_b ) ).

% UNIV_Times_UNIV
thf(fact_173_UNIV__Times__UNIV,axiom,
    ( ( produc2915425143180021232od_b_b @ top_top_set_b
      @ ^ [Uu: b] : top_to7498756471699006487od_b_b )
    = top_to3066526200275256831od_b_b ) ).

% UNIV_Times_UNIV
thf(fact_174_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_b_nat @ top_top_set_b
      @ ^ [Uu: b] : top_top_set_nat )
    = top_to8949910960566930148_b_nat ) ).

% UNIV_Times_UNIV
thf(fact_175_UNIV__Times__UNIV,axiom,
    ( ( produc8027630620858748621od_b_b @ top_top_set_nat
      @ ^ [Uu: nat] : top_to7498756471699006487od_b_b )
    = top_to3802939291301058418od_b_b ) ).

% UNIV_Times_UNIV
thf(fact_176_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_nat_b @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_b )
    = top_to2683632821903171722_nat_b ) ).

% UNIV_Times_UNIV
thf(fact_177_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_178_UNIV__Times__UNIV,axiom,
    ( ( produc870331913724930228_ereal @ top_top_set_nat
      @ ^ [Uu: nat] : top_to5683747375963461374_ereal )
    = top_to6634112653661286105_ereal ) ).

% UNIV_Times_UNIV
thf(fact_179_UNIV__Times__UNIV,axiom,
    ( ( produc4220900302008768982al_nat @ top_to5683747375963461374_ereal
      @ ^ [Uu: extended_ereal] : top_top_set_nat )
    = top_to7896853287916821811al_nat ) ).

% UNIV_Times_UNIV
thf(fact_180_UNIV__Times__UNIV,axiom,
    ( ( produc8095709571603465288_ereal @ top_to5683747375963461374_ereal
      @ ^ [Uu: extended_ereal] : top_to5683747375963461374_ereal )
    = top_to3798671025730093271_ereal ) ).

% UNIV_Times_UNIV
thf(fact_181_Times__empty,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B )
        = bot_bo2792761326896053555od_b_b )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bot_set_b ) ) ) ).

% Times_empty
thf(fact_182_Times__empty,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b] :
      ( ( ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : B )
        = bot_bo4493299828277062486od_b_b )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bo2792761326896053555od_b_b ) ) ) ).

% Times_empty
thf(fact_183_Times__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_184_Times__empty,axiom,
    ! [A: set_nat,B: set_b] :
      ( ( ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B )
        = bot_bo8379049445785516142_nat_b )
      = ( ( A = bot_bot_set_nat )
        | ( B = bot_bot_set_b ) ) ) ).

% Times_empty
thf(fact_185_Times__empty,axiom,
    ! [A: set_b,B: set_Product_prod_b_b] :
      ( ( ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : B )
        = bot_bo3230273597238993691od_b_b )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bo2792761326896053555od_b_b ) ) ) ).

% Times_empty
thf(fact_186_Times__empty,axiom,
    ! [A: set_b,B: set_nat] :
      ( ( ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B )
        = bot_bo5421955547594498760_b_nat )
      = ( ( A = bot_bot_set_b )
        | ( B = bot_bot_set_nat ) ) ) ).

% Times_empty
thf(fact_187_Times__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ( produc7828571232000974650nnreal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B )
        = bot_bo345837629027619229nnreal )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( B = bot_bo4854962954004695426nnreal ) ) ) ).

% Times_empty
thf(fact_188_Times__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Extended_ereal] :
      ( ( ( produc8409004985040558924_ereal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B )
        = bot_bo4235046875916767061_ereal )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( B = bot_bo8367695208629047834_ereal ) ) ) ).

% Times_empty
thf(fact_189_Times__empty,axiom,
    ! [A: set_Extended_ereal,B: set_Ex3793607809372303086nnreal] :
      ( ( ( produc1405777165569238334nnreal @ A
          @ ^ [Uu: extended_ereal] : B )
        = bot_bo1801955979378842239nnreal )
      = ( ( A = bot_bo8367695208629047834_ereal )
        | ( B = bot_bo4854962954004695426nnreal ) ) ) ).

% Times_empty
thf(fact_190_Times__empty,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ( produc8095709571603465288_ereal @ A
          @ ^ [Uu: extended_ereal] : B )
        = bot_bo4002835157671732723_ereal )
      = ( ( A = bot_bo8367695208629047834_ereal )
        | ( B = bot_bo8367695208629047834_ereal ) ) ) ).

% Times_empty
thf(fact_191_Sigma__empty2,axiom,
    ! [A: set_b] :
      ( ( product_Sigma_b_b @ A
        @ ^ [Uu: b] : bot_bot_set_b )
      = bot_bo2792761326896053555od_b_b ) ).

% Sigma_empty2
thf(fact_192_Sigma__empty2,axiom,
    ! [A: set_nat] :
      ( ( produc8027630620858748621od_b_b @ A
        @ ^ [Uu: nat] : bot_bo2792761326896053555od_b_b )
      = bot_bo4493299828277062486od_b_b ) ).

% Sigma_empty2
thf(fact_193_Sigma__empty2,axiom,
    ! [A: set_nat] :
      ( ( produc457027306803732586at_nat @ A
        @ ^ [Uu: nat] : bot_bot_set_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Sigma_empty2
thf(fact_194_Sigma__empty2,axiom,
    ! [A: set_nat] :
      ( ( product_Sigma_nat_b @ A
        @ ^ [Uu: nat] : bot_bot_set_b )
      = bot_bo8379049445785516142_nat_b ) ).

% Sigma_empty2
thf(fact_195_Sigma__empty2,axiom,
    ! [A: set_b] :
      ( ( produc2915425143180021232od_b_b @ A
        @ ^ [Uu: b] : bot_bo2792761326896053555od_b_b )
      = bot_bo3230273597238993691od_b_b ) ).

% Sigma_empty2
thf(fact_196_Sigma__empty2,axiom,
    ! [A: set_b] :
      ( ( product_Sigma_b_nat @ A
        @ ^ [Uu: b] : bot_bot_set_nat )
      = bot_bo5421955547594498760_b_nat ) ).

% Sigma_empty2
thf(fact_197_times__subset__iff,axiom,
    ! [A: set_b,C: set_b,B: set_b,D: set_b] :
      ( ( ord_le182087997850975847od_b_b
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : C )
        @ ( product_Sigma_b_b @ B
          @ ^ [Uu: b] : D ) )
      = ( ( A = bot_bot_set_b )
        | ( C = bot_bot_set_b )
        | ( ( ord_less_eq_set_b @ A @ B )
          & ( ord_less_eq_set_b @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_198_times__subset__iff,axiom,
    ! [A: set_nat,C: set_Product_prod_b_b,B: set_nat,D: set_Product_prod_b_b] :
      ( ( ord_le2005546836280662306od_b_b
        @ ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : C )
        @ ( produc8027630620858748621od_b_b @ B
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bo2792761326896053555od_b_b )
        | ( ( ord_less_eq_set_nat @ A @ B )
          & ( ord_le182087997850975847od_b_b @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_199_times__subset__iff,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_le3146513528884898305at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : C )
        @ ( produc457027306803732586at_nat @ B
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_nat @ A @ B )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_200_times__subset__iff,axiom,
    ! [A: set_nat,C: set_b,B: set_nat,D: set_b] :
      ( ( ord_le7995947752535495226_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : C )
        @ ( product_Sigma_nat_b @ B
          @ ^ [Uu: nat] : D ) )
      = ( ( A = bot_bot_set_nat )
        | ( C = bot_bot_set_b )
        | ( ( ord_less_eq_set_nat @ A @ B )
          & ( ord_less_eq_set_b @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_201_times__subset__iff,axiom,
    ! [A: set_b,C: set_Product_prod_b_b,B: set_b,D: set_Product_prod_b_b] :
      ( ( ord_le8131691160565715023od_b_b
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : C )
        @ ( produc2915425143180021232od_b_b @ B
          @ ^ [Uu: b] : D ) )
      = ( ( A = bot_bot_set_b )
        | ( C = bot_bo2792761326896053555od_b_b )
        | ( ( ord_less_eq_set_b @ A @ B )
          & ( ord_le182087997850975847od_b_b @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_202_times__subset__iff,axiom,
    ! [A: set_b,C: set_nat,B: set_b,D: set_nat] :
      ( ( ord_le5038853854344477844_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : C )
        @ ( product_Sigma_b_nat @ B
          @ ^ [Uu: b] : D ) )
      = ( ( A = bot_bot_set_b )
        | ( C = bot_bot_set_nat )
        | ( ( ord_less_eq_set_b @ A @ B )
          & ( ord_less_eq_set_nat @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_203_times__subset__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,D: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3346733166600019153nnreal
        @ ( produc7828571232000974650nnreal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : C )
        @ ( produc7828571232000974650nnreal @ B
          @ ^ [Uu: extend8495563244428889912nnreal] : D ) )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( C = bot_bo4854962954004695426nnreal )
        | ( ( ord_le6787938422905777998nnreal @ A @ B )
          & ( ord_le6787938422905777998nnreal @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_204_times__subset__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,C: set_Extended_ereal,B: set_Ex3793607809372303086nnreal,D: set_Extended_ereal] :
      ( ( ord_le3522949091209373473_ereal
        @ ( produc8409004985040558924_ereal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : C )
        @ ( produc8409004985040558924_ereal @ B
          @ ^ [Uu: extend8495563244428889912nnreal] : D ) )
      = ( ( A = bot_bo4854962954004695426nnreal )
        | ( C = bot_bo8367695208629047834_ereal )
        | ( ( ord_le6787938422905777998nnreal @ A @ B )
          & ( ord_le1644982726543182158_ereal @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_205_times__subset__iff,axiom,
    ! [A: set_Extended_ereal,C: set_Ex3793607809372303086nnreal,B: set_Extended_ereal,D: set_Ex3793607809372303086nnreal] :
      ( ( ord_le1089858194671448651nnreal
        @ ( produc1405777165569238334nnreal @ A
          @ ^ [Uu: extended_ereal] : C )
        @ ( produc1405777165569238334nnreal @ B
          @ ^ [Uu: extended_ereal] : D ) )
      = ( ( A = bot_bo8367695208629047834_ereal )
        | ( C = bot_bo4854962954004695426nnreal )
        | ( ( ord_le1644982726543182158_ereal @ A @ B )
          & ( ord_le6787938422905777998nnreal @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_206_times__subset__iff,axiom,
    ! [A: set_Extended_ereal,C: set_Extended_ereal,B: set_Extended_ereal,D: set_Extended_ereal] :
      ( ( ord_le8239133294219471655_ereal
        @ ( produc8095709571603465288_ereal @ A
          @ ^ [Uu: extended_ereal] : C )
        @ ( produc8095709571603465288_ereal @ B
          @ ^ [Uu: extended_ereal] : D ) )
      = ( ( A = bot_bo8367695208629047834_ereal )
        | ( C = bot_bo8367695208629047834_ereal )
        | ( ( ord_le1644982726543182158_ereal @ A @ B )
          & ( ord_le1644982726543182158_ereal @ C @ D ) ) ) ) ).

% times_subset_iff
thf(fact_207_finite__has__minimal,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A )
               => ( ( ord_le8460144461188290721at_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_208_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_209_finite__has__minimal,axiom,
    ! [A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ? [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ A )
            & ! [Xa: extended_ereal] :
                ( ( member2350847679896131959_ereal @ Xa @ A )
               => ( ( ord_le1083603963089353582_ereal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_210_finite__has__minimal,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ? [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
            & ! [Xa: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa @ A )
               => ( ( ord_le3935885782089961368nnreal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_211_finite__has__minimal,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ? [X2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ X2 @ A )
            & ! [Xa: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ Xa @ A )
               => ( ( ord_le6787938422905777998nnreal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_212_finite__has__minimal,axiom,
    ! [A: set_se6634062954251873166_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ? [X2: set_Extended_ereal] :
            ( ( member5519481007471526743_ereal @ X2 @ A )
            & ! [Xa: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ Xa @ A )
               => ( ( ord_le1644982726543182158_ereal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_213_finite__has__maximal,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A )
               => ( ( ord_le8460144461188290721at_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_214_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_215_finite__has__maximal,axiom,
    ! [A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ? [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ A )
            & ! [Xa: extended_ereal] :
                ( ( member2350847679896131959_ereal @ Xa @ A )
               => ( ( ord_le1083603963089353582_ereal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_216_finite__has__maximal,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ? [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
            & ! [Xa: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa @ A )
               => ( ( ord_le3935885782089961368nnreal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_217_finite__has__maximal,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ? [X2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ X2 @ A )
            & ! [Xa: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ Xa @ A )
               => ( ( ord_le6787938422905777998nnreal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_218_finite__has__maximal,axiom,
    ! [A: set_se6634062954251873166_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ? [X2: set_Extended_ereal] :
            ( ( member5519481007471526743_ereal @ X2 @ A )
            & ! [Xa: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ Xa @ A )
               => ( ( ord_le1644982726543182158_ereal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_219_Finite__Set_Ofinite__set,axiom,
    ( ( finite8153119339416810480od_b_b @ top_to1177433774625531895od_b_b )
    = ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b ) ) ).

% Finite_Set.finite_set
thf(fact_220_Finite__Set_Ofinite__set,axiom,
    ( ( finite_finite_set_b @ top_top_set_set_b )
    = ( finite_finite_b @ top_top_set_b ) ) ).

% Finite_Set.finite_set
thf(fact_221_Finite__Set_Ofinite__set,axiom,
    ( ( finite8041041396987327577od_b_b @ top_to1798758190975514664od_b_b )
    = ( finite7768965217515309219od_b_b @ top_to3802939291301058418od_b_b ) ) ).

% Finite_Set.finite_set
thf(fact_222_Finite__Set_Ofinite__set,axiom,
    ( ( finite9047747110432174090at_nat @ top_to7629004291339433233at_nat )
    = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).

% Finite_Set.finite_set
thf(fact_223_Finite__Set_Ofinite__set,axiom,
    ( ( finite9121790342551665777_nat_b @ top_to3915930344255043648_nat_b )
    = ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b ) ) ).

% Finite_Set.finite_set
thf(fact_224_Finite__Set_Ofinite__set,axiom,
    ( ( finite8128242158466110680od_b_b @ top_to4358540389053044959od_b_b )
    = ( finite4644902770518909432od_b_b @ top_to3066526200275256831od_b_b ) ) ).

% Finite_Set.finite_set
thf(fact_225_Finite__Set_Ofinite__set,axiom,
    ( ( finite6164696444360648395_b_nat @ top_to1896443468416587162_b_nat )
    = ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat ) ) ).

% Finite_Set.finite_set
thf(fact_226_Finite__Set_Ofinite__set,axiom,
    ( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% Finite_Set.finite_set
thf(fact_227_Finite__Set_Ofinite__set,axiom,
    ( ( finite2737741666826350167_ereal @ top_to4757929550322229470_ereal )
    = ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ).

% Finite_Set.finite_set
thf(fact_228_Sigma__mono,axiom,
    ! [A: set_b,C: set_b,B: b > set_b,D: b > set_b] :
      ( ( ord_less_eq_set_b @ A @ C )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( ord_less_eq_set_b @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le182087997850975847od_b_b @ ( product_Sigma_b_b @ A @ B ) @ ( product_Sigma_b_b @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_229_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B: nat > set_nat,D: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( produc457027306803732586at_nat @ A @ B ) @ ( produc457027306803732586at_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_230_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B: nat > set_b,D: nat > set_b] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_less_eq_set_b @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le7995947752535495226_nat_b @ ( product_Sigma_nat_b @ A @ B ) @ ( product_Sigma_nat_b @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_231_Sigma__mono,axiom,
    ! [A: set_b,C: set_b,B: b > set_nat,D: b > set_nat] :
      ( ( ord_less_eq_set_b @ A @ C )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( ord_less_eq_set_nat @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le5038853854344477844_b_nat @ ( product_Sigma_b_nat @ A @ B ) @ ( product_Sigma_b_nat @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_232_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B: nat > set_Ex3793607809372303086nnreal,D: nat > set_Ex3793607809372303086nnreal] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_le6787938422905777998nnreal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le1493227476259890537nnreal @ ( produc648375469580864466nnreal @ A @ B ) @ ( produc648375469580864466nnreal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_233_Sigma__mono,axiom,
    ! [A: set_b,C: set_b,B: b > set_Ex3793607809372303086nnreal,D: b > set_Ex3793607809372303086nnreal] :
      ( ( ord_less_eq_set_b @ A @ C )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( ord_le6787938422905777998nnreal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le5832031571377758204nnreal @ ( produc151940477230798575nnreal @ A @ B ) @ ( produc151940477230798575nnreal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_234_Sigma__mono,axiom,
    ! [A: set_nat,C: set_nat,B: nat > set_Extended_ereal,D: nat > set_Extended_ereal] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ( ord_le1644982726543182158_ereal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3920121919471048841_ereal @ ( produc870331913724930228_ereal @ A @ B ) @ ( produc870331913724930228_ereal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_235_Sigma__mono,axiom,
    ! [A: set_b,C: set_b,B: b > set_Extended_ereal,D: b > set_Extended_ereal] :
      ( ( ord_less_eq_set_b @ A @ C )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ( ord_le1644982726543182158_ereal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3110818134188293430_ereal @ ( produc4178612860395885143_ereal @ A @ B ) @ ( produc4178612860395885143_ereal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_236_Sigma__mono,axiom,
    ! [A: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal,D: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ C )
     => ( ! [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
           => ( ord_le6787938422905777998nnreal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3346733166600019153nnreal @ ( produc7828571232000974650nnreal @ A @ B ) @ ( produc7828571232000974650nnreal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_237_Sigma__mono,axiom,
    ! [A: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal > set_Extended_ereal,D: extend8495563244428889912nnreal > set_Extended_ereal] :
      ( ( ord_le6787938422905777998nnreal @ A @ C )
     => ( ! [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
           => ( ord_le1644982726543182158_ereal @ ( B @ X2 ) @ ( D @ X2 ) ) )
       => ( ord_le3522949091209373473_ereal @ ( produc8409004985040558924_ereal @ A @ B ) @ ( produc8409004985040558924_ereal @ C @ D ) ) ) ) ).

% Sigma_mono
thf(fact_238_Times__subset__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_b,B: set_b] :
      ( ( member_b @ X3 @ C )
     => ( ( ord_le182087997850975847od_b_b
          @ ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : C )
          @ ( product_Sigma_b_b @ B
            @ ^ [Uu: b] : C ) )
        = ( ord_less_eq_set_b @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_239_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le3146513528884898305at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : C )
          @ ( produc457027306803732586at_nat @ B
            @ ^ [Uu: nat] : C ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_240_Times__subset__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_nat,B: set_nat] :
      ( ( member_b @ X3 @ C )
     => ( ( ord_le7995947752535495226_nat_b
          @ ( product_Sigma_nat_b @ A
            @ ^ [Uu: nat] : C )
          @ ( product_Sigma_nat_b @ B
            @ ^ [Uu: nat] : C ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_241_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_b,B: set_b] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le5038853854344477844_b_nat
          @ ( product_Sigma_b_nat @ A
            @ ^ [Uu: b] : C )
          @ ( product_Sigma_b_nat @ B
            @ ^ [Uu: b] : C ) )
        = ( ord_less_eq_set_b @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_242_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le4875057589893456233al_nat
          @ ( produc5422075910247723986al_nat @ A
            @ ^ [Uu: extend8495563244428889912nnreal] : C )
          @ ( produc5422075910247723986al_nat @ B
            @ ^ [Uu: extend8495563244428889912nnreal] : C ) )
        = ( ord_le6787938422905777998nnreal @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_243_Times__subset__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( member_b @ X3 @ C )
     => ( ( ord_le1327023882693010642real_b
          @ ( produc6071992953914654589real_b @ A
            @ ^ [Uu: extend8495563244428889912nnreal] : C )
          @ ( produc6071992953914654589real_b @ B
            @ ^ [Uu: extend8495563244428889912nnreal] : C ) )
        = ( ord_le6787938422905777998nnreal @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_244_Times__subset__cancel2,axiom,
    ! [X3: nat,C: set_nat,A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( member_nat @ X3 @ C )
     => ( ( ord_le5182862553726584547al_nat
          @ ( produc4220900302008768982al_nat @ A
            @ ^ [Uu: extended_ereal] : C )
          @ ( produc4220900302008768982al_nat @ B
            @ ^ [Uu: extended_ereal] : C ) )
        = ( ord_le1644982726543182158_ereal @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_245_Times__subset__cancel2,axiom,
    ! [X3: b,C: set_b,A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( member_b @ X3 @ C )
     => ( ( ord_le7861226177329048024real_b
          @ ( produc2583502849437520505real_b @ A
            @ ^ [Uu: extended_ereal] : C )
          @ ( produc2583502849437520505real_b @ B
            @ ^ [Uu: extended_ereal] : C ) )
        = ( ord_le1644982726543182158_ereal @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_246_Times__subset__cancel2,axiom,
    ! [X3: product_prod_b_b,C: set_Product_prod_b_b,A: set_nat,B: set_nat] :
      ( ( member7862447936710763792od_b_b @ X3 @ C )
     => ( ( ord_le2005546836280662306od_b_b
          @ ( produc8027630620858748621od_b_b @ A
            @ ^ [Uu: nat] : C )
          @ ( produc8027630620858748621od_b_b @ B
            @ ^ [Uu: nat] : C ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_247_Times__subset__cancel2,axiom,
    ! [X3: product_prod_b_b,C: set_Product_prod_b_b,A: set_b,B: set_b] :
      ( ( member7862447936710763792od_b_b @ X3 @ C )
     => ( ( ord_le8131691160565715023od_b_b
          @ ( produc2915425143180021232od_b_b @ A
            @ ^ [Uu: b] : C )
          @ ( produc2915425143180021232od_b_b @ B
            @ ^ [Uu: b] : C ) )
        = ( ord_less_eq_set_b @ A @ B ) ) ) ).

% Times_subset_cancel2
thf(fact_248_Sigma__empty__iff,axiom,
    ! [I: set_b,X4: b > set_b] :
      ( ( ( product_Sigma_b_b @ I @ X4 )
        = bot_bo2792761326896053555od_b_b )
      = ( ! [X: b] :
            ( ( member_b @ X @ I )
           => ( ( X4 @ X )
              = bot_bot_set_b ) ) ) ) ).

% Sigma_empty_iff
thf(fact_249_Sigma__empty__iff,axiom,
    ! [I: set_nat,X4: nat > set_Product_prod_b_b] :
      ( ( ( produc8027630620858748621od_b_b @ I @ X4 )
        = bot_bo4493299828277062486od_b_b )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I )
           => ( ( X4 @ X )
              = bot_bo2792761326896053555od_b_b ) ) ) ) ).

% Sigma_empty_iff
thf(fact_250_Sigma__empty__iff,axiom,
    ! [I: set_nat,X4: nat > set_nat] :
      ( ( ( produc457027306803732586at_nat @ I @ X4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I )
           => ( ( X4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_251_Sigma__empty__iff,axiom,
    ! [I: set_nat,X4: nat > set_b] :
      ( ( ( product_Sigma_nat_b @ I @ X4 )
        = bot_bo8379049445785516142_nat_b )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ I )
           => ( ( X4 @ X )
              = bot_bot_set_b ) ) ) ) ).

% Sigma_empty_iff
thf(fact_252_Sigma__empty__iff,axiom,
    ! [I: set_b,X4: b > set_Product_prod_b_b] :
      ( ( ( produc2915425143180021232od_b_b @ I @ X4 )
        = bot_bo3230273597238993691od_b_b )
      = ( ! [X: b] :
            ( ( member_b @ X @ I )
           => ( ( X4 @ X )
              = bot_bo2792761326896053555od_b_b ) ) ) ) ).

% Sigma_empty_iff
thf(fact_253_Sigma__empty__iff,axiom,
    ! [I: set_b,X4: b > set_nat] :
      ( ( ( product_Sigma_b_nat @ I @ X4 )
        = bot_bo5421955547594498760_b_nat )
      = ( ! [X: b] :
            ( ( member_b @ X @ I )
           => ( ( X4 @ X )
              = bot_bot_set_nat ) ) ) ) ).

% Sigma_empty_iff
thf(fact_254_finite__prod,axiom,
    ( ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_prod
thf(fact_255_finite__prod,axiom,
    ( ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_256_finite__prod,axiom,
    ( ( finite6405620804434884703_ereal @ top_to3064337335865023206_ereal )
    = ( ( finite_finite_b @ top_top_set_b )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_prod
thf(fact_257_finite__prod,axiom,
    ( ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_prod
thf(fact_258_finite__prod,axiom,
    ( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_259_finite__prod,axiom,
    ( ( finite2240905580840527242_ereal @ top_to6634112653661286105_ereal )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_prod
thf(fact_260_finite__prod,axiom,
    ( ( finite7540987213771331073real_b @ top_to7814745379005777800real_b )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_prod
thf(fact_261_finite__prod,axiom,
    ( ( finite9115718422570557668al_nat @ top_to7896853287916821811al_nat )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_262_finite__prod,axiom,
    ( ( finite7645029973310554320_ereal @ top_to3798671025730093271_ereal )
    = ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
      & ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ) ).

% finite_prod
thf(fact_263_finite__prod,axiom,
    ( ( finite1344950077393947688_b_b_b @ top_to7562527894943373359_b_b_b )
    = ( ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b )
      & ( finite_finite_b @ top_top_set_b ) ) ) ).

% finite_prod
thf(fact_264_finite__Prod__UNIV,axiom,
    ( ( finite_finite_b @ top_top_set_b )
   => ( ( finite_finite_b @ top_top_set_b )
     => ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b ) ) ) ).

% finite_Prod_UNIV
thf(fact_265_finite__Prod__UNIV,axiom,
    ( ( finite_finite_b @ top_top_set_b )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_266_finite__Prod__UNIV,axiom,
    ( ( finite_finite_b @ top_top_set_b )
   => ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
     => ( finite6405620804434884703_ereal @ top_to3064337335865023206_ereal ) ) ) ).

% finite_Prod_UNIV
thf(fact_267_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_b @ top_top_set_b )
     => ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b ) ) ) ).

% finite_Prod_UNIV
thf(fact_268_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_269_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
     => ( finite2240905580840527242_ereal @ top_to6634112653661286105_ereal ) ) ) ).

% finite_Prod_UNIV
thf(fact_270_finite__Prod__UNIV,axiom,
    ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
   => ( ( finite_finite_b @ top_top_set_b )
     => ( finite7540987213771331073real_b @ top_to7814745379005777800real_b ) ) ) ).

% finite_Prod_UNIV
thf(fact_271_finite__Prod__UNIV,axiom,
    ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite9115718422570557668al_nat @ top_to7896853287916821811al_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_272_finite__Prod__UNIV,axiom,
    ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
   => ( ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
     => ( finite7645029973310554320_ereal @ top_to3798671025730093271_ereal ) ) ) ).

% finite_Prod_UNIV
thf(fact_273_finite__Prod__UNIV,axiom,
    ( ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b )
   => ( ( finite_finite_b @ top_top_set_b )
     => ( finite1344950077393947688_b_b_b @ top_to7562527894943373359_b_b_b ) ) ) ).

% finite_Prod_UNIV
thf(fact_274_finite__has__maximal2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
            & ( ord_le8460144461188290721at_nat @ A5 @ X2 )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A )
               => ( ( ord_le8460144461188290721at_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_275_finite__has__maximal2,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ A5 @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_276_finite__has__maximal2,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ A5 @ A )
       => ? [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ A )
            & ( ord_le1083603963089353582_ereal @ A5 @ X2 )
            & ! [Xa: extended_ereal] :
                ( ( member2350847679896131959_ereal @ Xa @ A )
               => ( ( ord_le1083603963089353582_ereal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_277_finite__has__maximal2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ A5 @ A )
       => ? [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
            & ( ord_le3935885782089961368nnreal @ A5 @ X2 )
            & ! [Xa: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa @ A )
               => ( ( ord_le3935885782089961368nnreal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_278_finite__has__maximal2,axiom,
    ! [A: set_se4580700918925141924nnreal,A5: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( member603777416030116741nnreal @ A5 @ A )
       => ? [X2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ X2 @ A )
            & ( ord_le6787938422905777998nnreal @ A5 @ X2 )
            & ! [Xa: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ Xa @ A )
               => ( ( ord_le6787938422905777998nnreal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_279_finite__has__maximal2,axiom,
    ! [A: set_se6634062954251873166_ereal,A5: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( member5519481007471526743_ereal @ A5 @ A )
       => ? [X2: set_Extended_ereal] :
            ( ( member5519481007471526743_ereal @ X2 @ A )
            & ( ord_le1644982726543182158_ereal @ A5 @ X2 )
            & ! [Xa: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ Xa @ A )
               => ( ( ord_le1644982726543182158_ereal @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_280_finite__has__minimal2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ? [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
            & ( ord_le8460144461188290721at_nat @ X2 @ A5 )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A )
               => ( ( ord_le8460144461188290721at_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_281_finite__has__minimal2,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A )
            & ( ord_less_eq_nat @ X2 @ A5 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_282_finite__has__minimal2,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ A5 @ A )
       => ? [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ A )
            & ( ord_le1083603963089353582_ereal @ X2 @ A5 )
            & ! [Xa: extended_ereal] :
                ( ( member2350847679896131959_ereal @ Xa @ A )
               => ( ( ord_le1083603963089353582_ereal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_283_finite__has__minimal2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ A5 @ A )
       => ? [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ A )
            & ( ord_le3935885782089961368nnreal @ X2 @ A5 )
            & ! [Xa: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ Xa @ A )
               => ( ( ord_le3935885782089961368nnreal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_284_finite__has__minimal2,axiom,
    ! [A: set_se4580700918925141924nnreal,A5: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( member603777416030116741nnreal @ A5 @ A )
       => ? [X2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ X2 @ A )
            & ( ord_le6787938422905777998nnreal @ X2 @ A5 )
            & ! [Xa: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ Xa @ A )
               => ( ( ord_le6787938422905777998nnreal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_285_finite__has__minimal2,axiom,
    ! [A: set_se6634062954251873166_ereal,A5: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( member5519481007471526743_ereal @ A5 @ A )
       => ? [X2: set_Extended_ereal] :
            ( ( member5519481007471526743_ereal @ X2 @ A )
            & ( ord_le1644982726543182158_ereal @ X2 @ A5 )
            & ! [Xa: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ Xa @ A )
               => ( ( ord_le1644982726543182158_ereal @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_286_finite__subset,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A @ B )
     => ( ( finite3757003017338540048od_b_b @ B )
       => ( finite3757003017338540048od_b_b @ A ) ) ) ).

% finite_subset
thf(fact_287_finite__subset,axiom,
    ! [A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( finite_finite_b @ B )
       => ( finite_finite_b @ A ) ) ) ).

% finite_subset
thf(fact_288_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_289_finite__subset,axiom,
    ! [A: set_Pr5139338970096277698od_b_b,B: set_Pr5139338970096277698od_b_b] :
      ( ( ord_le2005546836280662306od_b_b @ A @ B )
     => ( ( finite7768965217515309219od_b_b @ B )
       => ( finite7768965217515309219od_b_b @ A ) ) ) ).

% finite_subset
thf(fact_290_finite__subset,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( finite6177210948735845034at_nat @ B )
       => ( finite6177210948735845034at_nat @ A ) ) ) ).

% finite_subset
thf(fact_291_finite__subset,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,B: set_Pr4264375888882495962_nat_b] :
      ( ( ord_le7995947752535495226_nat_b @ A @ B )
     => ( ( finite659689794318260667_nat_b @ B )
       => ( finite659689794318260667_nat_b @ A ) ) ) ).

% finite_subset
thf(fact_292_finite__subset,axiom,
    ! [A: set_Pr4323519195528460463od_b_b,B: set_Pr4323519195528460463od_b_b] :
      ( ( ord_le8131691160565715023od_b_b @ A @ B )
     => ( ( finite4644902770518909432od_b_b @ B )
       => ( finite4644902770518909432od_b_b @ A ) ) ) ).

% finite_subset
thf(fact_293_finite__subset,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,B: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ A @ B )
     => ( ( finite7880342692102525205_b_nat @ B )
       => ( finite7880342692102525205_b_nat @ A ) ) ) ).

% finite_subset
thf(fact_294_finite__subset,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( finite3782138982310603983nnreal @ B )
       => ( finite3782138982310603983nnreal @ A ) ) ) ).

% finite_subset
thf(fact_295_finite__subset,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( finite7198162374296863863_ereal @ B )
       => ( finite7198162374296863863_ereal @ A ) ) ) ).

% finite_subset
thf(fact_296_infinite__super,axiom,
    ! [S: set_Product_prod_b_b,T: set_Product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ S @ T )
     => ( ~ ( finite3757003017338540048od_b_b @ S )
       => ~ ( finite3757003017338540048od_b_b @ T ) ) ) ).

% infinite_super
thf(fact_297_infinite__super,axiom,
    ! [S: set_b,T: set_b] :
      ( ( ord_less_eq_set_b @ S @ T )
     => ( ~ ( finite_finite_b @ S )
       => ~ ( finite_finite_b @ T ) ) ) ).

% infinite_super
thf(fact_298_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_299_infinite__super,axiom,
    ! [S: set_Pr5139338970096277698od_b_b,T: set_Pr5139338970096277698od_b_b] :
      ( ( ord_le2005546836280662306od_b_b @ S @ T )
     => ( ~ ( finite7768965217515309219od_b_b @ S )
       => ~ ( finite7768965217515309219od_b_b @ T ) ) ) ).

% infinite_super
thf(fact_300_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ T ) ) ) ).

% infinite_super
thf(fact_301_infinite__super,axiom,
    ! [S: set_Pr4264375888882495962_nat_b,T: set_Pr4264375888882495962_nat_b] :
      ( ( ord_le7995947752535495226_nat_b @ S @ T )
     => ( ~ ( finite659689794318260667_nat_b @ S )
       => ~ ( finite659689794318260667_nat_b @ T ) ) ) ).

% infinite_super
thf(fact_302_infinite__super,axiom,
    ! [S: set_Pr4323519195528460463od_b_b,T: set_Pr4323519195528460463od_b_b] :
      ( ( ord_le8131691160565715023od_b_b @ S @ T )
     => ( ~ ( finite4644902770518909432od_b_b @ S )
       => ~ ( finite4644902770518909432od_b_b @ T ) ) ) ).

% infinite_super
thf(fact_303_infinite__super,axiom,
    ! [S: set_Pr1307281990691478580_b_nat,T: set_Pr1307281990691478580_b_nat] :
      ( ( ord_le5038853854344477844_b_nat @ S @ T )
     => ( ~ ( finite7880342692102525205_b_nat @ S )
       => ~ ( finite7880342692102525205_b_nat @ T ) ) ) ).

% infinite_super
thf(fact_304_infinite__super,axiom,
    ! [S: set_Ex3793607809372303086nnreal,T: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ S @ T )
     => ( ~ ( finite3782138982310603983nnreal @ S )
       => ~ ( finite3782138982310603983nnreal @ T ) ) ) ).

% infinite_super
thf(fact_305_infinite__super,axiom,
    ! [S: set_Extended_ereal,T: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ S @ T )
     => ( ~ ( finite7198162374296863863_ereal @ S )
       => ~ ( finite7198162374296863863_ereal @ T ) ) ) ).

% infinite_super
thf(fact_306_rev__finite__subset,axiom,
    ! [B: set_Product_prod_b_b,A: set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ B )
     => ( ( ord_le182087997850975847od_b_b @ A @ B )
       => ( finite3757003017338540048od_b_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_307_rev__finite__subset,axiom,
    ! [B: set_b,A: set_b] :
      ( ( finite_finite_b @ B )
     => ( ( ord_less_eq_set_b @ A @ B )
       => ( finite_finite_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_308_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_309_rev__finite__subset,axiom,
    ! [B: set_Pr5139338970096277698od_b_b,A: set_Pr5139338970096277698od_b_b] :
      ( ( finite7768965217515309219od_b_b @ B )
     => ( ( ord_le2005546836280662306od_b_b @ A @ B )
       => ( finite7768965217515309219od_b_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_310_rev__finite__subset,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ B )
       => ( finite6177210948735845034at_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_311_rev__finite__subset,axiom,
    ! [B: set_Pr4264375888882495962_nat_b,A: set_Pr4264375888882495962_nat_b] :
      ( ( finite659689794318260667_nat_b @ B )
     => ( ( ord_le7995947752535495226_nat_b @ A @ B )
       => ( finite659689794318260667_nat_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_312_rev__finite__subset,axiom,
    ! [B: set_Pr4323519195528460463od_b_b,A: set_Pr4323519195528460463od_b_b] :
      ( ( finite4644902770518909432od_b_b @ B )
     => ( ( ord_le8131691160565715023od_b_b @ A @ B )
       => ( finite4644902770518909432od_b_b @ A ) ) ) ).

% rev_finite_subset
thf(fact_313_rev__finite__subset,axiom,
    ! [B: set_Pr1307281990691478580_b_nat,A: set_Pr1307281990691478580_b_nat] :
      ( ( finite7880342692102525205_b_nat @ B )
     => ( ( ord_le5038853854344477844_b_nat @ A @ B )
       => ( finite7880342692102525205_b_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_314_rev__finite__subset,axiom,
    ! [B: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ B )
     => ( ( ord_le6787938422905777998nnreal @ A @ B )
       => ( finite3782138982310603983nnreal @ A ) ) ) ).

% rev_finite_subset
thf(fact_315_rev__finite__subset,axiom,
    ! [B: set_Extended_ereal,A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ B )
     => ( ( ord_le1644982726543182158_ereal @ A @ B )
       => ( finite7198162374296863863_ereal @ A ) ) ) ).

% rev_finite_subset
thf(fact_316_mem__Collect__eq,axiom,
    ! [A5: nat,P: nat > $o] :
      ( ( member_nat @ A5 @ ( collect_nat @ P ) )
      = ( P @ A5 ) ) ).

% mem_Collect_eq
thf(fact_317_mem__Collect__eq,axiom,
    ! [A5: b,P: b > $o] :
      ( ( member_b @ A5 @ ( collect_b @ P ) )
      = ( P @ A5 ) ) ).

% mem_Collect_eq
thf(fact_318_mem__Collect__eq,axiom,
    ! [A5: product_prod_b_b,P: product_prod_b_b > $o] :
      ( ( member7862447936710763792od_b_b @ A5 @ ( collec548942219715005266od_b_b @ P ) )
      = ( P @ A5 ) ) ).

% mem_Collect_eq
thf(fact_319_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_320_Collect__mem__eq,axiom,
    ! [A: set_b] :
      ( ( collect_b
        @ ^ [X: b] : ( member_b @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_321_Collect__mem__eq,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_322_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_323_Collect__cong,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ! [X2: b] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_b @ P )
        = ( collect_b @ Q ) ) ) ).

% Collect_cong
thf(fact_324_Collect__cong,axiom,
    ! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ! [X2: product_prod_b_b] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec548942219715005266od_b_b @ P )
        = ( collec548942219715005266od_b_b @ Q ) ) ) ).

% Collect_cong
thf(fact_325_ex__new__if__finite,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ~ ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b )
     => ( ( finite3757003017338540048od_b_b @ A )
       => ? [A2: product_prod_b_b] :
            ~ ( member7862447936710763792od_b_b @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_326_ex__new__if__finite,axiom,
    ! [A: set_b] :
      ( ~ ( finite_finite_b @ top_top_set_b )
     => ( ( finite_finite_b @ A )
       => ? [A2: b] :
            ~ ( member_b @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_327_ex__new__if__finite,axiom,
    ! [A: set_Pr5139338970096277698od_b_b] :
      ( ~ ( finite7768965217515309219od_b_b @ top_to3802939291301058418od_b_b )
     => ( ( finite7768965217515309219od_b_b @ A )
       => ? [A2: produc1536031394801701132od_b_b] :
            ~ ( member5954771037092721571od_b_b @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_328_ex__new__if__finite,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
     => ( ( finite6177210948735845034at_nat @ A )
       => ? [A2: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_329_ex__new__if__finite,axiom,
    ! [A: set_Pr4264375888882495962_nat_b] :
      ( ~ ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b )
     => ( ( finite659689794318260667_nat_b @ A )
       => ? [A2: product_prod_nat_b] :
            ~ ( member8962352056413324475_nat_b @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_330_ex__new__if__finite,axiom,
    ! [A: set_Pr4323519195528460463od_b_b] :
      ( ~ ( finite4644902770518909432od_b_b @ top_to3066526200275256831od_b_b )
     => ( ( finite4644902770518909432od_b_b @ A )
       => ? [A2: produc2840042325109449167od_b_b] :
            ~ ( member1867066792959145720od_b_b @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_331_ex__new__if__finite,axiom,
    ! [A: set_Pr1307281990691478580_b_nat] :
      ( ~ ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat )
     => ( ( finite7880342692102525205_b_nat @ A )
       => ? [A2: product_prod_b_nat] :
            ~ ( member6959632917342813205_b_nat @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_332_ex__new__if__finite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A )
       => ? [A2: nat] :
            ~ ( member_nat @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_333_ex__new__if__finite,axiom,
    ! [A: set_Extended_ereal] :
      ( ~ ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal )
     => ( ( finite7198162374296863863_ereal @ A )
       => ? [A2: extended_ereal] :
            ~ ( member2350847679896131959_ereal @ A2 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_334_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_335_finite_OemptyI,axiom,
    finite3757003017338540048od_b_b @ bot_bo2792761326896053555od_b_b ).

% finite.emptyI
thf(fact_336_finite_OemptyI,axiom,
    finite_finite_b @ bot_bot_set_b ).

% finite.emptyI
thf(fact_337_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_338_finite_OemptyI,axiom,
    finite7768965217515309219od_b_b @ bot_bo4493299828277062486od_b_b ).

% finite.emptyI
thf(fact_339_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_340_finite_OemptyI,axiom,
    finite659689794318260667_nat_b @ bot_bo8379049445785516142_nat_b ).

% finite.emptyI
thf(fact_341_finite_OemptyI,axiom,
    finite4644902770518909432od_b_b @ bot_bo3230273597238993691od_b_b ).

% finite.emptyI
thf(fact_342_finite_OemptyI,axiom,
    finite7880342692102525205_b_nat @ bot_bo5421955547594498760_b_nat ).

% finite.emptyI
thf(fact_343_finite_OemptyI,axiom,
    finite3782138982310603983nnreal @ bot_bo4854962954004695426nnreal ).

% finite.emptyI
thf(fact_344_finite_OemptyI,axiom,
    finite7198162374296863863_ereal @ bot_bo8367695208629047834_ereal ).

% finite.emptyI
thf(fact_345_infinite__imp__nonempty,axiom,
    ! [S: set_Product_prod_b_b] :
      ( ~ ( finite3757003017338540048od_b_b @ S )
     => ( S != bot_bo2792761326896053555od_b_b ) ) ).

% infinite_imp_nonempty
thf(fact_346_infinite__imp__nonempty,axiom,
    ! [S: set_b] :
      ( ~ ( finite_finite_b @ S )
     => ( S != bot_bot_set_b ) ) ).

% infinite_imp_nonempty
thf(fact_347_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_348_infinite__imp__nonempty,axiom,
    ! [S: set_Pr5139338970096277698od_b_b] :
      ( ~ ( finite7768965217515309219od_b_b @ S )
     => ( S != bot_bo4493299828277062486od_b_b ) ) ).

% infinite_imp_nonempty
thf(fact_349_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_350_infinite__imp__nonempty,axiom,
    ! [S: set_Pr4264375888882495962_nat_b] :
      ( ~ ( finite659689794318260667_nat_b @ S )
     => ( S != bot_bo8379049445785516142_nat_b ) ) ).

% infinite_imp_nonempty
thf(fact_351_infinite__imp__nonempty,axiom,
    ! [S: set_Pr4323519195528460463od_b_b] :
      ( ~ ( finite4644902770518909432od_b_b @ S )
     => ( S != bot_bo3230273597238993691od_b_b ) ) ).

% infinite_imp_nonempty
thf(fact_352_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1307281990691478580_b_nat] :
      ( ~ ( finite7880342692102525205_b_nat @ S )
     => ( S != bot_bo5421955547594498760_b_nat ) ) ).

% infinite_imp_nonempty
thf(fact_353_infinite__imp__nonempty,axiom,
    ! [S: set_Ex3793607809372303086nnreal] :
      ( ~ ( finite3782138982310603983nnreal @ S )
     => ( S != bot_bo4854962954004695426nnreal ) ) ).

% infinite_imp_nonempty
thf(fact_354_infinite__imp__nonempty,axiom,
    ! [S: set_Extended_ereal] :
      ( ~ ( finite7198162374296863863_ereal @ S )
     => ( S != bot_bo8367695208629047834_ereal ) ) ).

% infinite_imp_nonempty
thf(fact_355_times__eq__iff,axiom,
    ! [A: set_b,B: set_b,C: set_b,D: set_b] :
      ( ( ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : B )
        = ( product_Sigma_b_b @ C
          @ ^ [Uu: b] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_b )
            | ( B = bot_bot_set_b ) )
          & ( ( C = bot_bot_set_b )
            | ( D = bot_bot_set_b ) ) ) ) ) ).

% times_eq_iff
thf(fact_356_times__eq__iff,axiom,
    ! [A: set_nat,B: set_Product_prod_b_b,C: set_nat,D: set_Product_prod_b_b] :
      ( ( ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : B )
        = ( produc8027630620858748621od_b_b @ C
          @ ^ [Uu: nat] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_nat )
            | ( B = bot_bo2792761326896053555od_b_b ) )
          & ( ( C = bot_bot_set_nat )
            | ( D = bot_bo2792761326896053555od_b_b ) ) ) ) ) ).

% times_eq_iff
thf(fact_357_times__eq__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B )
        = ( produc457027306803732586at_nat @ C
          @ ^ [Uu: nat] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_nat )
            | ( B = bot_bot_set_nat ) )
          & ( ( C = bot_bot_set_nat )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_358_times__eq__iff,axiom,
    ! [A: set_nat,B: set_b,C: set_nat,D: set_b] :
      ( ( ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : B )
        = ( product_Sigma_nat_b @ C
          @ ^ [Uu: nat] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_nat )
            | ( B = bot_bot_set_b ) )
          & ( ( C = bot_bot_set_nat )
            | ( D = bot_bot_set_b ) ) ) ) ) ).

% times_eq_iff
thf(fact_359_times__eq__iff,axiom,
    ! [A: set_b,B: set_Product_prod_b_b,C: set_b,D: set_Product_prod_b_b] :
      ( ( ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : B )
        = ( produc2915425143180021232od_b_b @ C
          @ ^ [Uu: b] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_b )
            | ( B = bot_bo2792761326896053555od_b_b ) )
          & ( ( C = bot_bot_set_b )
            | ( D = bot_bo2792761326896053555od_b_b ) ) ) ) ) ).

% times_eq_iff
thf(fact_360_times__eq__iff,axiom,
    ! [A: set_b,B: set_nat,C: set_b,D: set_nat] :
      ( ( ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : B )
        = ( product_Sigma_b_nat @ C
          @ ^ [Uu: b] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bot_set_b )
            | ( B = bot_bot_set_nat ) )
          & ( ( C = bot_bot_set_b )
            | ( D = bot_bot_set_nat ) ) ) ) ) ).

% times_eq_iff
thf(fact_361_times__eq__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal,D: set_Ex3793607809372303086nnreal] :
      ( ( ( produc7828571232000974650nnreal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B )
        = ( produc7828571232000974650nnreal @ C
          @ ^ [Uu: extend8495563244428889912nnreal] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bo4854962954004695426nnreal )
            | ( B = bot_bo4854962954004695426nnreal ) )
          & ( ( C = bot_bo4854962954004695426nnreal )
            | ( D = bot_bo4854962954004695426nnreal ) ) ) ) ) ).

% times_eq_iff
thf(fact_362_times__eq__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Extended_ereal,C: set_Ex3793607809372303086nnreal,D: set_Extended_ereal] :
      ( ( ( produc8409004985040558924_ereal @ A
          @ ^ [Uu: extend8495563244428889912nnreal] : B )
        = ( produc8409004985040558924_ereal @ C
          @ ^ [Uu: extend8495563244428889912nnreal] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bo4854962954004695426nnreal )
            | ( B = bot_bo8367695208629047834_ereal ) )
          & ( ( C = bot_bo4854962954004695426nnreal )
            | ( D = bot_bo8367695208629047834_ereal ) ) ) ) ) ).

% times_eq_iff
thf(fact_363_times__eq__iff,axiom,
    ! [A: set_Extended_ereal,B: set_Ex3793607809372303086nnreal,C: set_Extended_ereal,D: set_Ex3793607809372303086nnreal] :
      ( ( ( produc1405777165569238334nnreal @ A
          @ ^ [Uu: extended_ereal] : B )
        = ( produc1405777165569238334nnreal @ C
          @ ^ [Uu: extended_ereal] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bo8367695208629047834_ereal )
            | ( B = bot_bo4854962954004695426nnreal ) )
          & ( ( C = bot_bo8367695208629047834_ereal )
            | ( D = bot_bo4854962954004695426nnreal ) ) ) ) ) ).

% times_eq_iff
thf(fact_364_times__eq__iff,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C: set_Extended_ereal,D: set_Extended_ereal] :
      ( ( ( produc8095709571603465288_ereal @ A
          @ ^ [Uu: extended_ereal] : B )
        = ( produc8095709571603465288_ereal @ C
          @ ^ [Uu: extended_ereal] : D ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( ( A = bot_bo8367695208629047834_ereal )
            | ( B = bot_bo8367695208629047834_ereal ) )
          & ( ( C = bot_bo8367695208629047834_ereal )
            | ( D = bot_bo8367695208629047834_ereal ) ) ) ) ) ).

% times_eq_iff
thf(fact_365_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_b
            @ ^ [S2: b] : P )
          = top_top_set_b ) )
      & ( ~ P
       => ( ( collect_b
            @ ^ [S2: b] : P )
          = bot_bot_set_b ) ) ) ).

% Collect_const
thf(fact_366_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec548942219715005266od_b_b
            @ ^ [S2: product_prod_b_b] : P )
          = top_to7498756471699006487od_b_b ) )
      & ( ~ P
       => ( ( collec548942219715005266od_b_b
            @ ^ [S2: product_prod_b_b] : P )
          = bot_bo2792761326896053555od_b_b ) ) ) ).

% Collect_const
thf(fact_367_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec6648975593938027277nnreal
            @ ^ [S2: extend8495563244428889912nnreal] : P )
          = top_to7994903218803871134nnreal ) )
      & ( ~ P
       => ( ( collec6648975593938027277nnreal
            @ ^ [S2: extend8495563244428889912nnreal] : P )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% Collect_const
thf(fact_368_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S2: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S2: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_369_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec5835592288176408249_ereal
            @ ^ [S2: extended_ereal] : P )
          = top_to5683747375963461374_ereal ) )
      & ( ~ P
       => ( ( collec5835592288176408249_ereal
            @ ^ [S2: extended_ereal] : P )
          = bot_bo8367695208629047834_ereal ) ) ) ).

% Collect_const
thf(fact_370_finite__option__UNIV,axiom,
    ( ( finite1259493672748731734od_b_b @ top_to5215435299999671389od_b_b )
    = ( finite3757003017338540048od_b_b @ top_to7498756471699006487od_b_b ) ) ).

% finite_option_UNIV
thf(fact_371_finite__option__UNIV,axiom,
    ( ( finite1674126222631127406tion_b @ top_top_set_option_b )
    = ( finite_finite_b @ top_top_set_b ) ) ).

% finite_option_UNIV
thf(fact_372_finite__option__UNIV,axiom,
    ( ( finite8185795949293372275od_b_b @ top_to830859091422798914od_b_b )
    = ( finite7768965217515309219od_b_b @ top_to3802939291301058418od_b_b ) ) ).

% finite_option_UNIV
thf(fact_373_finite__option__UNIV,axiom,
    ( ( finite6732403688824079472at_nat @ top_to3251141154256563319at_nat )
    = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ).

% finite_option_UNIV
thf(fact_374_finite__option__UNIV,axiom,
    ( ( finite8448794514341901707_nat_b @ top_to196175057272917594_nat_b )
    = ( finite659689794318260667_nat_b @ top_to2683632821903171722_nat_b ) ) ).

% finite_option_UNIV
thf(fact_375_finite__option__UNIV,axiom,
    ( ( finite1495109547569289278od_b_b @ top_to7266328186411351877od_b_b )
    = ( finite4644902770518909432od_b_b @ top_to3066526200275256831od_b_b ) ) ).

% finite_option_UNIV
thf(fact_376_finite__option__UNIV,axiom,
    ( ( finite5491700616150884325_b_nat @ top_to7400060218289236916_b_nat )
    = ( finite7880342692102525205_b_nat @ top_to8949910960566930148_b_nat ) ) ).

% finite_option_UNIV
thf(fact_377_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_378_finite__option__UNIV,axiom,
    ( ( finite7804063092831063229_ereal @ top_to8408571170889687620_ereal )
    = ( finite7198162374296863863_ereal @ top_to5683747375963461374_ereal ) ) ).

% finite_option_UNIV
thf(fact_379_empty__subsetI,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A ) ).

% empty_subsetI
thf(fact_380_empty__subsetI,axiom,
    ! [A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ bot_bo8367695208629047834_ereal @ A ) ).

% empty_subsetI
thf(fact_381_subset__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% subset_empty
thf(fact_382_subset__empty,axiom,
    ! [A: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ bot_bo8367695208629047834_ereal )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% subset_empty
thf(fact_383_member__filter,axiom,
    ! [X3: nat,P: nat > $o,A: set_nat] :
      ( ( member_nat @ X3 @ ( filter_nat @ P @ A ) )
      = ( ( member_nat @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% member_filter
thf(fact_384_member__filter,axiom,
    ! [X3: b,P: b > $o,A: set_b] :
      ( ( member_b @ X3 @ ( filter_b @ P @ A ) )
      = ( ( member_b @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% member_filter
thf(fact_385_member__filter,axiom,
    ! [X3: product_prod_b_b,P: product_prod_b_b > $o,A: set_Product_prod_b_b] :
      ( ( member7862447936710763792od_b_b @ X3 @ ( filter1593123213581277208od_b_b @ P @ A ) )
      = ( ( member7862447936710763792od_b_b @ X3 @ A )
        & ( P @ X3 ) ) ) ).

% member_filter
thf(fact_386_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_387_empty__Collect__eq,axiom,
    ! [P: b > $o] :
      ( ( bot_bot_set_b
        = ( collect_b @ P ) )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_388_empty__Collect__eq,axiom,
    ! [P: product_prod_b_b > $o] :
      ( ( bot_bo2792761326896053555od_b_b
        = ( collec548942219715005266od_b_b @ P ) )
      = ( ! [X: product_prod_b_b] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_389_empty__Collect__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( bot_bo4854962954004695426nnreal
        = ( collec6648975593938027277nnreal @ P ) )
      = ( ! [X: extend8495563244428889912nnreal] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_390_empty__Collect__eq,axiom,
    ! [P: extended_ereal > $o] :
      ( ( bot_bo8367695208629047834_ereal
        = ( collec5835592288176408249_ereal @ P ) )
      = ( ! [X: extended_ereal] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_391_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_392_Collect__empty__eq,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( ! [X: b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_393_Collect__empty__eq,axiom,
    ! [P: product_prod_b_b > $o] :
      ( ( ( collec548942219715005266od_b_b @ P )
        = bot_bo2792761326896053555od_b_b )
      = ( ! [X: product_prod_b_b] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_394_Collect__empty__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( ! [X: extend8495563244428889912nnreal] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_395_Collect__empty__eq,axiom,
    ! [P: extended_ereal > $o] :
      ( ( ( collec5835592288176408249_ereal @ P )
        = bot_bo8367695208629047834_ereal )
      = ( ! [X: extended_ereal] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_396_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_397_all__not__in__conv,axiom,
    ! [A: set_b] :
      ( ( ! [X: b] :
            ~ ( member_b @ X @ A ) )
      = ( A = bot_bot_set_b ) ) ).

% all_not_in_conv
thf(fact_398_all__not__in__conv,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ( ! [X: product_prod_b_b] :
            ~ ( member7862447936710763792od_b_b @ X @ A ) )
      = ( A = bot_bo2792761326896053555od_b_b ) ) ).

% all_not_in_conv
thf(fact_399_all__not__in__conv,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ! [X: extend8495563244428889912nnreal] :
            ~ ( member7908768830364227535nnreal @ X @ A ) )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% all_not_in_conv
thf(fact_400_all__not__in__conv,axiom,
    ! [A: set_Extended_ereal] :
      ( ( ! [X: extended_ereal] :
            ~ ( member2350847679896131959_ereal @ X @ A ) )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% all_not_in_conv
thf(fact_401_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_402_empty__iff,axiom,
    ! [C2: b] :
      ~ ( member_b @ C2 @ bot_bot_set_b ) ).

% empty_iff
thf(fact_403_empty__iff,axiom,
    ! [C2: product_prod_b_b] :
      ~ ( member7862447936710763792od_b_b @ C2 @ bot_bo2792761326896053555od_b_b ) ).

% empty_iff
thf(fact_404_empty__iff,axiom,
    ! [C2: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ C2 @ bot_bo4854962954004695426nnreal ) ).

% empty_iff
thf(fact_405_empty__iff,axiom,
    ! [C2: extended_ereal] :
      ~ ( member2350847679896131959_ereal @ C2 @ bot_bo8367695208629047834_ereal ) ).

% empty_iff
thf(fact_406_subset__antisym,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( ord_le6787938422905777998nnreal @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_407_subset__antisym,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( ord_le1644982726543182158_ereal @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_408_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A )
         => ( member_nat @ X2 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_409_subsetI,axiom,
    ! [A: set_b,B: set_b] :
      ( ! [X2: b] :
          ( ( member_b @ X2 @ A )
         => ( member_b @ X2 @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% subsetI
thf(fact_410_subsetI,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b] :
      ( ! [X2: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ X2 @ A )
         => ( member7862447936710763792od_b_b @ X2 @ B ) )
     => ( ord_le182087997850975847od_b_b @ A @ B ) ) ).

% subsetI
thf(fact_411_subsetI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ! [X2: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ X2 @ A )
         => ( member7908768830364227535nnreal @ X2 @ B ) )
     => ( ord_le6787938422905777998nnreal @ A @ B ) ) ).

% subsetI
thf(fact_412_subsetI,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ! [X2: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X2 @ A )
         => ( member2350847679896131959_ereal @ X2 @ B ) )
     => ( ord_le1644982726543182158_ereal @ A @ B ) ) ).

% subsetI
thf(fact_413_UNIV__I,axiom,
    ! [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).

% UNIV_I
thf(fact_414_UNIV__I,axiom,
    ! [X3: product_prod_b_b] : ( member7862447936710763792od_b_b @ X3 @ top_to7498756471699006487od_b_b ) ).

% UNIV_I
thf(fact_415_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_416_UNIV__I,axiom,
    ! [X3: extended_ereal] : ( member2350847679896131959_ereal @ X3 @ top_to5683747375963461374_ereal ) ).

% UNIV_I
thf(fact_417_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B2: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A4 )
          @ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_418_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B2: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X: b] : ( member_b @ X @ A4 )
          @ ^ [X: b] : ( member_b @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_419_less__eq__set__def,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A4: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
          ( ord_le39139162152160566_b_b_o
          @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A4 )
          @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_420_less__eq__set__def,axiom,
    ( ord_le6787938422905777998nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
          ( ord_le7025323315894483639real_o
          @ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A4 )
          @ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_421_less__eq__set__def,axiom,
    ( ord_le1644982726543182158_ereal
    = ( ^ [A4: set_Extended_ereal,B2: set_Extended_ereal] :
          ( ord_le6694447793465728271real_o
          @ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A4 )
          @ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ B2 ) ) ) ) ).

% less_eq_set_def
thf(fact_422_UNIV__eq__I,axiom,
    ! [A: set_b] :
      ( ! [X2: b] : ( member_b @ X2 @ A )
     => ( top_top_set_b = A ) ) ).

% UNIV_eq_I
thf(fact_423_UNIV__eq__I,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ! [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ A )
     => ( top_to7498756471699006487od_b_b = A ) ) ).

% UNIV_eq_I
thf(fact_424_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] : ( member_nat @ X2 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_425_UNIV__eq__I,axiom,
    ! [A: set_Extended_ereal] :
      ( ! [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ A )
     => ( top_to5683747375963461374_ereal = A ) ) ).

% UNIV_eq_I
thf(fact_426_UNIV__witness,axiom,
    ? [X2: b] : ( member_b @ X2 @ top_top_set_b ) ).

% UNIV_witness
thf(fact_427_UNIV__witness,axiom,
    ? [X2: product_prod_b_b] : ( member7862447936710763792od_b_b @ X2 @ top_to7498756471699006487od_b_b ) ).

% UNIV_witness
thf(fact_428_UNIV__witness,axiom,
    ? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_429_UNIV__witness,axiom,
    ? [X2: extended_ereal] : ( member2350847679896131959_ereal @ X2 @ top_to5683747375963461374_ereal ) ).

% UNIV_witness
thf(fact_430_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ X3 @ A )
       => ( member_nat @ X3 @ B ) ) ) ).

% in_mono
thf(fact_431_in__mono,axiom,
    ! [A: set_b,B: set_b,X3: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ X3 @ A )
       => ( member_b @ X3 @ B ) ) ) ).

% in_mono
thf(fact_432_in__mono,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b,X3: product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A @ B )
     => ( ( member7862447936710763792od_b_b @ X3 @ A )
       => ( member7862447936710763792od_b_b @ X3 @ B ) ) ) ).

% in_mono
thf(fact_433_in__mono,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( member7908768830364227535nnreal @ X3 @ A )
       => ( member7908768830364227535nnreal @ X3 @ B ) ) ) ).

% in_mono
thf(fact_434_in__mono,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,X3: extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( member2350847679896131959_ereal @ X3 @ A )
       => ( member2350847679896131959_ereal @ X3 @ B ) ) ) ).

% in_mono
thf(fact_435_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C2 @ A )
       => ( member_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_436_subsetD,axiom,
    ! [A: set_b,B: set_b,C2: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ C2 @ A )
       => ( member_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_437_subsetD,axiom,
    ! [A: set_Product_prod_b_b,B: set_Product_prod_b_b,C2: product_prod_b_b] :
      ( ( ord_le182087997850975847od_b_b @ A @ B )
     => ( ( member7862447936710763792od_b_b @ C2 @ A )
       => ( member7862447936710763792od_b_b @ C2 @ B ) ) ) ).

% subsetD
thf(fact_438_subsetD,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( member7908768830364227535nnreal @ C2 @ A )
       => ( member7908768830364227535nnreal @ C2 @ B ) ) ) ).

% subsetD
thf(fact_439_subsetD,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C2: extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( member2350847679896131959_ereal @ C2 @ A )
       => ( member2350847679896131959_ereal @ C2 @ B ) ) ) ).

% subsetD
thf(fact_440_equalityE,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( A = B )
     => ~ ( ( ord_le6787938422905777998nnreal @ A @ B )
         => ~ ( ord_le6787938422905777998nnreal @ B @ A ) ) ) ).

% equalityE
thf(fact_441_equalityE,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ~ ( ( ord_le1644982726543182158_ereal @ A @ B )
         => ~ ( ord_le1644982726543182158_ereal @ B @ A ) ) ) ).

% equalityE
thf(fact_442_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B2: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A4 )
         => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_443_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B2: set_b] :
        ! [X: b] :
          ( ( member_b @ X @ A4 )
         => ( member_b @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_444_subset__eq,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A4: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
        ! [X: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ X @ A4 )
         => ( member7862447936710763792od_b_b @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_445_subset__eq,axiom,
    ( ord_le6787938422905777998nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
        ! [X: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ X @ A4 )
         => ( member7908768830364227535nnreal @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_446_subset__eq,axiom,
    ( ord_le1644982726543182158_ereal
    = ( ^ [A4: set_Extended_ereal,B2: set_Extended_ereal] :
        ! [X: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X @ A4 )
         => ( member2350847679896131959_ereal @ X @ B2 ) ) ) ) ).

% subset_eq
thf(fact_447_equalityD1,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( A = B )
     => ( ord_le6787938422905777998nnreal @ A @ B ) ) ).

% equalityD1
thf(fact_448_equalityD1,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ( ord_le1644982726543182158_ereal @ A @ B ) ) ).

% equalityD1
thf(fact_449_equalityD2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( A = B )
     => ( ord_le6787938422905777998nnreal @ B @ A ) ) ).

% equalityD2
thf(fact_450_equalityD2,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( A = B )
     => ( ord_le1644982726543182158_ereal @ B @ A ) ) ).

% equalityD2
thf(fact_451_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B2: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A4 )
         => ( member_nat @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_452_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B2: set_b] :
        ! [T2: b] :
          ( ( member_b @ T2 @ A4 )
         => ( member_b @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_453_subset__iff,axiom,
    ( ord_le182087997850975847od_b_b
    = ( ^ [A4: set_Product_prod_b_b,B2: set_Product_prod_b_b] :
        ! [T2: product_prod_b_b] :
          ( ( member7862447936710763792od_b_b @ T2 @ A4 )
         => ( member7862447936710763792od_b_b @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_454_subset__iff,axiom,
    ( ord_le6787938422905777998nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
        ! [T2: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ T2 @ A4 )
         => ( member7908768830364227535nnreal @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_455_subset__iff,axiom,
    ( ord_le1644982726543182158_ereal
    = ( ^ [A4: set_Extended_ereal,B2: set_Extended_ereal] :
        ! [T2: extended_ereal] :
          ( ( member2350847679896131959_ereal @ T2 @ A4 )
         => ( member2350847679896131959_ereal @ T2 @ B2 ) ) ) ) ).

% subset_iff
thf(fact_456_subset__refl,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ A ) ).

% subset_refl
thf(fact_457_subset__refl,axiom,
    ! [A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A @ A ) ).

% subset_refl
thf(fact_458_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_459_Collect__mono,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ! [X2: b] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).

% Collect_mono
thf(fact_460_Collect__mono,axiom,
    ! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ! [X2: product_prod_b_b] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le182087997850975847od_b_b @ ( collec548942219715005266od_b_b @ P ) @ ( collec548942219715005266od_b_b @ Q ) ) ) ).

% Collect_mono
thf(fact_461_Collect__mono,axiom,
    ! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ! [X2: extend8495563244428889912nnreal] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le6787938422905777998nnreal @ ( collec6648975593938027277nnreal @ P ) @ ( collec6648975593938027277nnreal @ Q ) ) ) ).

% Collect_mono
thf(fact_462_Collect__mono,axiom,
    ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
      ( ! [X2: extended_ereal] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) ) ) ).

% Collect_mono
thf(fact_463_subset__trans,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( ord_le6787938422905777998nnreal @ B @ C )
       => ( ord_le6787938422905777998nnreal @ A @ C ) ) ) ).

% subset_trans
thf(fact_464_subset__trans,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( ord_le1644982726543182158_ereal @ B @ C )
       => ( ord_le1644982726543182158_ereal @ A @ C ) ) ) ).

% subset_trans
thf(fact_465_set__eq__subset,axiom,
    ( ( ^ [Y: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] : ( Y = Z ) )
    = ( ^ [A4: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
          ( ( ord_le6787938422905777998nnreal @ A4 @ B2 )
          & ( ord_le6787938422905777998nnreal @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_466_set__eq__subset,axiom,
    ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
    = ( ^ [A4: set_Extended_ereal,B2: set_Extended_ereal] :
          ( ( ord_le1644982726543182158_ereal @ A4 @ B2 )
          & ( ord_le1644982726543182158_ereal @ B2 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_467_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_468_Collect__mono__iff,axiom,
    ! [P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
      = ( ! [X: b] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_469_Collect__mono__iff,axiom,
    ! [P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ( ord_le182087997850975847od_b_b @ ( collec548942219715005266od_b_b @ P ) @ ( collec548942219715005266od_b_b @ Q ) )
      = ( ! [X: product_prod_b_b] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_470_Collect__mono__iff,axiom,
    ! [P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ( ord_le6787938422905777998nnreal @ ( collec6648975593938027277nnreal @ P ) @ ( collec6648975593938027277nnreal @ Q ) )
      = ( ! [X: extend8495563244428889912nnreal] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_471_Collect__mono__iff,axiom,
    ! [P: extended_ereal > $o,Q: extended_ereal > $o] :
      ( ( ord_le1644982726543182158_ereal @ ( collec5835592288176408249_ereal @ P ) @ ( collec5835592288176408249_ereal @ Q ) )
      = ( ! [X: extended_ereal] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_472_emptyE,axiom,
    ! [A5: nat] :
      ~ ( member_nat @ A5 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_473_emptyE,axiom,
    ! [A5: b] :
      ~ ( member_b @ A5 @ bot_bot_set_b ) ).

% emptyE
thf(fact_474_emptyE,axiom,
    ! [A5: product_prod_b_b] :
      ~ ( member7862447936710763792od_b_b @ A5 @ bot_bo2792761326896053555od_b_b ) ).

% emptyE
thf(fact_475_emptyE,axiom,
    ! [A5: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ A5 @ bot_bo4854962954004695426nnreal ) ).

% emptyE
thf(fact_476_emptyE,axiom,
    ! [A5: extended_ereal] :
      ~ ( member2350847679896131959_ereal @ A5 @ bot_bo8367695208629047834_ereal ) ).

% emptyE
thf(fact_477_equals0D,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A5 @ A ) ) ).

% equals0D
thf(fact_478_equals0D,axiom,
    ! [A: set_b,A5: b] :
      ( ( A = bot_bot_set_b )
     => ~ ( member_b @ A5 @ A ) ) ).

% equals0D
thf(fact_479_equals0D,axiom,
    ! [A: set_Product_prod_b_b,A5: product_prod_b_b] :
      ( ( A = bot_bo2792761326896053555od_b_b )
     => ~ ( member7862447936710763792od_b_b @ A5 @ A ) ) ).

% equals0D
thf(fact_480_equals0D,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( A = bot_bo4854962954004695426nnreal )
     => ~ ( member7908768830364227535nnreal @ A5 @ A ) ) ).

% equals0D
thf(fact_481_equals0D,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( A = bot_bo8367695208629047834_ereal )
     => ~ ( member2350847679896131959_ereal @ A5 @ A ) ) ).

% equals0D
thf(fact_482_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y2: nat] :
          ~ ( member_nat @ Y2 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_483_equals0I,axiom,
    ! [A: set_b] :
      ( ! [Y2: b] :
          ~ ( member_b @ Y2 @ A )
     => ( A = bot_bot_set_b ) ) ).

% equals0I
thf(fact_484_equals0I,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ! [Y2: product_prod_b_b] :
          ~ ( member7862447936710763792od_b_b @ Y2 @ A )
     => ( A = bot_bo2792761326896053555od_b_b ) ) ).

% equals0I
thf(fact_485_equals0I,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ! [Y2: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ Y2 @ A )
     => ( A = bot_bo4854962954004695426nnreal ) ) ).

% equals0I
thf(fact_486_equals0I,axiom,
    ! [A: set_Extended_ereal] :
      ( ! [Y2: extended_ereal] :
          ~ ( member2350847679896131959_ereal @ Y2 @ A )
     => ( A = bot_bo8367695208629047834_ereal ) ) ).

% equals0I
thf(fact_487_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_488_ex__in__conv,axiom,
    ! [A: set_b] :
      ( ( ? [X: b] : ( member_b @ X @ A ) )
      = ( A != bot_bot_set_b ) ) ).

% ex_in_conv
thf(fact_489_ex__in__conv,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ( ? [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ A ) )
      = ( A != bot_bo2792761326896053555od_b_b ) ) ).

% ex_in_conv
thf(fact_490_ex__in__conv,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ? [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A ) )
      = ( A != bot_bo4854962954004695426nnreal ) ) ).

% ex_in_conv
thf(fact_491_ex__in__conv,axiom,
    ! [A: set_Extended_ereal] :
      ( ( ? [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ A ) )
      = ( A != bot_bo8367695208629047834_ereal ) ) ).

% ex_in_conv
thf(fact_492_UNIV__def,axiom,
    ( top_top_set_b
    = ( collect_b
      @ ^ [X: b] : $true ) ) ).

% UNIV_def
thf(fact_493_UNIV__def,axiom,
    ( top_to7498756471699006487od_b_b
    = ( collec548942219715005266od_b_b
      @ ^ [X: product_prod_b_b] : $true ) ) ).

% UNIV_def
thf(fact_494_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_495_UNIV__def,axiom,
    ( top_to5683747375963461374_ereal
    = ( collec5835592288176408249_ereal
      @ ^ [X: extended_ereal] : $true ) ) ).

% UNIV_def
thf(fact_496_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_497_Collect__subset,axiom,
    ! [A: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_498_Collect__subset,axiom,
    ! [A: set_Product_prod_b_b,P: product_prod_b_b > $o] :
      ( ord_le182087997850975847od_b_b
      @ ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_499_Collect__subset,axiom,
    ! [A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ord_le6787938422905777998nnreal
      @ ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_500_Collect__subset,axiom,
    ! [A: set_Extended_ereal,P: extended_ereal > $o] :
      ( ord_le1644982726543182158_ereal
      @ ( collec5835592288176408249_ereal
        @ ^ [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ A )
            & ( P @ X ) ) )
      @ A ) ).

% Collect_subset
thf(fact_501_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_502_empty__def,axiom,
    ( bot_bot_set_b
    = ( collect_b
      @ ^ [X: b] : $false ) ) ).

% empty_def
thf(fact_503_empty__def,axiom,
    ( bot_bo2792761326896053555od_b_b
    = ( collec548942219715005266od_b_b
      @ ^ [X: product_prod_b_b] : $false ) ) ).

% empty_def
thf(fact_504_empty__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal
      @ ^ [X: extend8495563244428889912nnreal] : $false ) ) ).

% empty_def
thf(fact_505_empty__def,axiom,
    ( bot_bo8367695208629047834_ereal
    = ( collec5835592288176408249_ereal
      @ ^ [X: extended_ereal] : $false ) ) ).

% empty_def
thf(fact_506_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P2: nat > $o,A4: set_nat] :
          ( collect_nat
          @ ^ [A3: nat] :
              ( ( member_nat @ A3 @ A4 )
              & ( P2 @ A3 ) ) ) ) ) ).

% Set.filter_def
thf(fact_507_Set_Ofilter__def,axiom,
    ( filter_b
    = ( ^ [P2: b > $o,A4: set_b] :
          ( collect_b
          @ ^ [A3: b] :
              ( ( member_b @ A3 @ A4 )
              & ( P2 @ A3 ) ) ) ) ) ).

% Set.filter_def
thf(fact_508_Set_Ofilter__def,axiom,
    ( filter1593123213581277208od_b_b
    = ( ^ [P2: product_prod_b_b > $o,A4: set_Product_prod_b_b] :
          ( collec548942219715005266od_b_b
          @ ^ [A3: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ A3 @ A4 )
              & ( P2 @ A3 ) ) ) ) ) ).

% Set.filter_def
thf(fact_509_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_510_subset__UNIV,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ top_to7994903218803871134nnreal ) ).

% subset_UNIV
thf(fact_511_subset__UNIV,axiom,
    ! [A: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A @ top_to5683747375963461374_ereal ) ).

% subset_UNIV
thf(fact_512_empty__not__UNIV,axiom,
    bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).

% empty_not_UNIV
thf(fact_513_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_514_empty__not__UNIV,axiom,
    bot_bo8367695208629047834_ereal != top_to5683747375963461374_ereal ).

% empty_not_UNIV
thf(fact_515_iso__tuple__UNIV__I,axiom,
    ! [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).

% iso_tuple_UNIV_I
thf(fact_516_iso__tuple__UNIV__I,axiom,
    ! [X3: product_prod_b_b] : ( member7862447936710763792od_b_b @ X3 @ top_to7498756471699006487od_b_b ) ).

% iso_tuple_UNIV_I
thf(fact_517_iso__tuple__UNIV__I,axiom,
    ! [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_518_iso__tuple__UNIV__I,axiom,
    ! [X3: extended_ereal] : ( member2350847679896131959_ereal @ X3 @ top_to5683747375963461374_ereal ) ).

% iso_tuple_UNIV_I
thf(fact_519_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_520_order__refl,axiom,
    ! [X3: extended_ereal] : ( ord_le1083603963089353582_ereal @ X3 @ X3 ) ).

% order_refl
thf(fact_521_order__refl,axiom,
    ! [X3: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X3 @ X3 ) ).

% order_refl
thf(fact_522_order__refl,axiom,
    ! [X3: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ X3 @ X3 ) ).

% order_refl
thf(fact_523_order__refl,axiom,
    ! [X3: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ X3 @ X3 ) ).

% order_refl
thf(fact_524_dual__order_Orefl,axiom,
    ! [A5: nat] : ( ord_less_eq_nat @ A5 @ A5 ) ).

% dual_order.refl
thf(fact_525_dual__order_Orefl,axiom,
    ! [A5: extended_ereal] : ( ord_le1083603963089353582_ereal @ A5 @ A5 ) ).

% dual_order.refl
thf(fact_526_dual__order_Orefl,axiom,
    ! [A5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A5 @ A5 ) ).

% dual_order.refl
thf(fact_527_dual__order_Orefl,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A5 @ A5 ) ).

% dual_order.refl
thf(fact_528_dual__order_Orefl,axiom,
    ! [A5: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A5 @ A5 ) ).

% dual_order.refl
thf(fact_529_finite__transitivity__chain,axiom,
    ! [A: set_Product_prod_b_b,R: product_prod_b_b > product_prod_b_b > $o] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ! [X2: product_prod_b_b] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: product_prod_b_b,Y2: product_prod_b_b,Z2: product_prod_b_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X2 @ A )
               => ? [Y3: product_prod_b_b] :
                    ( ( member7862447936710763792od_b_b @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo2792761326896053555od_b_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_530_finite__transitivity__chain,axiom,
    ! [A: set_b,R: b > b > $o] :
      ( ( finite_finite_b @ A )
     => ( ! [X2: b] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: b,Y2: b,Z2: b] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: b] :
                ( ( member_b @ X2 @ A )
               => ? [Y3: b] :
                    ( ( member_b @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bot_set_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_531_finite__transitivity__chain,axiom,
    ! [A: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X2: nat] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A )
               => ? [Y3: nat] :
                    ( ( member_nat @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_532_finite__transitivity__chain,axiom,
    ! [A: set_Pr5139338970096277698od_b_b,R: produc1536031394801701132od_b_b > produc1536031394801701132od_b_b > $o] :
      ( ( finite7768965217515309219od_b_b @ A )
     => ( ! [X2: produc1536031394801701132od_b_b] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: produc1536031394801701132od_b_b,Y2: produc1536031394801701132od_b_b,Z2: produc1536031394801701132od_b_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: produc1536031394801701132od_b_b] :
                ( ( member5954771037092721571od_b_b @ X2 @ A )
               => ? [Y3: produc1536031394801701132od_b_b] :
                    ( ( member5954771037092721571od_b_b @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo4493299828277062486od_b_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_533_finite__transitivity__chain,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A )
               => ? [Y3: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_534_finite__transitivity__chain,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,R: product_prod_nat_b > product_prod_nat_b > $o] :
      ( ( finite659689794318260667_nat_b @ A )
     => ( ! [X2: product_prod_nat_b] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: product_prod_nat_b,Y2: product_prod_nat_b,Z2: product_prod_nat_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: product_prod_nat_b] :
                ( ( member8962352056413324475_nat_b @ X2 @ A )
               => ? [Y3: product_prod_nat_b] :
                    ( ( member8962352056413324475_nat_b @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo8379049445785516142_nat_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_535_finite__transitivity__chain,axiom,
    ! [A: set_Pr4323519195528460463od_b_b,R: produc2840042325109449167od_b_b > produc2840042325109449167od_b_b > $o] :
      ( ( finite4644902770518909432od_b_b @ A )
     => ( ! [X2: produc2840042325109449167od_b_b] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: produc2840042325109449167od_b_b,Y2: produc2840042325109449167od_b_b,Z2: produc2840042325109449167od_b_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: produc2840042325109449167od_b_b] :
                ( ( member1867066792959145720od_b_b @ X2 @ A )
               => ? [Y3: produc2840042325109449167od_b_b] :
                    ( ( member1867066792959145720od_b_b @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo3230273597238993691od_b_b ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_536_finite__transitivity__chain,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,R: product_prod_b_nat > product_prod_b_nat > $o] :
      ( ( finite7880342692102525205_b_nat @ A )
     => ( ! [X2: product_prod_b_nat] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: product_prod_b_nat,Y2: product_prod_b_nat,Z2: product_prod_b_nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: product_prod_b_nat] :
                ( ( member6959632917342813205_b_nat @ X2 @ A )
               => ? [Y3: product_prod_b_nat] :
                    ( ( member6959632917342813205_b_nat @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo5421955547594498760_b_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_537_finite__transitivity__chain,axiom,
    ! [A: set_Ex3793607809372303086nnreal,R: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ! [X2: extend8495563244428889912nnreal] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X2 @ A )
               => ? [Y3: extend8495563244428889912nnreal] :
                    ( ( member7908768830364227535nnreal @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo4854962954004695426nnreal ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_538_finite__transitivity__chain,axiom,
    ! [A: set_Extended_ereal,R: extended_ereal > extended_ereal > $o] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ! [X2: extended_ereal] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal,Z2: extended_ereal] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [X2: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X2 @ A )
               => ? [Y3: extended_ereal] :
                    ( ( member2350847679896131959_ereal @ Y3 @ A )
                    & ( R @ X2 @ Y3 ) ) )
           => ( A = bot_bo8367695208629047834_ereal ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_539_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] :
          ~ ( member_nat @ X2 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_540_subset__emptyI,axiom,
    ! [A: set_b] :
      ( ! [X2: b] :
          ~ ( member_b @ X2 @ A )
     => ( ord_less_eq_set_b @ A @ bot_bot_set_b ) ) ).

% subset_emptyI
thf(fact_541_subset__emptyI,axiom,
    ! [A: set_Product_prod_b_b] :
      ( ! [X2: product_prod_b_b] :
          ~ ( member7862447936710763792od_b_b @ X2 @ A )
     => ( ord_le182087997850975847od_b_b @ A @ bot_bo2792761326896053555od_b_b ) ) ).

% subset_emptyI
thf(fact_542_subset__emptyI,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ! [X2: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ X2 @ A )
     => ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% subset_emptyI
thf(fact_543_subset__emptyI,axiom,
    ! [A: set_Extended_ereal] :
      ( ! [X2: extended_ereal] :
          ~ ( member2350847679896131959_ereal @ X2 @ A )
     => ( ord_le1644982726543182158_ereal @ A @ bot_bo8367695208629047834_ereal ) ) ).

% subset_emptyI
thf(fact_544_finite__indexed__bound,axiom,
    ! [A: set_b,P: b > nat > $o] :
      ( ( finite_finite_b @ A )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: b] :
            ( ( member_b @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_545_finite__indexed__bound,axiom,
    ! [A: set_nat,P: nat > nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: nat] :
            ( ( member_nat @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_546_finite__indexed__bound,axiom,
    ! [A: set_b,P: b > extended_ereal > $o] :
      ( ( finite_finite_b @ A )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ? [X_12: extended_ereal] : ( P @ X2 @ X_12 ) )
       => ? [M: extended_ereal] :
          ! [X5: b] :
            ( ( member_b @ X5 @ A )
           => ? [K: extended_ereal] :
                ( ( ord_le1083603963089353582_ereal @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_547_finite__indexed__bound,axiom,
    ! [A: set_nat,P: nat > extended_ereal > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ? [X_12: extended_ereal] : ( P @ X2 @ X_12 ) )
       => ? [M: extended_ereal] :
          ! [X5: nat] :
            ( ( member_nat @ X5 @ A )
           => ? [K: extended_ereal] :
                ( ( ord_le1083603963089353582_ereal @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_548_finite__indexed__bound,axiom,
    ! [A: set_b,P: b > extend8495563244428889912nnreal > $o] :
      ( ( finite_finite_b @ A )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ A )
           => ? [X_12: extend8495563244428889912nnreal] : ( P @ X2 @ X_12 ) )
       => ? [M: extend8495563244428889912nnreal] :
          ! [X5: b] :
            ( ( member_b @ X5 @ A )
           => ? [K: extend8495563244428889912nnreal] :
                ( ( ord_le3935885782089961368nnreal @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_549_finite__indexed__bound,axiom,
    ! [A: set_nat,P: nat > extend8495563244428889912nnreal > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A )
           => ? [X_12: extend8495563244428889912nnreal] : ( P @ X2 @ X_12 ) )
       => ? [M: extend8495563244428889912nnreal] :
          ! [X5: nat] :
            ( ( member_nat @ X5 @ A )
           => ? [K: extend8495563244428889912nnreal] :
                ( ( ord_le3935885782089961368nnreal @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_550_finite__indexed__bound,axiom,
    ! [A: set_Product_prod_b_b,P: product_prod_b_b > nat > $o] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ! [X2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_551_finite__indexed__bound,axiom,
    ! [A: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > nat > $o] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_552_finite__indexed__bound,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,P: product_prod_nat_b > nat > $o] :
      ( ( finite659689794318260667_nat_b @ A )
     => ( ! [X2: product_prod_nat_b] :
            ( ( member8962352056413324475_nat_b @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: product_prod_nat_b] :
            ( ( member8962352056413324475_nat_b @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_553_finite__indexed__bound,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,P: product_prod_b_nat > nat > $o] :
      ( ( finite7880342692102525205_b_nat @ A )
     => ( ! [X2: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X2 @ A )
           => ? [X_12: nat] : ( P @ X2 @ X_12 ) )
       => ? [M: nat] :
          ! [X5: product_prod_b_nat] :
            ( ( member6959632917342813205_b_nat @ X5 @ A )
           => ? [K: nat] :
                ( ( ord_less_eq_nat @ K @ M )
                & ( P @ X5 @ K ) ) ) ) ) ).

% finite_indexed_bound
thf(fact_554_top__set__def,axiom,
    ( top_top_set_b
    = ( collect_b @ top_top_b_o ) ) ).

% top_set_def
thf(fact_555_top__set__def,axiom,
    ( top_to7498756471699006487od_b_b
    = ( collec548942219715005266od_b_b @ top_to7135874014580417414_b_b_o ) ) ).

% top_set_def
thf(fact_556_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_557_top__set__def,axiom,
    ( top_to5683747375963461374_ereal
    = ( collec5835592288176408249_ereal @ top_to6999531812125281119real_o ) ) ).

% top_set_def
thf(fact_558_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_559_bot__set__def,axiom,
    ( bot_bot_set_b
    = ( collect_b @ bot_bot_b_o ) ) ).

% bot_set_def
thf(fact_560_bot__set__def,axiom,
    ( bot_bo2792761326896053555od_b_b
    = ( collec548942219715005266od_b_b @ bot_bo2608278733301331306_b_b_o ) ) ).

% bot_set_def
thf(fact_561_bot__set__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal @ bot_bo412624608084785539real_o ) ) ).

% bot_set_def
thf(fact_562_bot__set__def,axiom,
    ( bot_bo8367695208629047834_ereal
    = ( collec5835592288176408249_ereal @ bot_bo5519581617326455619real_o ) ) ).

% bot_set_def
thf(fact_563_order__antisym__conv,axiom,
    ! [Y4: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y4 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y4 )
        = ( X3 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_564_order__antisym__conv,axiom,
    ! [Y4: extended_ereal,X3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ Y4 @ X3 )
     => ( ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
        = ( X3 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_565_order__antisym__conv,axiom,
    ! [Y4: extend8495563244428889912nnreal,X3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ Y4 @ X3 )
     => ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
        = ( X3 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_566_order__antisym__conv,axiom,
    ! [Y4: set_Ex3793607809372303086nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ Y4 @ X3 )
     => ( ( ord_le6787938422905777998nnreal @ X3 @ Y4 )
        = ( X3 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_567_order__antisym__conv,axiom,
    ! [Y4: set_Extended_ereal,X3: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ Y4 @ X3 )
     => ( ( ord_le1644982726543182158_ereal @ X3 @ Y4 )
        = ( X3 = Y4 ) ) ) ).

% order_antisym_conv
thf(fact_568_linorder__le__cases,axiom,
    ! [X3: nat,Y4: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y4 )
     => ( ord_less_eq_nat @ Y4 @ X3 ) ) ).

% linorder_le_cases
thf(fact_569_linorder__le__cases,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ~ ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
     => ( ord_le1083603963089353582_ereal @ Y4 @ X3 ) ) ).

% linorder_le_cases
thf(fact_570_linorder__le__cases,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ~ ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
     => ( ord_le3935885782089961368nnreal @ Y4 @ X3 ) ) ).

% linorder_le_cases
thf(fact_571_ord__le__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_572_ord__le__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > extended_ereal,C2: extended_ereal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_573_ord__le__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_574_ord__le__eq__subst,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > nat,C2: nat] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_575_ord__le__eq__subst,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_576_ord__le__eq__subst,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_577_ord__le__eq__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C2: nat] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_578_ord__le__eq__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_579_ord__le__eq__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_580_ord__le__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le6787938422905777998nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_581_ord__eq__le__subst,axiom,
    ! [A5: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_582_ord__eq__le__subst,axiom,
    ! [A5: extended_ereal,F: nat > extended_ereal,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_583_ord__eq__le__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_584_ord__eq__le__subst,axiom,
    ! [A5: nat,F: extended_ereal > nat,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_585_ord__eq__le__subst,axiom,
    ! [A5: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_586_ord__eq__le__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_587_ord__eq__le__subst,axiom,
    ! [A5: nat,F: extend8495563244428889912nnreal > nat,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_588_ord__eq__le__subst,axiom,
    ! [A5: extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_589_ord__eq__le__subst,axiom,
    ! [A5: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_590_ord__eq__le__subst,axiom,
    ! [A5: set_Ex3793607809372303086nnreal,F: nat > set_Ex3793607809372303086nnreal,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le6787938422905777998nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_591_linorder__linear,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y4 )
      | ( ord_less_eq_nat @ Y4 @ X3 ) ) ).

% linorder_linear
thf(fact_592_linorder__linear,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
      | ( ord_le1083603963089353582_ereal @ Y4 @ X3 ) ) ).

% linorder_linear
thf(fact_593_linorder__linear,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
      | ( ord_le3935885782089961368nnreal @ Y4 @ X3 ) ) ).

% linorder_linear
thf(fact_594_order__eq__refl,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( X3 = Y4 )
     => ( ord_less_eq_nat @ X3 @ Y4 ) ) ).

% order_eq_refl
thf(fact_595_order__eq__refl,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( X3 = Y4 )
     => ( ord_le1083603963089353582_ereal @ X3 @ Y4 ) ) ).

% order_eq_refl
thf(fact_596_order__eq__refl,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( X3 = Y4 )
     => ( ord_le3935885782089961368nnreal @ X3 @ Y4 ) ) ).

% order_eq_refl
thf(fact_597_order__eq__refl,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
      ( ( X3 = Y4 )
     => ( ord_le6787938422905777998nnreal @ X3 @ Y4 ) ) ).

% order_eq_refl
thf(fact_598_order__eq__refl,axiom,
    ! [X3: set_Extended_ereal,Y4: set_Extended_ereal] :
      ( ( X3 = Y4 )
     => ( ord_le1644982726543182158_ereal @ X3 @ Y4 ) ) ).

% order_eq_refl
thf(fact_599_order__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_600_order__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > extended_ereal,C2: extended_ereal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_le1083603963089353582_ereal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_601_order__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_602_order__subst2,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > nat,C2: nat] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_603_order__subst2,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ord_le1083603963089353582_ereal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_604_order__subst2,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_605_order__subst2,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C2: nat] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_606_order__subst2,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ord_le1083603963089353582_ereal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_607_order__subst2,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ord_le3935885782089961368nnreal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_608_order__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_le6787938422905777998nnreal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le6787938422905777998nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le6787938422905777998nnreal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_609_order__subst1,axiom,
    ! [A5: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_610_order__subst1,axiom,
    ! [A5: nat,F: extended_ereal > nat,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_less_eq_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_611_order__subst1,axiom,
    ! [A5: nat,F: extend8495563244428889912nnreal > nat,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_less_eq_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_612_order__subst1,axiom,
    ! [A5: extended_ereal,F: nat > extended_ereal,B3: nat,C2: nat] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_613_order__subst1,axiom,
    ! [A5: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_614_order__subst1,axiom,
    ! [A5: extended_ereal,F: extend8495563244428889912nnreal > extended_ereal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le1083603963089353582_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1083603963089353582_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_615_order__subst1,axiom,
    ! [A5: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B3: nat,C2: nat] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_616_order__subst1,axiom,
    ! [A5: extend8495563244428889912nnreal,F: extended_ereal > extend8495563244428889912nnreal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ ( F @ B3 ) )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1083603963089353582_ereal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_617_order__subst1,axiom,
    ! [A5: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ ( F @ B3 ) )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ! [X2: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
              ( ( ord_le3935885782089961368nnreal @ X2 @ Y2 )
             => ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le3935885782089961368nnreal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_618_order__subst1,axiom,
    ! [A5: nat,F: set_Ex3793607809372303086nnreal > nat,B3: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_less_eq_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_le6787938422905777998nnreal @ B3 @ C2 )
       => ( ! [X2: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] :
              ( ( ord_le6787938422905777998nnreal @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_619_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A3 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_620_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [A3: extended_ereal,B4: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A3 @ B4 )
          & ( ord_le1083603963089353582_ereal @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_621_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y = Z ) )
    = ( ^ [A3: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A3 @ B4 )
          & ( ord_le3935885782089961368nnreal @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_622_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] : ( Y = Z ) )
    = ( ^ [A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
          ( ( ord_le6787938422905777998nnreal @ A3 @ B4 )
          & ( ord_le6787938422905777998nnreal @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_623_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
    = ( ^ [A3: set_Extended_ereal,B4: set_Extended_ereal] :
          ( ( ord_le1644982726543182158_ereal @ A3 @ B4 )
          & ( ord_le1644982726543182158_ereal @ B4 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_624_antisym,axiom,
    ! [A5: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A5 )
       => ( A5 = B3 ) ) ) ).

% antisym
thf(fact_625_antisym,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ A5 )
       => ( A5 = B3 ) ) ) ).

% antisym
thf(fact_626_antisym,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ A5 )
       => ( A5 = B3 ) ) ) ).

% antisym
thf(fact_627_antisym,axiom,
    ! [A5: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A5 @ B3 )
     => ( ( ord_le6787938422905777998nnreal @ B3 @ A5 )
       => ( A5 = B3 ) ) ) ).

% antisym
thf(fact_628_antisym,axiom,
    ! [A5: set_Extended_ereal,B3: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A5 @ B3 )
     => ( ( ord_le1644982726543182158_ereal @ B3 @ A5 )
       => ( A5 = B3 ) ) ) ).

% antisym
thf(fact_629_dual__order_Otrans,axiom,
    ! [B3: nat,A5: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A5 )
     => ( ( ord_less_eq_nat @ C2 @ B3 )
       => ( ord_less_eq_nat @ C2 @ A5 ) ) ) ).

% dual_order.trans
thf(fact_630_dual__order_Otrans,axiom,
    ! [B3: extended_ereal,A5: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B3 @ A5 )
     => ( ( ord_le1083603963089353582_ereal @ C2 @ B3 )
       => ( ord_le1083603963089353582_ereal @ C2 @ A5 ) ) ) ).

% dual_order.trans
thf(fact_631_dual__order_Otrans,axiom,
    ! [B3: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B3 @ A5 )
     => ( ( ord_le3935885782089961368nnreal @ C2 @ B3 )
       => ( ord_le3935885782089961368nnreal @ C2 @ A5 ) ) ) ).

% dual_order.trans
thf(fact_632_dual__order_Otrans,axiom,
    ! [B3: set_Ex3793607809372303086nnreal,A5: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ B3 @ A5 )
     => ( ( ord_le6787938422905777998nnreal @ C2 @ B3 )
       => ( ord_le6787938422905777998nnreal @ C2 @ A5 ) ) ) ).

% dual_order.trans
thf(fact_633_dual__order_Otrans,axiom,
    ! [B3: set_Extended_ereal,A5: set_Extended_ereal,C2: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ B3 @ A5 )
     => ( ( ord_le1644982726543182158_ereal @ C2 @ B3 )
       => ( ord_le1644982726543182158_ereal @ C2 @ A5 ) ) ) ).

% dual_order.trans
thf(fact_634_dual__order_Oantisym,axiom,
    ! [B3: nat,A5: nat] :
      ( ( ord_less_eq_nat @ B3 @ A5 )
     => ( ( ord_less_eq_nat @ A5 @ B3 )
       => ( A5 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_635_dual__order_Oantisym,axiom,
    ! [B3: extended_ereal,A5: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ B3 @ A5 )
     => ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
       => ( A5 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_636_dual__order_Oantisym,axiom,
    ! [B3: extend8495563244428889912nnreal,A5: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ B3 @ A5 )
     => ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
       => ( A5 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_637_dual__order_Oantisym,axiom,
    ! [B3: set_Ex3793607809372303086nnreal,A5: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ B3 @ A5 )
     => ( ( ord_le6787938422905777998nnreal @ A5 @ B3 )
       => ( A5 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_638_dual__order_Oantisym,axiom,
    ! [B3: set_Extended_ereal,A5: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ B3 @ A5 )
     => ( ( ord_le1644982726543182158_ereal @ A5 @ B3 )
       => ( A5 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_639_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [A3: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_640_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [A3: extended_ereal,B4: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ B4 @ A3 )
          & ( ord_le1083603963089353582_ereal @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_641_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y = Z ) )
    = ( ^ [A3: extend8495563244428889912nnreal,B4: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ B4 @ A3 )
          & ( ord_le3935885782089961368nnreal @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_642_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] : ( Y = Z ) )
    = ( ^ [A3: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
          ( ( ord_le6787938422905777998nnreal @ B4 @ A3 )
          & ( ord_le6787938422905777998nnreal @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_643_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
    = ( ^ [A3: set_Extended_ereal,B4: set_Extended_ereal] :
          ( ( ord_le1644982726543182158_ereal @ B4 @ A3 )
          & ( ord_le1644982726543182158_ereal @ A3 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_644_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A5: nat,B3: nat] :
      ( ! [A2: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A2 @ B5 )
         => ( P @ A2 @ B5 ) )
     => ( ! [A2: nat,B5: nat] :
            ( ( P @ B5 @ A2 )
           => ( P @ A2 @ B5 ) )
       => ( P @ A5 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_645_linorder__wlog,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A5: extended_ereal,B3: extended_ereal] :
      ( ! [A2: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ A2 @ B5 )
         => ( P @ A2 @ B5 ) )
     => ( ! [A2: extended_ereal,B5: extended_ereal] :
            ( ( P @ B5 @ A2 )
           => ( P @ A2 @ B5 ) )
       => ( P @ A5 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_646_linorder__wlog,axiom,
    ! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
      ( ! [A2: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ A2 @ B5 )
         => ( P @ A2 @ B5 ) )
     => ( ! [A2: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
            ( ( P @ B5 @ A2 )
           => ( P @ A2 @ B5 ) )
       => ( P @ A5 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_647_order__trans,axiom,
    ! [X3: nat,Y4: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ Z3 )
       => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_648_order__trans,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal,Z3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
     => ( ( ord_le1083603963089353582_ereal @ Y4 @ Z3 )
       => ( ord_le1083603963089353582_ereal @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_649_order__trans,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ Z3 )
       => ( ord_le3935885782089961368nnreal @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_650_order__trans,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal,Z3: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X3 @ Y4 )
     => ( ( ord_le6787938422905777998nnreal @ Y4 @ Z3 )
       => ( ord_le6787938422905777998nnreal @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_651_order__trans,axiom,
    ! [X3: set_Extended_ereal,Y4: set_Extended_ereal,Z3: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ X3 @ Y4 )
     => ( ( ord_le1644982726543182158_ereal @ Y4 @ Z3 )
       => ( ord_le1644982726543182158_ereal @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_652_order_Otrans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A5 @ C2 ) ) ) ).

% order.trans
thf(fact_653_order_Otrans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ord_le1083603963089353582_ereal @ A5 @ C2 ) ) ) ).

% order.trans
thf(fact_654_order_Otrans,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ord_le3935885782089961368nnreal @ A5 @ C2 ) ) ) ).

% order.trans
thf(fact_655_order_Otrans,axiom,
    ! [A5: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A5 @ B3 )
     => ( ( ord_le6787938422905777998nnreal @ B3 @ C2 )
       => ( ord_le6787938422905777998nnreal @ A5 @ C2 ) ) ) ).

% order.trans
thf(fact_656_order_Otrans,axiom,
    ! [A5: set_Extended_ereal,B3: set_Extended_ereal,C2: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A5 @ B3 )
     => ( ( ord_le1644982726543182158_ereal @ B3 @ C2 )
       => ( ord_le1644982726543182158_ereal @ A5 @ C2 ) ) ) ).

% order.trans
thf(fact_657_order__antisym,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y4 )
     => ( ( ord_less_eq_nat @ Y4 @ X3 )
       => ( X3 = Y4 ) ) ) ).

% order_antisym
thf(fact_658_order__antisym,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
     => ( ( ord_le1083603963089353582_ereal @ Y4 @ X3 )
       => ( X3 = Y4 ) ) ) ).

% order_antisym
thf(fact_659_order__antisym,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
     => ( ( ord_le3935885782089961368nnreal @ Y4 @ X3 )
       => ( X3 = Y4 ) ) ) ).

% order_antisym
thf(fact_660_order__antisym,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X3 @ Y4 )
     => ( ( ord_le6787938422905777998nnreal @ Y4 @ X3 )
       => ( X3 = Y4 ) ) ) ).

% order_antisym
thf(fact_661_order__antisym,axiom,
    ! [X3: set_Extended_ereal,Y4: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ X3 @ Y4 )
     => ( ( ord_le1644982726543182158_ereal @ Y4 @ X3 )
       => ( X3 = Y4 ) ) ) ).

% order_antisym
thf(fact_662_ord__le__eq__trans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_eq_nat @ A5 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_663_ord__le__eq__trans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le1083603963089353582_ereal @ A5 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_664_ord__le__eq__trans,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le3935885782089961368nnreal @ A5 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_665_ord__le__eq__trans,axiom,
    ! [A5: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le6787938422905777998nnreal @ A5 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_666_ord__le__eq__trans,axiom,
    ! [A5: set_Extended_ereal,B3: set_Extended_ereal,C2: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le1644982726543182158_ereal @ A5 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_667_ord__eq__le__trans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( A5 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A5 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_668_ord__eq__le__trans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5 = B3 )
     => ( ( ord_le1083603963089353582_ereal @ B3 @ C2 )
       => ( ord_le1083603963089353582_ereal @ A5 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_669_ord__eq__le__trans,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal,C2: extend8495563244428889912nnreal] :
      ( ( A5 = B3 )
     => ( ( ord_le3935885782089961368nnreal @ B3 @ C2 )
       => ( ord_le3935885782089961368nnreal @ A5 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_670_ord__eq__le__trans,axiom,
    ! [A5: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal,C2: set_Ex3793607809372303086nnreal] :
      ( ( A5 = B3 )
     => ( ( ord_le6787938422905777998nnreal @ B3 @ C2 )
       => ( ord_le6787938422905777998nnreal @ A5 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_671_ord__eq__le__trans,axiom,
    ! [A5: set_Extended_ereal,B3: set_Extended_ereal,C2: set_Extended_ereal] :
      ( ( A5 = B3 )
     => ( ( ord_le1644982726543182158_ereal @ B3 @ C2 )
       => ( ord_le1644982726543182158_ereal @ A5 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_672_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
    = ( ^ [X: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_673_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: extended_ereal,Z: extended_ereal] : ( Y = Z ) )
    = ( ^ [X: extended_ereal,Y5: extended_ereal] :
          ( ( ord_le1083603963089353582_ereal @ X @ Y5 )
          & ( ord_le1083603963089353582_ereal @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_674_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y = Z ) )
    = ( ^ [X: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] :
          ( ( ord_le3935885782089961368nnreal @ X @ Y5 )
          & ( ord_le3935885782089961368nnreal @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_675_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_Ex3793607809372303086nnreal,Z: set_Ex3793607809372303086nnreal] : ( Y = Z ) )
    = ( ^ [X: set_Ex3793607809372303086nnreal,Y5: set_Ex3793607809372303086nnreal] :
          ( ( ord_le6787938422905777998nnreal @ X @ Y5 )
          & ( ord_le6787938422905777998nnreal @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_676_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y: set_Extended_ereal,Z: set_Extended_ereal] : ( Y = Z ) )
    = ( ^ [X: set_Extended_ereal,Y5: set_Extended_ereal] :
          ( ( ord_le1644982726543182158_ereal @ X @ Y5 )
          & ( ord_le1644982726543182158_ereal @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_677_le__cases3,axiom,
    ! [X3: nat,Y4: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y4 )
       => ~ ( ord_less_eq_nat @ Y4 @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y4 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y4 ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y4 )
             => ~ ( ord_less_eq_nat @ Y4 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y4 @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_678_le__cases3,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal,Z3: extended_ereal] :
      ( ( ( ord_le1083603963089353582_ereal @ X3 @ Y4 )
       => ~ ( ord_le1083603963089353582_ereal @ Y4 @ Z3 ) )
     => ( ( ( ord_le1083603963089353582_ereal @ Y4 @ X3 )
         => ~ ( ord_le1083603963089353582_ereal @ X3 @ Z3 ) )
       => ( ( ( ord_le1083603963089353582_ereal @ X3 @ Z3 )
           => ~ ( ord_le1083603963089353582_ereal @ Z3 @ Y4 ) )
         => ( ( ( ord_le1083603963089353582_ereal @ Z3 @ Y4 )
             => ~ ( ord_le1083603963089353582_ereal @ Y4 @ X3 ) )
           => ( ( ( ord_le1083603963089353582_ereal @ Y4 @ Z3 )
               => ~ ( ord_le1083603963089353582_ereal @ Z3 @ X3 ) )
             => ~ ( ( ord_le1083603963089353582_ereal @ Z3 @ X3 )
                 => ~ ( ord_le1083603963089353582_ereal @ X3 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_679_le__cases3,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] :
      ( ( ( ord_le3935885782089961368nnreal @ X3 @ Y4 )
       => ~ ( ord_le3935885782089961368nnreal @ Y4 @ Z3 ) )
     => ( ( ( ord_le3935885782089961368nnreal @ Y4 @ X3 )
         => ~ ( ord_le3935885782089961368nnreal @ X3 @ Z3 ) )
       => ( ( ( ord_le3935885782089961368nnreal @ X3 @ Z3 )
           => ~ ( ord_le3935885782089961368nnreal @ Z3 @ Y4 ) )
         => ( ( ( ord_le3935885782089961368nnreal @ Z3 @ Y4 )
             => ~ ( ord_le3935885782089961368nnreal @ Y4 @ X3 ) )
           => ( ( ( ord_le3935885782089961368nnreal @ Y4 @ Z3 )
               => ~ ( ord_le3935885782089961368nnreal @ Z3 @ X3 ) )
             => ~ ( ( ord_le3935885782089961368nnreal @ Z3 @ X3 )
                 => ~ ( ord_le3935885782089961368nnreal @ X3 @ Y4 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_680_nle__le,axiom,
    ! [A5: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A5 @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A5 )
        & ( B3 != A5 ) ) ) ).

% nle_le
thf(fact_681_nle__le,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ~ ( ord_le1083603963089353582_ereal @ A5 @ B3 ) )
      = ( ( ord_le1083603963089353582_ereal @ B3 @ A5 )
        & ( B3 != A5 ) ) ) ).

% nle_le
thf(fact_682_nle__le,axiom,
    ! [A5: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
      ( ( ~ ( ord_le3935885782089961368nnreal @ A5 @ B3 ) )
      = ( ( ord_le3935885782089961368nnreal @ B3 @ A5 )
        & ( B3 != A5 ) ) ) ).

% nle_le
thf(fact_683_Collect__restrict,axiom,
    ! [X4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X4 )
            & ( P @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_684_Collect__restrict,axiom,
    ! [X4: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ X4 )
            & ( P @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_685_Collect__restrict,axiom,
    ! [X4: set_Product_prod_b_b,P: product_prod_b_b > $o] :
      ( ord_le182087997850975847od_b_b
      @ ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X @ X4 )
            & ( P @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_686_Collect__restrict,axiom,
    ! [X4: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ord_le6787938422905777998nnreal
      @ ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ X4 )
            & ( P @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_687_Collect__restrict,axiom,
    ! [X4: set_Extended_ereal,P: extended_ereal > $o] :
      ( ord_le1644982726543182158_ereal
      @ ( collec5835592288176408249_ereal
        @ ^ [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ X4 )
            & ( P @ X ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_688_prop__restrict,axiom,
    ! [X3: nat,Z4: set_nat,X4: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X4 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_689_prop__restrict,axiom,
    ! [X3: b,Z4: set_b,X4: set_b,P: b > $o] :
      ( ( member_b @ X3 @ Z4 )
     => ( ( ord_less_eq_set_b @ Z4
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ X4 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_690_prop__restrict,axiom,
    ! [X3: product_prod_b_b,Z4: set_Product_prod_b_b,X4: set_Product_prod_b_b,P: product_prod_b_b > $o] :
      ( ( member7862447936710763792od_b_b @ X3 @ Z4 )
     => ( ( ord_le182087997850975847od_b_b @ Z4
          @ ( collec548942219715005266od_b_b
            @ ^ [X: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X @ X4 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_691_prop__restrict,axiom,
    ! [X3: extend8495563244428889912nnreal,Z4: set_Ex3793607809372303086nnreal,X4: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ( member7908768830364227535nnreal @ X3 @ Z4 )
     => ( ( ord_le6787938422905777998nnreal @ Z4
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ X4 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_692_prop__restrict,axiom,
    ! [X3: extended_ereal,Z4: set_Extended_ereal,X4: set_Extended_ereal,P: extended_ereal > $o] :
      ( ( member2350847679896131959_ereal @ X3 @ Z4 )
     => ( ( ord_le1644982726543182158_ereal @ Z4
          @ ( collec5835592288176408249_ereal
            @ ^ [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ X4 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_693_top_Oextremum__uniqueI,axiom,
    ! [A5: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A5 )
     => ( A5 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_694_top_Oextremum__uniqueI,axiom,
    ! [A5: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ top_to6662034908053899550_ereal @ A5 )
     => ( A5 = top_to6662034908053899550_ereal ) ) ).

% top.extremum_uniqueI
thf(fact_695_top_Oextremum__uniqueI,axiom,
    ! [A5: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A5 )
     => ( A5 = top_to1496364449551166952nnreal ) ) ).

% top.extremum_uniqueI
thf(fact_696_top_Oextremum__uniqueI,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A5 )
     => ( A5 = top_to7994903218803871134nnreal ) ) ).

% top.extremum_uniqueI
thf(fact_697_top_Oextremum__uniqueI,axiom,
    ! [A5: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ top_to5683747375963461374_ereal @ A5 )
     => ( A5 = top_to5683747375963461374_ereal ) ) ).

% top.extremum_uniqueI
thf(fact_698_top_Oextremum__unique,axiom,
    ! [A5: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A5 )
      = ( A5 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_699_top_Oextremum__unique,axiom,
    ! [A5: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ top_to6662034908053899550_ereal @ A5 )
      = ( A5 = top_to6662034908053899550_ereal ) ) ).

% top.extremum_unique
thf(fact_700_top_Oextremum__unique,axiom,
    ! [A5: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ top_to1496364449551166952nnreal @ A5 )
      = ( A5 = top_to1496364449551166952nnreal ) ) ).

% top.extremum_unique
thf(fact_701_top_Oextremum__unique,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A5 )
      = ( A5 = top_to7994903218803871134nnreal ) ) ).

% top.extremum_unique
thf(fact_702_top_Oextremum__unique,axiom,
    ! [A5: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ top_to5683747375963461374_ereal @ A5 )
      = ( A5 = top_to5683747375963461374_ereal ) ) ).

% top.extremum_unique
thf(fact_703_top__greatest,axiom,
    ! [A5: set_nat] : ( ord_less_eq_set_nat @ A5 @ top_top_set_nat ) ).

% top_greatest
thf(fact_704_top__greatest,axiom,
    ! [A5: extended_ereal] : ( ord_le1083603963089353582_ereal @ A5 @ top_to6662034908053899550_ereal ) ).

% top_greatest
thf(fact_705_top__greatest,axiom,
    ! [A5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A5 @ top_to1496364449551166952nnreal ) ).

% top_greatest
thf(fact_706_top__greatest,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A5 @ top_to7994903218803871134nnreal ) ).

% top_greatest
thf(fact_707_top__greatest,axiom,
    ! [A5: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ A5 @ top_to5683747375963461374_ereal ) ).

% top_greatest
thf(fact_708_bot_Oextremum__uniqueI,axiom,
    ! [A5: nat] :
      ( ( ord_less_eq_nat @ A5 @ bot_bot_nat )
     => ( A5 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_709_bot_Oextremum__uniqueI,axiom,
    ! [A5: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ bot_bo2710585358178759738_ereal )
     => ( A5 = bot_bo2710585358178759738_ereal ) ) ).

% bot.extremum_uniqueI
thf(fact_710_bot_Oextremum__uniqueI,axiom,
    ! [A5: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ bot_bo841427958541957580nnreal )
     => ( A5 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_711_bot_Oextremum__uniqueI,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A5 @ bot_bo4854962954004695426nnreal )
     => ( A5 = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_712_bot_Oextremum__uniqueI,axiom,
    ! [A5: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A5 @ bot_bo8367695208629047834_ereal )
     => ( A5 = bot_bo8367695208629047834_ereal ) ) ).

% bot.extremum_uniqueI
thf(fact_713_bot_Oextremum__unique,axiom,
    ! [A5: nat] :
      ( ( ord_less_eq_nat @ A5 @ bot_bot_nat )
      = ( A5 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_714_bot_Oextremum__unique,axiom,
    ! [A5: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ A5 @ bot_bo2710585358178759738_ereal )
      = ( A5 = bot_bo2710585358178759738_ereal ) ) ).

% bot.extremum_unique
thf(fact_715_bot_Oextremum__unique,axiom,
    ! [A5: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A5 @ bot_bo841427958541957580nnreal )
      = ( A5 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_unique
thf(fact_716_bot_Oextremum__unique,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A5 @ bot_bo4854962954004695426nnreal )
      = ( A5 = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_unique
thf(fact_717_bot_Oextremum__unique,axiom,
    ! [A5: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A5 @ bot_bo8367695208629047834_ereal )
      = ( A5 = bot_bo8367695208629047834_ereal ) ) ).

% bot.extremum_unique
thf(fact_718_bot_Oextremum,axiom,
    ! [A5: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A5 ) ).

% bot.extremum
thf(fact_719_bot_Oextremum,axiom,
    ! [A5: extended_ereal] : ( ord_le1083603963089353582_ereal @ bot_bo2710585358178759738_ereal @ A5 ) ).

% bot.extremum
thf(fact_720_bot_Oextremum,axiom,
    ! [A5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A5 ) ).

% bot.extremum
thf(fact_721_bot_Oextremum,axiom,
    ! [A5: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A5 ) ).

% bot.extremum
thf(fact_722_bot_Oextremum,axiom,
    ! [A5: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ bot_bo8367695208629047834_ereal @ A5 ) ).

% bot.extremum
thf(fact_723_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_724_pred__subset__eq,axiom,
    ! [R: set_b,S: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X: b] : ( member_b @ X @ R )
        @ ^ [X: b] : ( member_b @ X @ S ) )
      = ( ord_less_eq_set_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_725_pred__subset__eq,axiom,
    ! [R: set_Product_prod_b_b,S: set_Product_prod_b_b] :
      ( ( ord_le39139162152160566_b_b_o
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ R )
        @ ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ S ) )
      = ( ord_le182087997850975847od_b_b @ R @ S ) ) ).

% pred_subset_eq
thf(fact_726_pred__subset__eq,axiom,
    ! [R: set_Ex3793607809372303086nnreal,S: set_Ex3793607809372303086nnreal] :
      ( ( ord_le7025323315894483639real_o
        @ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ R )
        @ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ S ) )
      = ( ord_le6787938422905777998nnreal @ R @ S ) ) ).

% pred_subset_eq
thf(fact_727_pred__subset__eq,axiom,
    ! [R: set_Extended_ereal,S: set_Extended_ereal] :
      ( ( ord_le6694447793465728271real_o
        @ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ R )
        @ ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ S ) )
      = ( ord_le1644982726543182158_ereal @ R @ S ) ) ).

% pred_subset_eq
thf(fact_728_Set_Ois__empty__def,axiom,
    ( is_emp182806100662350310nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal] : ( A4 = bot_bo4854962954004695426nnreal ) ) ) ).

% Set.is_empty_def
thf(fact_729_Set_Ois__empty__def,axiom,
    ( is_emp6845480677552319904_ereal
    = ( ^ [A4: set_Extended_ereal] : ( A4 = bot_bo8367695208629047834_ereal ) ) ) ).

% Set.is_empty_def
thf(fact_730_subset__Collect__iff,axiom,
    ! [B: set_nat,A: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ B
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_731_subset__Collect__iff,axiom,
    ! [B: set_b,A: set_b,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ( ord_less_eq_set_b @ B
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: b] :
              ( ( member_b @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_732_subset__Collect__iff,axiom,
    ! [B: set_Product_prod_b_b,A: set_Product_prod_b_b,P: product_prod_b_b > $o] :
      ( ( ord_le182087997850975847od_b_b @ B @ A )
     => ( ( ord_le182087997850975847od_b_b @ B
          @ ( collec548942219715005266od_b_b
            @ ^ [X: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_733_subset__Collect__iff,axiom,
    ! [B: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
      ( ( ord_le6787938422905777998nnreal @ B @ A )
     => ( ( ord_le6787938422905777998nnreal @ B
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_734_subset__Collect__iff,axiom,
    ! [B: set_Extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
      ( ( ord_le1644982726543182158_ereal @ B @ A )
     => ( ( ord_le1644982726543182158_ereal @ B
          @ ( collec5835592288176408249_ereal
            @ ^ [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
                & ( P @ X ) ) ) )
        = ( ! [X: extended_ereal] :
              ( ( member2350847679896131959_ereal @ X @ B )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_735_subset__CollectI,axiom,
    ! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ B )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_736_subset__CollectI,axiom,
    ! [B: set_b,A: set_b,Q: b > $o,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ! [X2: b] :
            ( ( member_b @ X2 @ B )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_less_eq_set_b
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ B )
                & ( Q @ X ) ) )
          @ ( collect_b
            @ ^ [X: b] :
                ( ( member_b @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_737_subset__CollectI,axiom,
    ! [B: set_Product_prod_b_b,A: set_Product_prod_b_b,Q: product_prod_b_b > $o,P: product_prod_b_b > $o] :
      ( ( ord_le182087997850975847od_b_b @ B @ A )
     => ( ! [X2: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X2 @ B )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le182087997850975847od_b_b
          @ ( collec548942219715005266od_b_b
            @ ^ [X: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec548942219715005266od_b_b
            @ ^ [X: product_prod_b_b] :
                ( ( member7862447936710763792od_b_b @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_738_subset__CollectI,axiom,
    ! [B: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,Q: extend8495563244428889912nnreal > $o,P: extend8495563244428889912nnreal > $o] :
      ( ( ord_le6787938422905777998nnreal @ B @ A )
     => ( ! [X2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X2 @ B )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le6787938422905777998nnreal
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec6648975593938027277nnreal
            @ ^ [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_739_subset__CollectI,axiom,
    ! [B: set_Extended_ereal,A: set_Extended_ereal,Q: extended_ereal > $o,P: extended_ereal > $o] :
      ( ( ord_le1644982726543182158_ereal @ B @ A )
     => ( ! [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ B )
           => ( ( Q @ X2 )
             => ( P @ X2 ) ) )
       => ( ord_le1644982726543182158_ereal
          @ ( collec5835592288176408249_ereal
            @ ^ [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ B )
                & ( Q @ X ) ) )
          @ ( collec5835592288176408249_ereal
            @ ^ [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_740_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_741_conj__subset__def,axiom,
    ! [A: set_b,P: b > $o,Q: b > $o] :
      ( ( ord_less_eq_set_b @ A
        @ ( collect_b
          @ ^ [X: b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_b @ A @ ( collect_b @ P ) )
        & ( ord_less_eq_set_b @ A @ ( collect_b @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_742_conj__subset__def,axiom,
    ! [A: set_Product_prod_b_b,P: product_prod_b_b > $o,Q: product_prod_b_b > $o] :
      ( ( ord_le182087997850975847od_b_b @ A
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le182087997850975847od_b_b @ A @ ( collec548942219715005266od_b_b @ P ) )
        & ( ord_le182087997850975847od_b_b @ A @ ( collec548942219715005266od_b_b @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_743_conj__subset__def,axiom,
    ! [A: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o,Q: extend8495563244428889912nnreal > $o] :
      ( ( ord_le6787938422905777998nnreal @ A
        @ ( collec6648975593938027277nnreal
          @ ^ [X: extend8495563244428889912nnreal] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le6787938422905777998nnreal @ A @ ( collec6648975593938027277nnreal @ P ) )
        & ( ord_le6787938422905777998nnreal @ A @ ( collec6648975593938027277nnreal @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_744_conj__subset__def,axiom,
    ! [A: set_Extended_ereal,P: extended_ereal > $o,Q: extended_ereal > $o] :
      ( ( ord_le1644982726543182158_ereal @ A
        @ ( collec5835592288176408249_ereal
          @ ^ [X: extended_ereal] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le1644982726543182158_ereal @ A @ ( collec5835592288176408249_ereal @ P ) )
        & ( ord_le1644982726543182158_ereal @ A @ ( collec5835592288176408249_ereal @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_745_Fpow__def,axiom,
    ( finite724154798228260685od_b_b
    = ( ^ [A4: set_Product_prod_b_b] :
          ( collec1108733003560609586od_b_b
          @ ^ [X6: set_Product_prod_b_b] :
              ( ( ord_le182087997850975847od_b_b @ X6 @ A4 )
              & ( finite3757003017338540048od_b_b @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_746_Fpow__def,axiom,
    ( finite_Fpow_b
    = ( ^ [A4: set_b] :
          ( collect_set_b
          @ ^ [X6: set_b] :
              ( ( ord_less_eq_set_b @ X6 @ A4 )
              & ( finite_finite_b @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_747_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A4: set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( ord_less_eq_set_nat @ X6 @ A4 )
              & ( finite_finite_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_748_Fpow__def,axiom,
    ( finite5796909905237389606od_b_b
    = ( ^ [A4: set_Pr5139338970096277698od_b_b] :
          ( collec1400520969827591831od_b_b
          @ ^ [X6: set_Pr5139338970096277698od_b_b] :
              ( ( ord_le2005546836280662306od_b_b @ X6 @ A4 )
              & ( finite7768965217515309219od_b_b @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_749_Fpow__def,axiom,
    ( finite4599829005654479975at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [X6: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X6 @ A4 )
              & ( finite6177210948735845034at_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_750_Fpow__def,axiom,
    ( finite7362295380003814718_nat_b
    = ( ^ [A4: set_Pr4264375888882495962_nat_b] :
          ( collec5983057837050141871_nat_b
          @ ^ [X6: set_Pr4264375888882495962_nat_b] :
              ( ( ord_le7995947752535495226_nat_b @ X6 @ A4 )
              & ( finite659689794318260667_nat_b @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_751_Fpow__def,axiom,
    ( finite5390238786891289653od_b_b
    = ( ^ [A4: set_Pr4323519195528460463od_b_b] :
          ( collec4149886324997387802od_b_b
          @ ^ [X6: set_Pr4323519195528460463od_b_b] :
              ( ( ord_le8131691160565715023od_b_b @ X6 @ A4 )
              & ( finite4644902770518909432od_b_b @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_752_Fpow__def,axiom,
    ( finite5359576240933303448_b_nat
    = ( ^ [A4: set_Pr1307281990691478580_b_nat] :
          ( collec3025963938859124489_b_nat
          @ ^ [X6: set_Pr1307281990691478580_b_nat] :
              ( ( ord_le5038853854344477844_b_nat @ X6 @ A4 )
              & ( finite7880342692102525205_b_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_753_Fpow__def,axiom,
    ( finite4759999794119668562nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal] :
          ( collec4858231573021281987nnreal
          @ ^ [X6: set_Ex3793607809372303086nnreal] :
              ( ( ord_le6787938422905777998nnreal @ X6 @ A4 )
              & ( finite3782138982310603983nnreal @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_754_Fpow__def,axiom,
    ( finite2137394461708460340_ereal
    = ( ^ [A4: set_Extended_ereal] :
          ( collec85322473871370393_ereal
          @ ^ [X6: set_Extended_ereal] :
              ( ( ord_le1644982726543182158_ereal @ X6 @ A4 )
              & ( finite7198162374296863863_ereal @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_755_arg__min__least,axiom,
    ! [S: set_b,Y4: b,F: b > nat] :
      ( ( finite_finite_b @ S )
     => ( ( S != bot_bot_set_b )
       => ( ( member_b @ Y4 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7575731748627795062_b_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_756_arg__min__least,axiom,
    ! [S: set_nat,Y4: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y4 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_757_arg__min__least,axiom,
    ! [S: set_Ex3793607809372303086nnreal,Y4: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat] :
      ( ( finite3782138982310603983nnreal @ S )
     => ( ( S != bot_bo4854962954004695426nnreal )
       => ( ( member7908768830364227535nnreal @ Y4 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic2638263054070015267al_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_758_arg__min__least,axiom,
    ! [S: set_Extended_ereal,Y4: extended_ereal,F: extended_ereal > nat] :
      ( ( finite7198162374296863863_ereal @ S )
     => ( ( S != bot_bo8367695208629047834_ereal )
       => ( ( member2350847679896131959_ereal @ Y4 @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic5936838755584868421al_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_759_arg__min__least,axiom,
    ! [S: set_b,Y4: b,F: b > extended_ereal] :
      ( ( finite_finite_b @ S )
     => ( ( S != bot_bot_set_b )
       => ( ( member_b @ Y4 @ S )
         => ( ord_le1083603963089353582_ereal @ ( F @ ( lattic7657629326291102120_ereal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_760_arg__min__least,axiom,
    ! [S: set_nat,Y4: nat,F: nat > extended_ereal] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y4 @ S )
         => ( ord_le1083603963089353582_ereal @ ( F @ ( lattic2586270367301029667_ereal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_761_arg__min__least,axiom,
    ! [S: set_Ex3793607809372303086nnreal,Y4: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extended_ereal] :
      ( ( finite3782138982310603983nnreal @ S )
     => ( ( S != bot_bo4854962954004695426nnreal )
       => ( ( member7908768830364227535nnreal @ Y4 @ S )
         => ( ord_le1083603963089353582_ereal @ ( F @ ( lattic1702846024332642747_ereal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_762_arg__min__least,axiom,
    ! [S: set_Extended_ereal,Y4: extended_ereal,F: extended_ereal > extended_ereal] :
      ( ( finite7198162374296863863_ereal @ S )
     => ( ( S != bot_bo8367695208629047834_ereal )
       => ( ( member2350847679896131959_ereal @ Y4 @ S )
         => ( ord_le1083603963089353582_ereal @ ( F @ ( lattic3718249791026499097_ereal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_763_arg__min__least,axiom,
    ! [S: set_b,Y4: b,F: b > extend8495563244428889912nnreal] :
      ( ( finite_finite_b @ S )
     => ( ( S != bot_bot_set_b )
       => ( ( member_b @ Y4 @ S )
         => ( ord_le3935885782089961368nnreal @ ( F @ ( lattic1709475118623791070nnreal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_764_arg__min__least,axiom,
    ! [S: set_nat,Y4: nat,F: nat > extend8495563244428889912nnreal] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y4 @ S )
         => ( ord_le3935885782089961368nnreal @ ( F @ ( lattic7087934650257931555nnreal @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).

% arg_min_least
thf(fact_765_Greatest__equality,axiom,
    ! [P: extended_ereal > $o,X3: extended_ereal] :
      ( ( P @ X3 )
     => ( ! [Y2: extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le1083603963089353582_ereal @ Y2 @ X3 ) )
       => ( ( order_5661308041690931239_ereal @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_766_Greatest__equality,axiom,
    ! [P: extend8495563244428889912nnreal > $o,X3: extend8495563244428889912nnreal] :
      ( ( P @ X3 )
     => ( ! [Y2: extend8495563244428889912nnreal] :
            ( ( P @ Y2 )
           => ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) )
       => ( ( order_7545170809120406815nnreal @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_767_Greatest__equality,axiom,
    ! [P: set_Ex3793607809372303086nnreal > $o,X3: set_Ex3793607809372303086nnreal] :
      ( ( P @ X3 )
     => ( ! [Y2: set_Ex3793607809372303086nnreal] :
            ( ( P @ Y2 )
           => ( ord_le6787938422905777998nnreal @ Y2 @ X3 ) )
       => ( ( order_673430698331062485nnreal @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_768_Greatest__equality,axiom,
    ! [P: set_Extended_ereal > $o,X3: set_Extended_ereal] :
      ( ( P @ X3 )
     => ( ! [Y2: set_Extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le1644982726543182158_ereal @ Y2 @ X3 ) )
       => ( ( order_9053233744620718599_ereal @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_769_Greatest__equality,axiom,
    ! [P: nat > $o,X3: nat] :
      ( ( P @ X3 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X3 ) )
       => ( ( order_Greatest_nat @ P )
          = X3 ) ) ) ).

% Greatest_equality
thf(fact_770_GreatestI2__order,axiom,
    ! [P: extended_ereal > $o,X3: extended_ereal,Q: extended_ereal > $o] :
      ( ( P @ X3 )
     => ( ! [Y2: extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le1083603963089353582_ereal @ Y2 @ X3 ) )
       => ( ! [X2: extended_ereal] :
              ( ( P @ X2 )
             => ( ! [Y3: extended_ereal] :
                    ( ( P @ Y3 )
                   => ( ord_le1083603963089353582_ereal @ Y3 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_5661308041690931239_ereal @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_771_GreatestI2__order,axiom,
    ! [P: extend8495563244428889912nnreal > $o,X3: extend8495563244428889912nnreal,Q: extend8495563244428889912nnreal > $o] :
      ( ( P @ X3 )
     => ( ! [Y2: extend8495563244428889912nnreal] :
            ( ( P @ Y2 )
           => ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) )
       => ( ! [X2: extend8495563244428889912nnreal] :
              ( ( P @ X2 )
             => ( ! [Y3: extend8495563244428889912nnreal] :
                    ( ( P @ Y3 )
                   => ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_7545170809120406815nnreal @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_772_GreatestI2__order,axiom,
    ! [P: set_Ex3793607809372303086nnreal > $o,X3: set_Ex3793607809372303086nnreal,Q: set_Ex3793607809372303086nnreal > $o] :
      ( ( P @ X3 )
     => ( ! [Y2: set_Ex3793607809372303086nnreal] :
            ( ( P @ Y2 )
           => ( ord_le6787938422905777998nnreal @ Y2 @ X3 ) )
       => ( ! [X2: set_Ex3793607809372303086nnreal] :
              ( ( P @ X2 )
             => ( ! [Y3: set_Ex3793607809372303086nnreal] :
                    ( ( P @ Y3 )
                   => ( ord_le6787938422905777998nnreal @ Y3 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_673430698331062485nnreal @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_773_GreatestI2__order,axiom,
    ! [P: set_Extended_ereal > $o,X3: set_Extended_ereal,Q: set_Extended_ereal > $o] :
      ( ( P @ X3 )
     => ( ! [Y2: set_Extended_ereal] :
            ( ( P @ Y2 )
           => ( ord_le1644982726543182158_ereal @ Y2 @ X3 ) )
       => ( ! [X2: set_Extended_ereal] :
              ( ( P @ X2 )
             => ( ! [Y3: set_Extended_ereal] :
                    ( ( P @ Y3 )
                   => ( ord_le1644982726543182158_ereal @ Y3 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_9053233744620718599_ereal @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_774_GreatestI2__order,axiom,
    ! [P: nat > $o,X3: nat,Q: nat > $o] :
      ( ( P @ X3 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ X3 ) )
       => ( ! [X2: nat] :
              ( ( P @ X2 )
             => ( ! [Y3: nat] :
                    ( ( P @ Y3 )
                   => ( ord_less_eq_nat @ Y3 @ X2 ) )
               => ( Q @ X2 ) ) )
         => ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).

% GreatestI2_order
thf(fact_775_finite__quotient,axiom,
    ! [A: set_Product_prod_b_b,R2: set_Pr3901141605387707591od_b_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ( ord_le7993227757367499879od_b_b @ R2
          @ ( produc5716436088342041352od_b_b @ A
            @ ^ [Uu: product_prod_b_b] : A ) )
       => ( finite8153119339416810480od_b_b @ ( equiv_4051174557841558190od_b_b @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_776_finite__quotient,axiom,
    ! [A: set_Pr5139338970096277698od_b_b,R2: set_Pr2857578619867412697od_b_b] :
      ( ( finite7768965217515309219od_b_b @ A )
     => ( ( ord_le5024618550196289657od_b_b @ R2
          @ ( produc2402687488715291874od_b_b @ A
            @ ^ [Uu: produc1536031394801701132od_b_b] : A ) )
       => ( finite8041041396987327577od_b_b @ ( equiv_6452974403491352709od_b_b @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_777_finite__quotient,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( ord_le3000389064537975527at_nat @ R2
          @ ( produc2761391749766926216at_nat @ A
            @ ^ [Uu: product_prod_nat_nat] : A ) )
       => ( finite9047747110432174090at_nat @ ( equiv_3811336339175339080at_nat @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_778_finite__quotient,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,R2: set_Pr4934845982659317257_nat_b] :
      ( ( finite659689794318260667_nat_b @ A )
     => ( ( ord_le4141943563446670761_nat_b @ R2
          @ ( produc7450675564766945298_nat_b @ A
            @ ^ [Uu: product_prod_nat_b] : A ) )
       => ( finite9121790342551665777_nat_b @ ( equiv_3858820710734844829_nat_b @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_779_finite__quotient,axiom,
    ! [A: set_Pr4323519195528460463od_b_b,R2: set_Pr2813452432853373639od_b_b] :
      ( ( finite4644902770518909432od_b_b @ A )
     => ( ( ord_le6270780334114785895od_b_b @ R2
          @ ( produc3615177219224589064od_b_b @ A
            @ ^ [Uu: produc2840042325109449167od_b_b] : A ) )
       => ( finite8128242158466110680od_b_b @ ( equiv_921398034223189654od_b_b @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_780_finite__quotient,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,R2: set_Pr5241429435540800445_b_nat] :
      ( ( finite7880342692102525205_b_nat @ A )
     => ( ( ord_le4448527016328153949_b_nat @ R2
          @ ( produc4599712156513991878_b_nat @ A
            @ ^ [Uu: product_prod_b_nat] : A ) )
       => ( finite6164696444360648395_b_nat @ ( equiv_1856101571664333559_b_nat @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_781_finite__quotient,axiom,
    ! [A: set_b,R2: set_Product_prod_b_b] :
      ( ( finite_finite_b @ A )
     => ( ( ord_le182087997850975847od_b_b @ R2
          @ ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : A ) )
       => ( finite_finite_set_b @ ( equiv_quotient_b @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_782_finite__quotient,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : A ) )
       => ( finite1152437895449049373et_nat @ ( equiv_quotient_nat @ A @ R2 ) ) ) ) ).

% finite_quotient
thf(fact_783_quotient__empty,axiom,
    ! [R2: set_Pr3708903437949654321nnreal] :
      ( ( equiv_4072829958317279409nnreal @ bot_bo4854962954004695426nnreal @ R2 )
      = bot_bo2988155216863113784nnreal ) ).

% quotient_empty
thf(fact_784_quotient__empty,axiom,
    ! [R2: set_Pr2129990008675586951_ereal] :
      ( ( equiv_8383376530702616597_ereal @ bot_bo8367695208629047834_ereal @ R2 )
      = bot_bo7400643019497942010_ereal ) ).

% quotient_empty
thf(fact_785_quotient__is__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal,R2: set_Pr3708903437949654321nnreal] :
      ( ( ( equiv_4072829958317279409nnreal @ A @ R2 )
        = bot_bo2988155216863113784nnreal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% quotient_is_empty
thf(fact_786_quotient__is__empty,axiom,
    ! [A: set_Extended_ereal,R2: set_Pr2129990008675586951_ereal] :
      ( ( ( equiv_8383376530702616597_ereal @ A @ R2 )
        = bot_bo7400643019497942010_ereal )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% quotient_is_empty
thf(fact_787_quotient__is__empty2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,R2: set_Pr3708903437949654321nnreal] :
      ( ( bot_bo2988155216863113784nnreal
        = ( equiv_4072829958317279409nnreal @ A @ R2 ) )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% quotient_is_empty2
thf(fact_788_quotient__is__empty2,axiom,
    ! [A: set_Extended_ereal,R2: set_Pr2129990008675586951_ereal] :
      ( ( bot_bo7400643019497942010_ereal
        = ( equiv_8383376530702616597_ereal @ A @ R2 ) )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% quotient_is_empty2
thf(fact_789_Fpow__mono,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ord_le3366939622266546180nnreal @ ( finite4759999794119668562nnreal @ A ) @ ( finite4759999794119668562nnreal @ B ) ) ) ).

% Fpow_mono
thf(fact_790_Fpow__mono,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ord_le5287700718633833262_ereal @ ( finite2137394461708460340_ereal @ A ) @ ( finite2137394461708460340_ereal @ B ) ) ) ).

% Fpow_mono
thf(fact_791_empty__in__Fpow,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( finite4759999794119668562nnreal @ A ) ) ).

% empty_in_Fpow
thf(fact_792_empty__in__Fpow,axiom,
    ! [A: set_Extended_ereal] : ( member5519481007471526743_ereal @ bot_bo8367695208629047834_ereal @ ( finite2137394461708460340_ereal @ A ) ) ).

% empty_in_Fpow
thf(fact_793_finite__equiv__class,axiom,
    ! [A: set_Product_prod_b_b,R2: set_Pr3901141605387707591od_b_b,X4: set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ A )
     => ( ( ord_le7993227757367499879od_b_b @ R2
          @ ( produc5716436088342041352od_b_b @ A
            @ ^ [Uu: product_prod_b_b] : A ) )
       => ( ( member1252001552157608176od_b_b @ X4 @ ( equiv_4051174557841558190od_b_b @ A @ R2 ) )
         => ( finite3757003017338540048od_b_b @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_794_finite__equiv__class,axiom,
    ! [A: set_Pr5139338970096277698od_b_b,R2: set_Pr2857578619867412697od_b_b,X4: set_Pr5139338970096277698od_b_b] :
      ( ( finite7768965217515309219od_b_b @ A )
     => ( ( ord_le5024618550196289657od_b_b @ R2
          @ ( produc2402687488715291874od_b_b @ A
            @ ^ [Uu: produc1536031394801701132od_b_b] : A ) )
       => ( ( member3392960547284361049od_b_b @ X4 @ ( equiv_6452974403491352709od_b_b @ A @ R2 ) )
         => ( finite7768965217515309219od_b_b @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_795_finite__equiv__class,axiom,
    ! [A: set_Pr1261947904930325089at_nat,R2: set_Pr8693737435421807431at_nat,X4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( ord_le3000389064537975527at_nat @ R2
          @ ( produc2761391749766926216at_nat @ A
            @ ^ [Uu: product_prod_nat_nat] : A ) )
       => ( ( member2643936169264416010at_nat @ X4 @ ( equiv_3811336339175339080at_nat @ A @ R2 ) )
         => ( finite6177210948735845034at_nat @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_796_finite__equiv__class,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,R2: set_Pr4934845982659317257_nat_b,X4: set_Pr4264375888882495962_nat_b] :
      ( ( finite659689794318260667_nat_b @ A )
     => ( ( ord_le4141943563446670761_nat_b @ R2
          @ ( produc7450675564766945298_nat_b @ A
            @ ^ [Uu: product_prod_nat_b] : A ) )
       => ( ( member7364522358610929521_nat_b @ X4 @ ( equiv_3858820710734844829_nat_b @ A @ R2 ) )
         => ( finite659689794318260667_nat_b @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_797_finite__equiv__class,axiom,
    ! [A: set_Pr4323519195528460463od_b_b,R2: set_Pr2813452432853373639od_b_b,X4: set_Pr4323519195528460463od_b_b] :
      ( ( finite4644902770518909432od_b_b @ A )
     => ( ( ord_le6270780334114785895od_b_b @ R2
          @ ( produc3615177219224589064od_b_b @ A
            @ ^ [Uu: produc2840042325109449167od_b_b] : A ) )
       => ( ( member3004383007682734552od_b_b @ X4 @ ( equiv_921398034223189654od_b_b @ A @ R2 ) )
         => ( finite4644902770518909432od_b_b @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_798_finite__equiv__class,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,R2: set_Pr5241429435540800445_b_nat,X4: set_Pr1307281990691478580_b_nat] :
      ( ( finite7880342692102525205_b_nat @ A )
     => ( ( ord_le4448527016328153949_b_nat @ R2
          @ ( produc4599712156513991878_b_nat @ A
            @ ^ [Uu: product_prod_b_nat] : A ) )
       => ( ( member4407428460419912139_b_nat @ X4 @ ( equiv_1856101571664333559_b_nat @ A @ R2 ) )
         => ( finite7880342692102525205_b_nat @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_799_finite__equiv__class,axiom,
    ! [A: set_b,R2: set_Product_prod_b_b,X4: set_b] :
      ( ( finite_finite_b @ A )
     => ( ( ord_le182087997850975847od_b_b @ R2
          @ ( product_Sigma_b_b @ A
            @ ^ [Uu: b] : A ) )
       => ( ( member_set_b @ X4 @ ( equiv_quotient_b @ A @ R2 ) )
         => ( finite_finite_b @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_800_finite__equiv__class,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X4: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_le3146513528884898305at_nat @ R2
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : A ) )
       => ( ( member_set_nat @ X4 @ ( equiv_quotient_nat @ A @ R2 ) )
         => ( finite_finite_nat @ X4 ) ) ) ) ).

% finite_equiv_class
thf(fact_801_top__empty__eq,axiom,
    ( top_top_b_o
    = ( ^ [X: b] : ( member_b @ X @ top_top_set_b ) ) ) ).

% top_empty_eq
thf(fact_802_top__empty__eq,axiom,
    ( top_to7135874014580417414_b_b_o
    = ( ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ top_to7498756471699006487od_b_b ) ) ) ).

% top_empty_eq
thf(fact_803_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_804_top__empty__eq,axiom,
    ( top_to6999531812125281119real_o
    = ( ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ top_to5683747375963461374_ereal ) ) ) ).

% top_empty_eq
thf(fact_805_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_806_bot__empty__eq,axiom,
    ( bot_bot_b_o
    = ( ^ [X: b] : ( member_b @ X @ bot_bot_set_b ) ) ) ).

% bot_empty_eq
thf(fact_807_bot__empty__eq,axiom,
    ( bot_bo2608278733301331306_b_b_o
    = ( ^ [X: product_prod_b_b] : ( member7862447936710763792od_b_b @ X @ bot_bo2792761326896053555od_b_b ) ) ) ).

% bot_empty_eq
thf(fact_808_bot__empty__eq,axiom,
    ( bot_bo412624608084785539real_o
    = ( ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ bot_bo4854962954004695426nnreal ) ) ) ).

% bot_empty_eq
thf(fact_809_bot__empty__eq,axiom,
    ( bot_bo5519581617326455619real_o
    = ( ^ [X: extended_ereal] : ( member2350847679896131959_ereal @ X @ bot_bo8367695208629047834_ereal ) ) ) ).

% bot_empty_eq
thf(fact_810_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_811_Collect__empty__eq__bot,axiom,
    ! [P: b > $o] :
      ( ( ( collect_b @ P )
        = bot_bot_set_b )
      = ( P = bot_bot_b_o ) ) ).

% Collect_empty_eq_bot
thf(fact_812_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_b_b > $o] :
      ( ( ( collec548942219715005266od_b_b @ P )
        = bot_bo2792761326896053555od_b_b )
      = ( P = bot_bo2608278733301331306_b_b_o ) ) ).

% Collect_empty_eq_bot
thf(fact_813_Collect__empty__eq__bot,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( P = bot_bo412624608084785539real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_814_Collect__empty__eq__bot,axiom,
    ! [P: extended_ereal > $o] :
      ( ( ( collec5835592288176408249_ereal @ P )
        = bot_bo8367695208629047834_ereal )
      = ( P = bot_bo5519581617326455619real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_815_Sup__fin_Osubset__imp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( finite6177210948735845034at_nat @ B )
         => ( ord_le8460144461188290721at_nat @ ( lattic3087465103008755000at_nat @ A ) @ ( lattic3087465103008755000at_nat @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_816_Sup__fin_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_817_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se4580700918925141924nnreal,B: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ B )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( finite3583719589609615493nnreal @ B )
         => ( ord_le6787938422905777998nnreal @ ( lattic6502854510321989495nnreal @ A ) @ ( lattic6502854510321989495nnreal @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_818_Sup__fin_Osubset__imp,axiom,
    ! [A: set_se6634062954251873166_ereal,B: set_se6634062954251873166_ereal] :
      ( ( ord_le5287700718633833262_ereal @ A @ B )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( finite2737741666826350167_ereal @ B )
         => ( ord_le1644982726543182158_ereal @ ( lattic6967077786232488677_ereal @ A ) @ ( lattic6967077786232488677_ereal @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_819_Sup__fin_Osubset__imp,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ B )
         => ( ord_le3935885782089961368nnreal @ ( lattic4971367313934416833nnreal @ A ) @ ( lattic4971367313934416833nnreal @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_820_Sup__fin_Osubset__imp,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ B )
         => ( ord_le1083603963089353582_ereal @ ( lattic6963256781428740357_ereal @ A ) @ ( lattic6963256781428740357_ereal @ B ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_821_Inf__fin_Osubset__imp,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( finite6177210948735845034at_nat @ B )
         => ( ord_le8460144461188290721at_nat @ ( lattic4585598177315101342at_nat @ B ) @ ( lattic4585598177315101342at_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_822_Inf__fin_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_823_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se4580700918925141924nnreal,B: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ B )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( finite3583719589609615493nnreal @ B )
         => ( ord_le6787938422905777998nnreal @ ( lattic8983332664287545745nnreal @ B ) @ ( lattic8983332664287545745nnreal @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_824_Inf__fin_Osubset__imp,axiom,
    ! [A: set_se6634062954251873166_ereal,B: set_se6634062954251873166_ereal] :
      ( ( ord_le5287700718633833262_ereal @ A @ B )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( finite2737741666826350167_ereal @ B )
         => ( ord_le1644982726543182158_ereal @ ( lattic3876247248944996939_ereal @ B ) @ ( lattic3876247248944996939_ereal @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_825_Inf__fin_Osubset__imp,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ B )
         => ( ord_le3935885782089961368nnreal @ ( lattic6966437970300483035nnreal @ B ) @ ( lattic6966437970300483035nnreal @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_826_Inf__fin_Osubset__imp,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ B )
         => ( ord_le1083603963089353582_ereal @ ( lattic1367271666432936043_ereal @ B ) @ ( lattic1367271666432936043_ereal @ A ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_827_Max_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ).

% Max.subset_imp
thf(fact_828_Max_Osubset__imp,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ B )
         => ( ord_le3935885782089961368nnreal @ ( lattic933167949679527817nnreal @ A ) @ ( lattic933167949679527817nnreal @ B ) ) ) ) ) ).

% Max.subset_imp
thf(fact_829_Max_Osubset__imp,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ B )
         => ( ord_le1083603963089353582_ereal @ ( lattic5789169447084241725_ereal @ A ) @ ( lattic5789169447084241725_ereal @ B ) ) ) ) ) ).

% Max.subset_imp
thf(fact_830_Max__mono,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ N )
     => ( ( M2 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M2 ) @ ( lattic8265883725875713057ax_nat @ N ) ) ) ) ) ).

% Max_mono
thf(fact_831_Max__mono,axiom,
    ! [M2: set_Ex3793607809372303086nnreal,N: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ M2 @ N )
     => ( ( M2 != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ N )
         => ( ord_le3935885782089961368nnreal @ ( lattic933167949679527817nnreal @ M2 ) @ ( lattic933167949679527817nnreal @ N ) ) ) ) ) ).

% Max_mono
thf(fact_832_Max__mono,axiom,
    ! [M2: set_Extended_ereal,N: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ M2 @ N )
     => ( ( M2 != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ N )
         => ( ord_le1083603963089353582_ereal @ ( lattic5789169447084241725_ereal @ M2 ) @ ( lattic5789169447084241725_ereal @ N ) ) ) ) ) ).

% Max_mono
thf(fact_833_Max_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_834_Max_Obounded__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ ( lattic5789169447084241725_ereal @ A ) @ X3 )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_835_Max_Obounded__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( lattic933167949679527817nnreal @ A ) @ X3 )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X @ X3 ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_836_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ord_le8460144461188290721at_nat @ ( lattic4585598177315101342at_nat @ A ) @ ( lattic3087465103008755000at_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_837_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_838_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ord_le1083603963089353582_ereal @ ( lattic1367271666432936043_ereal @ A ) @ ( lattic6963256781428740357_ereal @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_839_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ord_le3935885782089961368nnreal @ ( lattic6966437970300483035nnreal @ A ) @ ( lattic4971367313934416833nnreal @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_840_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ord_le6787938422905777998nnreal @ ( lattic8983332664287545745nnreal @ A ) @ ( lattic6502854510321989495nnreal @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_841_Inf__fin__le__Sup__fin,axiom,
    ! [A: set_se6634062954251873166_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ord_le1644982726543182158_ereal @ ( lattic3876247248944996939_ereal @ A ) @ ( lattic6967077786232488677_ereal @ A ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_842_Max__ge,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ X3 @ A )
       => ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).

% Max_ge
thf(fact_843_Max__ge,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ X3 @ A )
       => ( ord_le1083603963089353582_ereal @ X3 @ ( lattic5789169447084241725_ereal @ A ) ) ) ) ).

% Max_ge
thf(fact_844_Max__ge,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ X3 @ A )
       => ( ord_le3935885782089961368nnreal @ X3 @ ( lattic933167949679527817nnreal @ A ) ) ) ) ).

% Max_ge
thf(fact_845_Max__eqI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [Y2: nat] :
            ( ( member_nat @ Y2 @ A )
           => ( ord_less_eq_nat @ Y2 @ X3 ) )
       => ( ( member_nat @ X3 @ A )
         => ( ( lattic8265883725875713057ax_nat @ A )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_846_Max__eqI,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ! [Y2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ Y2 @ A )
           => ( ord_le1083603963089353582_ereal @ Y2 @ X3 ) )
       => ( ( member2350847679896131959_ereal @ X3 @ A )
         => ( ( lattic5789169447084241725_ereal @ A )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_847_Max__eqI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ! [Y2: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ Y2 @ A )
           => ( ord_le3935885782089961368nnreal @ Y2 @ X3 ) )
       => ( ( member7908768830364227535nnreal @ X3 @ A )
         => ( ( lattic933167949679527817nnreal @ A )
            = X3 ) ) ) ) ).

% Max_eqI
thf(fact_848_Max__eq__if,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( ord_less_eq_nat @ X2 @ Xa ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ B )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ A )
                    & ( ord_less_eq_nat @ X2 @ Xa ) ) )
           => ( ( lattic8265883725875713057ax_nat @ A )
              = ( lattic8265883725875713057ax_nat @ B ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_849_Max__eq__if,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( finite7198162374296863863_ereal @ B )
       => ( ! [X2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ X2 @ A )
             => ? [Xa: extended_ereal] :
                  ( ( member2350847679896131959_ereal @ Xa @ B )
                  & ( ord_le1083603963089353582_ereal @ X2 @ Xa ) ) )
         => ( ! [X2: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X2 @ B )
               => ? [Xa: extended_ereal] :
                    ( ( member2350847679896131959_ereal @ Xa @ A )
                    & ( ord_le1083603963089353582_ereal @ X2 @ Xa ) ) )
           => ( ( lattic5789169447084241725_ereal @ A )
              = ( lattic5789169447084241725_ereal @ B ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_850_Max__eq__if,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( finite3782138982310603983nnreal @ B )
       => ( ! [X2: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X2 @ A )
             => ? [Xa: extend8495563244428889912nnreal] :
                  ( ( member7908768830364227535nnreal @ Xa @ B )
                  & ( ord_le3935885782089961368nnreal @ X2 @ Xa ) ) )
         => ( ! [X2: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X2 @ B )
               => ? [Xa: extend8495563244428889912nnreal] :
                    ( ( member7908768830364227535nnreal @ Xa @ A )
                    & ( ord_le3935885782089961368nnreal @ X2 @ Xa ) ) )
           => ( ( lattic933167949679527817nnreal @ A )
              = ( lattic933167949679527817nnreal @ B ) ) ) ) ) ) ).

% Max_eq_if
thf(fact_851_Max_OcoboundedI,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ( ord_less_eq_nat @ A5 @ ( lattic8265883725875713057ax_nat @ A ) ) ) ) ).

% Max.coboundedI
thf(fact_852_Max_OcoboundedI,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ A5 @ A )
       => ( ord_le1083603963089353582_ereal @ A5 @ ( lattic5789169447084241725_ereal @ A ) ) ) ) ).

% Max.coboundedI
thf(fact_853_Max_OcoboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ A5 @ A )
       => ( ord_le3935885782089961368nnreal @ A5 @ ( lattic933167949679527817nnreal @ A ) ) ) ) ).

% Max.coboundedI
thf(fact_854_Max__in,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).

% Max_in
thf(fact_855_Max__in,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( member7908768830364227535nnreal @ ( lattic933167949679527817nnreal @ A ) @ A ) ) ) ).

% Max_in
thf(fact_856_Max__in,axiom,
    ! [A: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( member2350847679896131959_ereal @ ( lattic5789169447084241725_ereal @ A ) @ A ) ) ) ).

% Max_in
thf(fact_857_Inf__fin_OcoboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ( ord_le8460144461188290721at_nat @ ( lattic4585598177315101342at_nat @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_858_Inf__fin_OcoboundedI,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_859_Inf__fin_OcoboundedI,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ A5 @ A )
       => ( ord_le1083603963089353582_ereal @ ( lattic1367271666432936043_ereal @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_860_Inf__fin_OcoboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ A5 @ A )
       => ( ord_le3935885782089961368nnreal @ ( lattic6966437970300483035nnreal @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_861_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se4580700918925141924nnreal,A5: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( member603777416030116741nnreal @ A5 @ A )
       => ( ord_le6787938422905777998nnreal @ ( lattic8983332664287545745nnreal @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_862_Inf__fin_OcoboundedI,axiom,
    ! [A: set_se6634062954251873166_ereal,A5: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( member5519481007471526743_ereal @ A5 @ A )
       => ( ord_le1644982726543182158_ereal @ ( lattic3876247248944996939_ereal @ A ) @ A5 ) ) ) ).

% Inf_fin.coboundedI
thf(fact_863_Sup__fin_OcoboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ( ord_le8460144461188290721at_nat @ A5 @ ( lattic3087465103008755000at_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_864_Sup__fin_OcoboundedI,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ( ord_less_eq_nat @ A5 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_865_Sup__fin_OcoboundedI,axiom,
    ! [A: set_Extended_ereal,A5: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( member2350847679896131959_ereal @ A5 @ A )
       => ( ord_le1083603963089353582_ereal @ A5 @ ( lattic6963256781428740357_ereal @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_866_Sup__fin_OcoboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A5: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( member7908768830364227535nnreal @ A5 @ A )
       => ( ord_le3935885782089961368nnreal @ A5 @ ( lattic4971367313934416833nnreal @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_867_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se4580700918925141924nnreal,A5: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( member603777416030116741nnreal @ A5 @ A )
       => ( ord_le6787938422905777998nnreal @ A5 @ ( lattic6502854510321989495nnreal @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_868_Sup__fin_OcoboundedI,axiom,
    ! [A: set_se6634062954251873166_ereal,A5: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( member5519481007471526743_ereal @ A5 @ A )
       => ( ord_le1644982726543182158_ereal @ A5 @ ( lattic6967077786232488677_ereal @ A ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_869_Max__eq__iff,axiom,
    ! [A: set_nat,M3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A )
            = M3 )
          = ( ( member_nat @ M3 @ A )
            & ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_870_Max__eq__iff,axiom,
    ! [A: set_Extended_ereal,M3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ( lattic5789169447084241725_ereal @ A )
            = M3 )
          = ( ( member2350847679896131959_ereal @ M3 @ A )
            & ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X @ M3 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_871_Max__eq__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ( lattic933167949679527817nnreal @ A )
            = M3 )
          = ( ( member7908768830364227535nnreal @ M3 @ A )
            & ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X @ M3 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_872_Max__ge__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic8265883725875713057ax_nat @ A ) )
          = ( ? [X: nat] :
                ( ( member_nat @ X @ A )
                & ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_873_Max__ge__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ X3 @ ( lattic5789169447084241725_ereal @ A ) )
          = ( ? [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
                & ( ord_le1083603963089353582_ereal @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_874_Max__ge__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ X3 @ ( lattic933167949679527817nnreal @ A ) )
          = ( ? [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
                & ( ord_le3935885782089961368nnreal @ X3 @ X ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_875_eq__Max__iff,axiom,
    ! [A: set_nat,M3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( M3
            = ( lattic8265883725875713057ax_nat @ A ) )
          = ( ( member_nat @ M3 @ A )
            & ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_876_eq__Max__iff,axiom,
    ! [A: set_Extended_ereal,M3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( M3
            = ( lattic5789169447084241725_ereal @ A ) )
          = ( ( member2350847679896131959_ereal @ M3 @ A )
            & ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X @ M3 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_877_eq__Max__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( M3
            = ( lattic933167949679527817nnreal @ A ) )
          = ( ( member7908768830364227535nnreal @ M3 @ A )
            & ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X @ M3 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_878_Max_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
         => ! [A6: nat] :
              ( ( member_nat @ A6 @ A )
             => ( ord_less_eq_nat @ A6 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_879_Max_OboundedE,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ ( lattic5789169447084241725_ereal @ A ) @ X3 )
         => ! [A6: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A6 @ A )
             => ( ord_le1083603963089353582_ereal @ A6 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_880_Max_OboundedE,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( lattic933167949679527817nnreal @ A ) @ X3 )
         => ! [A6: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A6 @ A )
             => ( ord_le3935885782089961368nnreal @ A6 @ X3 ) ) ) ) ) ).

% Max.boundedE
thf(fact_881_Max_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A2: nat] :
              ( ( member_nat @ A2 @ A )
             => ( ord_less_eq_nat @ A2 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_882_Max_OboundedI,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ! [A2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A2 @ A )
             => ( ord_le1083603963089353582_ereal @ A2 @ X3 ) )
         => ( ord_le1083603963089353582_ereal @ ( lattic5789169447084241725_ereal @ A ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_883_Max_OboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ! [A2: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A2 @ A )
             => ( ord_le3935885782089961368nnreal @ A2 @ X3 ) )
         => ( ord_le3935885782089961368nnreal @ ( lattic933167949679527817nnreal @ A ) @ X3 ) ) ) ) ).

% Max.boundedI
thf(fact_884_Inf__fin_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( ord_le8460144461188290721at_nat @ X3 @ ( lattic4585598177315101342at_nat @ A ) )
         => ! [A6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A6 @ A )
             => ( ord_le8460144461188290721at_nat @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_885_Inf__fin_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
         => ! [A6: nat] :
              ( ( member_nat @ A6 @ A )
             => ( ord_less_eq_nat @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_886_Inf__fin_OboundedE,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ X3 @ ( lattic1367271666432936043_ereal @ A ) )
         => ! [A6: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A6 @ A )
             => ( ord_le1083603963089353582_ereal @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_887_Inf__fin_OboundedE,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ X3 @ ( lattic6966437970300483035nnreal @ A ) )
         => ! [A6: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A6 @ A )
             => ( ord_le3935885782089961368nnreal @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_888_Inf__fin_OboundedE,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( ord_le6787938422905777998nnreal @ X3 @ ( lattic8983332664287545745nnreal @ A ) )
         => ! [A6: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ A6 @ A )
             => ( ord_le6787938422905777998nnreal @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_889_Inf__fin_OboundedE,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( ord_le1644982726543182158_ereal @ X3 @ ( lattic3876247248944996939_ereal @ A ) )
         => ! [A6: set_Extended_ereal] :
              ( ( member5519481007471526743_ereal @ A6 @ A )
             => ( ord_le1644982726543182158_ereal @ X3 @ A6 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_890_Inf__fin_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ! [A2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A2 @ A )
             => ( ord_le8460144461188290721at_nat @ X3 @ A2 ) )
         => ( ord_le8460144461188290721at_nat @ X3 @ ( lattic4585598177315101342at_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_891_Inf__fin_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A2: nat] :
              ( ( member_nat @ A2 @ A )
             => ( ord_less_eq_nat @ X3 @ A2 ) )
         => ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_892_Inf__fin_OboundedI,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ! [A2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A2 @ A )
             => ( ord_le1083603963089353582_ereal @ X3 @ A2 ) )
         => ( ord_le1083603963089353582_ereal @ X3 @ ( lattic1367271666432936043_ereal @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_893_Inf__fin_OboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ! [A2: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A2 @ A )
             => ( ord_le3935885782089961368nnreal @ X3 @ A2 ) )
         => ( ord_le3935885782089961368nnreal @ X3 @ ( lattic6966437970300483035nnreal @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_894_Inf__fin_OboundedI,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ! [A2: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ A2 @ A )
             => ( ord_le6787938422905777998nnreal @ X3 @ A2 ) )
         => ( ord_le6787938422905777998nnreal @ X3 @ ( lattic8983332664287545745nnreal @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_895_Inf__fin_OboundedI,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ! [A2: set_Extended_ereal] :
              ( ( member5519481007471526743_ereal @ A2 @ A )
             => ( ord_le1644982726543182158_ereal @ X3 @ A2 ) )
         => ( ord_le1644982726543182158_ereal @ X3 @ ( lattic3876247248944996939_ereal @ A ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_896_Sup__fin_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( ord_le8460144461188290721at_nat @ ( lattic3087465103008755000at_nat @ A ) @ X3 )
         => ! [A6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A6 @ A )
             => ( ord_le8460144461188290721at_nat @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_897_Sup__fin_OboundedE,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
         => ! [A6: nat] :
              ( ( member_nat @ A6 @ A )
             => ( ord_less_eq_nat @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_898_Sup__fin_OboundedE,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ ( lattic6963256781428740357_ereal @ A ) @ X3 )
         => ! [A6: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A6 @ A )
             => ( ord_le1083603963089353582_ereal @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_899_Sup__fin_OboundedE,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( lattic4971367313934416833nnreal @ A ) @ X3 )
         => ! [A6: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A6 @ A )
             => ( ord_le3935885782089961368nnreal @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_900_Sup__fin_OboundedE,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( ord_le6787938422905777998nnreal @ ( lattic6502854510321989495nnreal @ A ) @ X3 )
         => ! [A6: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ A6 @ A )
             => ( ord_le6787938422905777998nnreal @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_901_Sup__fin_OboundedE,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( ord_le1644982726543182158_ereal @ ( lattic6967077786232488677_ereal @ A ) @ X3 )
         => ! [A6: set_Extended_ereal] :
              ( ( member5519481007471526743_ereal @ A6 @ A )
             => ( ord_le1644982726543182158_ereal @ A6 @ X3 ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_902_Sup__fin_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ! [A2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A2 @ A )
             => ( ord_le8460144461188290721at_nat @ A2 @ X3 ) )
         => ( ord_le8460144461188290721at_nat @ ( lattic3087465103008755000at_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_903_Sup__fin_OboundedI,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ! [A2: nat] :
              ( ( member_nat @ A2 @ A )
             => ( ord_less_eq_nat @ A2 @ X3 ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_904_Sup__fin_OboundedI,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ! [A2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ A2 @ A )
             => ( ord_le1083603963089353582_ereal @ A2 @ X3 ) )
         => ( ord_le1083603963089353582_ereal @ ( lattic6963256781428740357_ereal @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_905_Sup__fin_OboundedI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ! [A2: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ A2 @ A )
             => ( ord_le3935885782089961368nnreal @ A2 @ X3 ) )
         => ( ord_le3935885782089961368nnreal @ ( lattic4971367313934416833nnreal @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_906_Sup__fin_OboundedI,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ! [A2: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ A2 @ A )
             => ( ord_le6787938422905777998nnreal @ A2 @ X3 ) )
         => ( ord_le6787938422905777998nnreal @ ( lattic6502854510321989495nnreal @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_907_Sup__fin_OboundedI,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ! [A2: set_Extended_ereal] :
              ( ( member5519481007471526743_ereal @ A2 @ A )
             => ( ord_le1644982726543182158_ereal @ A2 @ X3 ) )
         => ( ord_le1644982726543182158_ereal @ ( lattic6967077786232488677_ereal @ A ) @ X3 ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_908_Inf__fin_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( ord_le8460144461188290721at_nat @ X3 @ ( lattic4585598177315101342at_nat @ A ) )
          = ( ! [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A )
               => ( ord_le8460144461188290721at_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_909_Inf__fin_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A ) )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_910_Inf__fin_Obounded__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ X3 @ ( lattic1367271666432936043_ereal @ A ) )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_911_Inf__fin_Obounded__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ X3 @ ( lattic6966437970300483035nnreal @ A ) )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_912_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( ord_le6787938422905777998nnreal @ X3 @ ( lattic8983332664287545745nnreal @ A ) )
          = ( ! [X: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ X @ A )
               => ( ord_le6787938422905777998nnreal @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_913_Inf__fin_Obounded__iff,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( ord_le1644982726543182158_ereal @ X3 @ ( lattic3876247248944996939_ereal @ A ) )
          = ( ! [X: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ X @ A )
               => ( ord_le1644982726543182158_ereal @ X3 @ X ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_914_Sup__fin_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ( ( ord_le8460144461188290721at_nat @ ( lattic3087465103008755000at_nat @ A ) @ X3 )
          = ( ! [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A )
               => ( ord_le8460144461188290721at_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_915_Sup__fin_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_916_Sup__fin_Obounded__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ ( lattic6963256781428740357_ereal @ A ) @ X3 )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_917_Sup__fin_Obounded__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ ( lattic4971367313934416833nnreal @ A ) @ X3 )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_918_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se4580700918925141924nnreal,X3: set_Ex3793607809372303086nnreal] :
      ( ( finite3583719589609615493nnreal @ A )
     => ( ( A != bot_bo2988155216863113784nnreal )
       => ( ( ord_le6787938422905777998nnreal @ ( lattic6502854510321989495nnreal @ A ) @ X3 )
          = ( ! [X: set_Ex3793607809372303086nnreal] :
                ( ( member603777416030116741nnreal @ X @ A )
               => ( ord_le6787938422905777998nnreal @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_919_Sup__fin_Obounded__iff,axiom,
    ! [A: set_se6634062954251873166_ereal,X3: set_Extended_ereal] :
      ( ( finite2737741666826350167_ereal @ A )
     => ( ( A != bot_bo7400643019497942010_ereal )
       => ( ( ord_le1644982726543182158_ereal @ ( lattic6967077786232488677_ereal @ A ) @ X3 )
          = ( ! [X: set_Extended_ereal] :
                ( ( member5519481007471526743_ereal @ X @ A )
               => ( ord_le1644982726543182158_ereal @ X @ X3 ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_920_dual__Min,axiom,
    ( ( lattices_Min_nat
      @ ^ [X: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X ) )
    = lattic8265883725875713057ax_nat ) ).

% dual_Min
thf(fact_921_dual__Min,axiom,
    ( ( lattic7357840519375772703_ereal
      @ ^ [X: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y5 @ X ) )
    = lattic5789169447084241725_ereal ) ).

% dual_Min
thf(fact_922_dual__Min,axiom,
    ( ( lattic336639396402831143nnreal
      @ ^ [X: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y5 @ X ) )
    = lattic933167949679527817nnreal ) ).

% dual_Min
thf(fact_923_Max__const,axiom,
    ! [A: set_nat,C2: extend8495563244428889912nnreal] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic933167949679527817nnreal
            @ ( image_8459861568512453903nnreal
              @ ^ [Uu: nat] : C2
              @ A ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_924_Max__const,axiom,
    ! [A: set_nat,C2: extended_ereal] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic5789169447084241725_ereal
            @ ( image_4309273772856505399_ereal
              @ ^ [Uu: nat] : C2
              @ A ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_925_Max__const,axiom,
    ! [A: set_Extended_ereal,C2: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( lattic5789169447084241725_ereal
            @ ( image_6042159593519690757_ereal
              @ ^ [Uu: extended_ereal] : C2
              @ A ) )
          = C2 ) ) ) ).

% Max_const
thf(fact_926_Inf__fin_Osubset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
            = ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_927_Inf__fin_Osubset,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( B != bot_bo2099793752762293965at_nat )
       => ( ( ord_le3146513528884898305at_nat @ B @ A )
         => ( ( inf_in4240866628268492783at_nat @ ( lattic4585598177315101342at_nat @ B ) @ ( lattic4585598177315101342at_nat @ A ) )
            = ( lattic4585598177315101342at_nat @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_928_Inf__fin_Osubset,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( B != bot_bo4854962954004695426nnreal )
       => ( ( ord_le6787938422905777998nnreal @ B @ A )
         => ( ( inf_in7439215052339218890nnreal @ ( lattic6966437970300483035nnreal @ B ) @ ( lattic6966437970300483035nnreal @ A ) )
            = ( lattic6966437970300483035nnreal @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_929_Inf__fin_Osubset,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( B != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1644982726543182158_ereal @ B @ A )
         => ( ( inf_in2794916579150040252_ereal @ ( lattic1367271666432936043_ereal @ B ) @ ( lattic1367271666432936043_ereal @ A ) )
            = ( lattic1367271666432936043_ereal @ A ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_930_Sup__fin_Osubset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( B != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B @ A )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
            = ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_931_Sup__fin_Osubset,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( B != bot_bo2099793752762293965at_nat )
       => ( ( ord_le3146513528884898305at_nat @ B @ A )
         => ( ( sup_su4120719815643632853at_nat @ ( lattic3087465103008755000at_nat @ B ) @ ( lattic3087465103008755000at_nat @ A ) )
            = ( lattic3087465103008755000at_nat @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_932_Sup__fin_Osubset,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( B != bot_bo4854962954004695426nnreal )
       => ( ( ord_le6787938422905777998nnreal @ B @ A )
         => ( ( sup_su6922871097908479076nnreal @ ( lattic4971367313934416833nnreal @ B ) @ ( lattic4971367313934416833nnreal @ A ) )
            = ( lattic4971367313934416833nnreal @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_933_Sup__fin_Osubset,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( B != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1644982726543182158_ereal @ B @ A )
         => ( ( sup_su7653423775389492130_ereal @ ( lattic6963256781428740357_ereal @ B ) @ ( lattic6963256781428740357_ereal @ A ) )
            = ( lattic6963256781428740357_ereal @ A ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_934_Min__antimono,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ N )
     => ( ( M2 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).

% Min_antimono
thf(fact_935_Min__antimono,axiom,
    ! [M2: set_Ex3793607809372303086nnreal,N: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ M2 @ N )
     => ( ( M2 != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ N )
         => ( ord_le3935885782089961368nnreal @ ( lattic8839003927053164919nnreal @ N ) @ ( lattic8839003927053164919nnreal @ M2 ) ) ) ) ) ).

% Min_antimono
thf(fact_936_Min__antimono,axiom,
    ! [M2: set_Extended_ereal,N: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ M2 @ N )
     => ( ( M2 != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ N )
         => ( ord_le1083603963089353582_ereal @ ( lattic3885496046747141583_ereal @ N ) @ ( lattic3885496046747141583_ereal @ M2 ) ) ) ) ) ).

% Min_antimono
thf(fact_937_Min_Osubset__imp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).

% Min.subset_imp
thf(fact_938_Min_Osubset__imp,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( finite3782138982310603983nnreal @ B )
         => ( ord_le3935885782089961368nnreal @ ( lattic8839003927053164919nnreal @ B ) @ ( lattic8839003927053164919nnreal @ A ) ) ) ) ) ).

% Min.subset_imp
thf(fact_939_Min_Osubset__imp,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( finite7198162374296863863_ereal @ B )
         => ( ord_le1083603963089353582_ereal @ ( lattic3885496046747141583_ereal @ B ) @ ( lattic3885496046747141583_ereal @ A ) ) ) ) ) ).

% Min.subset_imp
thf(fact_940_Max__less__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le7381754540660121996nnreal @ ( lattic933167949679527817nnreal @ A ) @ X3 )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le7381754540660121996nnreal @ X @ X3 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_941_Max__less__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X3 )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_942_Max__less__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1188267648640031866_ereal @ ( lattic5789169447084241725_ereal @ A ) @ X3 )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1188267648640031866_ereal @ X @ X3 ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_943_Id__on__empty,axiom,
    ( ( id_on_2591757777509516123nnreal @ bot_bo4854962954004695426nnreal )
    = bot_bo345837629027619229nnreal ) ).

% Id_on_empty
thf(fact_944_Id__on__empty,axiom,
    ( ( id_on_Extended_ereal @ bot_bo8367695208629047834_ereal )
    = bot_bo4002835157671732723_ereal ) ).

% Id_on_empty
thf(fact_945_Int__UNIV,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B )
        = top_top_set_nat )
      = ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_946_Int__UNIV,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ( inf_in2779415704524776092_ereal @ A @ B )
        = top_to5683747375963461374_ereal )
      = ( ( A = top_to5683747375963461374_ereal )
        & ( B = top_to5683747375963461374_ereal ) ) ) ).

% Int_UNIV
thf(fact_947_image__eqI,axiom,
    ! [B3: extended_ereal,F: extended_ereal > extended_ereal,X3: extended_ereal,A: set_Extended_ereal] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member2350847679896131959_ereal @ X3 @ A )
       => ( member2350847679896131959_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_948_image__eqI,axiom,
    ! [B3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,X3: nat,A: set_nat] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_nat @ X3 @ A )
       => ( member7908768830364227535nnreal @ B3 @ ( image_8459861568512453903nnreal @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_949_image__eqI,axiom,
    ! [B3: extended_ereal,F: nat > extended_ereal,X3: nat,A: set_nat] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_nat @ X3 @ A )
       => ( member2350847679896131959_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_950_image__eqI,axiom,
    ! [B3: nat,F: nat > nat,X3: nat,A: set_nat] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_nat @ X3 @ A )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_951_image__eqI,axiom,
    ! [B3: b,F: nat > b,X3: nat,A: set_nat] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_nat @ X3 @ A )
       => ( member_b @ B3 @ ( image_nat_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_952_image__eqI,axiom,
    ! [B3: nat,F: b > nat,X3: b,A: set_b] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_b @ X3 @ A )
       => ( member_nat @ B3 @ ( image_b_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_953_image__eqI,axiom,
    ! [B3: b,F: b > b,X3: b,A: set_b] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_b @ X3 @ A )
       => ( member_b @ B3 @ ( image_b_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_954_image__eqI,axiom,
    ! [B3: product_prod_b_b,F: nat > product_prod_b_b,X3: nat,A: set_nat] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_nat @ X3 @ A )
       => ( member7862447936710763792od_b_b @ B3 @ ( image_6808858347418066896od_b_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_955_image__eqI,axiom,
    ! [B3: product_prod_b_b,F: b > product_prod_b_b,X3: b,A: set_b] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member_b @ X3 @ A )
       => ( member7862447936710763792od_b_b @ B3 @ ( image_3973729904588732333od_b_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_956_image__eqI,axiom,
    ! [B3: nat,F: product_prod_b_b > nat,X3: product_prod_b_b,A: set_Product_prod_b_b] :
      ( ( B3
        = ( F @ X3 ) )
     => ( ( member7862447936710763792od_b_b @ X3 @ A )
       => ( member_nat @ B3 @ ( image_6770982514055950706_b_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_957_Un__subset__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ ( sup_su1870425459839775770nnreal @ A @ B ) @ C )
      = ( ( ord_le6787938422905777998nnreal @ A @ C )
        & ( ord_le6787938422905777998nnreal @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_958_Un__subset__iff,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ ( sup_su2680283192902082946_ereal @ A @ B ) @ C )
      = ( ( ord_le1644982726543182158_ereal @ A @ C )
        & ( ord_le1644982726543182158_ereal @ B @ C ) ) ) ).

% Un_subset_iff
thf(fact_959_Int__subset__iff,axiom,
    ! [C: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ C @ ( inf_in3368558534146122112nnreal @ A @ B ) )
      = ( ( ord_le6787938422905777998nnreal @ C @ A )
        & ( ord_le6787938422905777998nnreal @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_960_Int__subset__iff,axiom,
    ! [C: set_Extended_ereal,A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ C @ ( inf_in2779415704524776092_ereal @ A @ B ) )
      = ( ( ord_le1644982726543182158_ereal @ C @ A )
        & ( ord_le1644982726543182158_ereal @ C @ B ) ) ) ).

% Int_subset_iff
thf(fact_961_psubsetI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( A != B )
       => ( ord_le7033131045148242242nnreal @ A @ B ) ) ) ).

% psubsetI
thf(fact_962_psubsetI,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ord_le1644982726543182158_ereal @ A @ B )
     => ( ( A != B )
       => ( ord_le5321083090456148570_ereal @ A @ B ) ) ) ).

% psubsetI
thf(fact_963_Un__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ( sup_su1870425459839775770nnreal @ A @ B )
        = bot_bo4854962954004695426nnreal )
      = ( ( A = bot_bo4854962954004695426nnreal )
        & ( B = bot_bo4854962954004695426nnreal ) ) ) ).

% Un_empty
thf(fact_964_Un__empty,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( ( sup_su2680283192902082946_ereal @ A @ B )
        = bot_bo8367695208629047834_ereal )
      = ( ( A = bot_bo8367695208629047834_ereal )
        & ( B = bot_bo8367695208629047834_ereal ) ) ) ).

% Un_empty
thf(fact_965_finite__Un,axiom,
    ! [F2: set_Product_prod_b_b,G: set_Product_prod_b_b] :
      ( ( finite3757003017338540048od_b_b @ ( sup_su2483643821041016987od_b_b @ F2 @ G ) )
      = ( ( finite3757003017338540048od_b_b @ F2 )
        & ( finite3757003017338540048od_b_b @ G ) ) ) ).

% finite_Un
thf(fact_966_finite__Un,axiom,
    ! [F2: set_b,G: set_b] :
      ( ( finite_finite_b @ ( sup_sup_set_b @ F2 @ G ) )
      = ( ( finite_finite_b @ F2 )
        & ( finite_finite_b @ G ) ) ) ).

% finite_Un
thf(fact_967_finite__Un,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G ) ) ) ).

% finite_Un
thf(fact_968_finite__Un,axiom,
    ! [F2: set_Pr5139338970096277698od_b_b,G: set_Pr5139338970096277698od_b_b] :
      ( ( finite7768965217515309219od_b_b @ ( sup_su7306626712404911598od_b_b @ F2 @ G ) )
      = ( ( finite7768965217515309219od_b_b @ F2 )
        & ( finite7768965217515309219od_b_b @ G ) ) ) ).

% finite_Un
thf(fact_969_finite__Un,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G ) )
      = ( ( finite6177210948735845034at_nat @ F2 )
        & ( finite6177210948735845034at_nat @ G ) ) ) ).

% finite_Un
thf(fact_970_finite__Un,axiom,
    ! [F2: set_Pr4264375888882495962_nat_b,G: set_Pr4264375888882495962_nat_b] :
      ( ( finite659689794318260667_nat_b @ ( sup_su9013224398775143174_nat_b @ F2 @ G ) )
      = ( ( finite659689794318260667_nat_b @ F2 )
        & ( finite659689794318260667_nat_b @ G ) ) ) ).

% finite_Un
thf(fact_971_finite__Un,axiom,
    ! [F2: set_Pr4323519195528460463od_b_b,G: set_Pr4323519195528460463od_b_b] :
      ( ( finite4644902770518909432od_b_b @ ( sup_su492463831250443907od_b_b @ F2 @ G ) )
      = ( ( finite4644902770518909432od_b_b @ F2 )
        & ( finite4644902770518909432od_b_b @ G ) ) ) ).

% finite_Un
thf(fact_972_finite__Un,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,G: set_Pr1307281990691478580_b_nat] :
      ( ( finite7880342692102525205_b_nat @ ( sup_su6056130500584125792_b_nat @ F2 @ G ) )
      = ( ( finite7880342692102525205_b_nat @ F2 )
        & ( finite7880342692102525205_b_nat @ G ) ) ) ).

% finite_Un
thf(fact_973_finite__Int,axiom,
    ! [F2: set_Product_prod_b_b,G: set_Product_prod_b_b] :
      ( ( ( finite3757003017338540048od_b_b @ F2 )
        | ( finite3757003017338540048od_b_b @ G ) )
     => ( finite3757003017338540048od_b_b @ ( inf_in8340392639285749429od_b_b @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_974_finite__Int,axiom,
    ! [F2: set_b,G: set_b] :
      ( ( ( finite_finite_b @ F2 )
        | ( finite_finite_b @ G ) )
     => ( finite_finite_b @ ( inf_inf_set_b @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_975_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_976_finite__Int,axiom,
    ! [F2: set_Pr5139338970096277698od_b_b,G: set_Pr5139338970096277698od_b_b] :
      ( ( ( finite7768965217515309219od_b_b @ F2 )
        | ( finite7768965217515309219od_b_b @ G ) )
     => ( finite7768965217515309219od_b_b @ ( inf_in5735908020845025108od_b_b @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_977_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F2 )
        | ( finite6177210948735845034at_nat @ G ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_978_finite__Int,axiom,
    ! [F2: set_Pr4264375888882495962_nat_b,G: set_Pr4264375888882495962_nat_b] :
      ( ( ( finite659689794318260667_nat_b @ F2 )
        | ( finite659689794318260667_nat_b @ G ) )
     => ( finite659689794318260667_nat_b @ ( inf_in3142974464590631532_nat_b @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_979_finite__Int,axiom,
    ! [F2: set_Pr4323519195528460463od_b_b,G: set_Pr4323519195528460463od_b_b] :
      ( ( ( finite4644902770518909432od_b_b @ F2 )
        | ( finite4644902770518909432od_b_b @ G ) )
     => ( finite4644902770518909432od_b_b @ ( inf_in5360946474388847261od_b_b @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_980_finite__Int,axiom,
    ! [F2: set_Pr1307281990691478580_b_nat,G: set_Pr1307281990691478580_b_nat] :
      ( ( ( finite7880342692102525205_b_nat @ F2 )
        | ( finite7880342692102525205_b_nat @ G ) )
     => ( finite7880342692102525205_b_nat @ ( inf_in185880566399614150_b_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_981_if__image__distrib,axiom,
    ! [P: extended_ereal > $o,F: extended_ereal > extended_ereal,G2: extended_ereal > extended_ereal,S: set_Extended_ereal] :
      ( ( image_6042159593519690757_ereal
        @ ^ [X: extended_ereal] : ( if_Extended_ereal @ ( P @ X ) @ ( F @ X ) @ ( G2 @ X ) )
        @ S )
      = ( sup_su2680283192902082946_ereal @ ( image_6042159593519690757_ereal @ F @ ( inf_in2779415704524776092_ereal @ S @ ( collec5835592288176408249_ereal @ P ) ) )
        @ ( image_6042159593519690757_ereal @ G2
          @ ( inf_in2779415704524776092_ereal @ S
            @ ( collec5835592288176408249_ereal
              @ ^ [X: extended_ereal] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_982_if__image__distrib,axiom,
    ! [P: nat > $o,F: nat > extend8495563244428889912nnreal,G2: nat > extend8495563244428889912nnreal,S: set_nat] :
      ( ( image_8459861568512453903nnreal
        @ ^ [X: nat] : ( if_Ext9135588136721118450nnreal @ ( P @ X ) @ ( F @ X ) @ ( G2 @ X ) )
        @ S )
      = ( sup_su1870425459839775770nnreal @ ( image_8459861568512453903nnreal @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_8459861568512453903nnreal @ G2
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X: nat] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_983_if__image__distrib,axiom,
    ! [P: nat > $o,F: nat > extended_ereal,G2: nat > extended_ereal,S: set_nat] :
      ( ( image_4309273772856505399_ereal
        @ ^ [X: nat] : ( if_Extended_ereal @ ( P @ X ) @ ( F @ X ) @ ( G2 @ X ) )
        @ S )
      = ( sup_su2680283192902082946_ereal @ ( image_4309273772856505399_ereal @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
        @ ( image_4309273772856505399_ereal @ G2
          @ ( inf_inf_set_nat @ S
            @ ( collect_nat
              @ ^ [X: nat] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_984_image__ident,axiom,
    ! [Y6: set_Extended_ereal] :
      ( ( image_6042159593519690757_ereal
        @ ^ [X: extended_ereal] : X
        @ Y6 )
      = Y6 ) ).

% image_ident
thf(fact_985_image__empty,axiom,
    ! [F: nat > extend8495563244428889912nnreal] :
      ( ( image_8459861568512453903nnreal @ F @ bot_bot_set_nat )
      = bot_bo4854962954004695426nnreal ) ).

% image_empty
thf(fact_986_image__empty,axiom,
    ! [F: nat > extended_ereal] :
      ( ( image_4309273772856505399_ereal @ F @ bot_bot_set_nat )
      = bot_bo8367695208629047834_ereal ) ).

% image_empty
thf(fact_987_image__empty,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( image_8394674774369097847nnreal @ F @ bot_bo4854962954004695426nnreal )
      = bot_bo4854962954004695426nnreal ) ).

% image_empty
thf(fact_988_image__empty,axiom,
    ! [F: extend8495563244428889912nnreal > extended_ereal] :
      ( ( image_6393943237584228047_ereal @ F @ bot_bo4854962954004695426nnreal )
      = bot_bo8367695208629047834_ereal ) ).

% image_empty
thf(fact_989_image__empty,axiom,
    ! [F: extended_ereal > extend8495563244428889912nnreal] :
      ( ( image_8614087454967683265nnreal @ F @ bot_bo8367695208629047834_ereal )
      = bot_bo4854962954004695426nnreal ) ).

% image_empty
thf(fact_990_image__empty,axiom,
    ! [F: extended_ereal > extended_ereal] :
      ( ( image_6042159593519690757_ereal @ F @ bot_bo8367695208629047834_ereal )
      = bot_bo8367695208629047834_ereal ) ).

% image_empty
thf(fact_991_empty__is__image,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( bot_bo4854962954004695426nnreal
        = ( image_8459861568512453903nnreal @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_992_empty__is__image,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( bot_bo4854962954004695426nnreal
        = ( image_8394674774369097847nnreal @ F @ A ) )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% empty_is_image
thf(fact_993_empty__is__image,axiom,
    ! [F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
      ( ( bot_bo4854962954004695426nnreal
        = ( image_8614087454967683265nnreal @ F @ A ) )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% empty_is_image
thf(fact_994_empty__is__image,axiom,
    ! [F: nat > extended_ereal,A: set_nat] :
      ( ( bot_bo8367695208629047834_ereal
        = ( image_4309273772856505399_ereal @ F @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% empty_is_image
thf(fact_995_empty__is__image,axiom,
    ! [F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
      ( ( bot_bo8367695208629047834_ereal
        = ( image_6393943237584228047_ereal @ F @ A ) )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% empty_is_image
thf(fact_996_empty__is__image,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
      ( ( bot_bo8367695208629047834_ereal
        = ( image_6042159593519690757_ereal @ F @ A ) )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% empty_is_image
thf(fact_997_image__is__empty,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( ( image_8459861568512453903nnreal @ F @ A )
        = bot_bo4854962954004695426nnreal )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_998_image__is__empty,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( ( image_8394674774369097847nnreal @ F @ A )
        = bot_bo4854962954004695426nnreal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% image_is_empty
thf(fact_999_image__is__empty,axiom,
    ! [F: extended_ereal > extend8495563244428889912nnreal,A: set_Extended_ereal] :
      ( ( ( image_8614087454967683265nnreal @ F @ A )
        = bot_bo4854962954004695426nnreal )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% image_is_empty
thf(fact_1000_image__is__empty,axiom,
    ! [F: nat > extended_ereal,A: set_nat] :
      ( ( ( image_4309273772856505399_ereal @ F @ A )
        = bot_bo8367695208629047834_ereal )
      = ( A = bot_bot_set_nat ) ) ).

% image_is_empty
thf(fact_1001_image__is__empty,axiom,
    ! [F: extend8495563244428889912nnreal > extended_ereal,A: set_Ex3793607809372303086nnreal] :
      ( ( ( image_6393943237584228047_ereal @ F @ A )
        = bot_bo8367695208629047834_ereal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% image_is_empty
thf(fact_1002_image__is__empty,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
      ( ( ( image_6042159593519690757_ereal @ F @ A )
        = bot_bo8367695208629047834_ereal )
      = ( A = bot_bo8367695208629047834_ereal ) ) ).

% image_is_empty
thf(fact_1003_finite__imageI,axiom,
    ! [F2: set_Extended_ereal,H: extended_ereal > extended_ereal] :
      ( ( finite7198162374296863863_ereal @ F2 )
     => ( finite7198162374296863863_ereal @ ( image_6042159593519690757_ereal @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1004_finite__imageI,axiom,
    ! [F2: set_b,H: b > b] :
      ( ( finite_finite_b @ F2 )
     => ( finite_finite_b @ ( image_b_b @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1005_finite__imageI,axiom,
    ! [F2: set_b,H: b > nat] :
      ( ( finite_finite_b @ F2 )
     => ( finite_finite_nat @ ( image_b_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1006_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > extend8495563244428889912nnreal] :
      ( ( finite_finite_nat @ F2 )
     => ( finite3782138982310603983nnreal @ ( image_8459861568512453903nnreal @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1007_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > extended_ereal] :
      ( ( finite_finite_nat @ F2 )
     => ( finite7198162374296863863_ereal @ ( image_4309273772856505399_ereal @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1008_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > b] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_b @ ( image_nat_b @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1009_finite__imageI,axiom,
    ! [F2: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F2 )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1010_finite__imageI,axiom,
    ! [F2: set_Product_prod_b_b,H: product_prod_b_b > b] :
      ( ( finite3757003017338540048od_b_b @ F2 )
     => ( finite_finite_b @ ( image_8398514867482601949_b_b_b @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1011_finite__imageI,axiom,
    ! [F2: set_Product_prod_b_b,H: product_prod_b_b > nat] :
      ( ( finite3757003017338540048od_b_b @ F2 )
     => ( finite_finite_nat @ ( image_6770982514055950706_b_nat @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1012_finite__imageI,axiom,
    ! [F2: set_b,H: b > product_prod_b_b] :
      ( ( finite_finite_b @ F2 )
     => ( finite3757003017338540048od_b_b @ ( image_3973729904588732333od_b_b @ H @ F2 ) ) ) ).

% finite_imageI
thf(fact_1013_inf__Sup__absorb,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ( ( inf_inf_nat @ A5 @ ( lattic1093996805478795353in_nat @ A ) )
          = A5 ) ) ) ).

% inf_Sup_absorb
thf(fact_1014_inf__Sup__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ( ( inf_in4240866628268492783at_nat @ A5 @ ( lattic3087465103008755000at_nat @ A ) )
          = A5 ) ) ) ).

% inf_Sup_absorb
thf(fact_1015_sup__Inf__absorb,axiom,
    ! [A: set_nat,A5: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A5 @ A )
       => ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A5 )
          = A5 ) ) ) ).

% sup_Inf_absorb
thf(fact_1016_sup__Inf__absorb,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A5: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( member8440522571783428010at_nat @ A5 @ A )
       => ( ( sup_su4120719815643632853at_nat @ ( lattic4585598177315101342at_nat @ A ) @ A5 )
          = A5 ) ) ) ).

% sup_Inf_absorb
thf(fact_1017_Min_Obounded__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1018_Min_Obounded__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1083603963089353582_ereal @ X3 @ ( lattic3885496046747141583_ereal @ A ) )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1083603963089353582_ereal @ X3 @ X ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1019_Min_Obounded__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le3935885782089961368nnreal @ X3 @ ( lattic8839003927053164919nnreal @ A ) )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le3935885782089961368nnreal @ X3 @ X ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1020_Min__gr__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( finite3782138982310603983nnreal @ A )
     => ( ( A != bot_bo4854962954004695426nnreal )
       => ( ( ord_le7381754540660121996nnreal @ X3 @ ( lattic8839003927053164919nnreal @ A ) )
          = ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A )
               => ( ord_le7381754540660121996nnreal @ X3 @ X ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1021_Min__gr__iff,axiom,
    ! [A: set_nat,X3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( ord_less_nat @ X3 @ ( lattic8721135487736765967in_nat @ A ) )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A )
               => ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1022_Min__gr__iff,axiom,
    ! [A: set_Extended_ereal,X3: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( ord_le1188267648640031866_ereal @ X3 @ ( lattic3885496046747141583_ereal @ A ) )
          = ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ A )
               => ( ord_le1188267648640031866_ereal @ X3 @ X ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1023_Min__const,axiom,
    ! [A: set_nat,C2: extend8495563244428889912nnreal] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic8839003927053164919nnreal
            @ ( image_8459861568512453903nnreal
              @ ^ [Uu: nat] : C2
              @ A ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_1024_Min__const,axiom,
    ! [A: set_nat,C2: extended_ereal] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ( ( lattic3885496046747141583_ereal
            @ ( image_4309273772856505399_ereal
              @ ^ [Uu: nat] : C2
              @ A ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_1025_Min__const,axiom,
    ! [A: set_Extended_ereal,C2: extended_ereal] :
      ( ( finite7198162374296863863_ereal @ A )
     => ( ( A != bot_bo8367695208629047834_ereal )
       => ( ( lattic3885496046747141583_ereal
            @ ( image_6042159593519690757_ereal
              @ ^ [Uu: extended_ereal] : C2
              @ A ) )
          = C2 ) ) ) ).

% Min_const
thf(fact_1026_image__Int__subset,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat,B: set_nat] : ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_in3368558534146122112nnreal @ ( image_8459861568512453903nnreal @ F @ A ) @ ( image_8459861568512453903nnreal @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1027_image__Int__subset,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,B: set_Extended_ereal] : ( ord_le1644982726543182158_ereal @ ( image_6042159593519690757_ereal @ F @ ( inf_in2779415704524776092_ereal @ A @ B ) ) @ ( inf_in2779415704524776092_ereal @ ( image_6042159593519690757_ereal @ F @ A ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1028_image__Int__subset,axiom,
    ! [F: nat > extended_ereal,A: set_nat,B: set_nat] : ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_in2779415704524776092_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ ( image_4309273772856505399_ereal @ F @ B ) ) ) ).

% image_Int_subset
thf(fact_1029_Un__Int__assoc__eq,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
      ( ( ( sup_su1870425459839775770nnreal @ ( inf_in3368558534146122112nnreal @ A @ B ) @ C )
        = ( inf_in3368558534146122112nnreal @ A @ ( sup_su1870425459839775770nnreal @ B @ C ) ) )
      = ( ord_le6787938422905777998nnreal @ C @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_1030_Un__Int__assoc__eq,axiom,
    ! [A: set_Extended_ereal,B: set_Extended_ereal,C: set_Extended_ereal] :
      ( ( ( sup_su2680283192902082946_ereal @ ( inf_in2779415704524776092_ereal @ A @ B ) @ C )
        = ( inf_in2779415704524776092_ereal @ A @ ( sup_su2680283192902082946_ereal @ B @ C ) ) )
      = ( ord_le1644982726543182158_ereal @ C @ A ) ) ).

% Un_Int_assoc_eq
thf(fact_1031_order__less__imp__not__less,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ~ ( ord_less_nat @ Y4 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1032_order__less__imp__not__less,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ~ ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1033_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ( Y4 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1034_order__less__imp__not__eq2,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( Y4 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1035_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ( X3 != Y4 ) ) ).

% order_less_imp_not_eq
thf(fact_1036_order__less__imp__not__eq,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( X3 != Y4 ) ) ).

% order_less_imp_not_eq
thf(fact_1037_linorder__less__linear,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
      | ( X3 = Y4 )
      | ( ord_less_nat @ Y4 @ X3 ) ) ).

% linorder_less_linear
thf(fact_1038_linorder__less__linear,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
      | ( X3 = Y4 )
      | ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ).

% linorder_less_linear
thf(fact_1039_order__less__imp__triv,axiom,
    ! [X3: nat,Y4: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ( ( ord_less_nat @ Y4 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1040_order__less__imp__triv,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal,P: $o] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( ( ord_le1188267648640031866_ereal @ Y4 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1041_order__less__not__sym,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ~ ( ord_less_nat @ Y4 @ X3 ) ) ).

% order_less_not_sym
thf(fact_1042_order__less__not__sym,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ~ ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ).

% order_less_not_sym
thf(fact_1043_order__less__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1044_order__less__subst2,axiom,
    ! [A5: nat,B3: nat,F: nat > extended_ereal,C2: extended_ereal] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( ord_le1188267648640031866_ereal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1045_order__less__subst2,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > nat,C2: nat] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1046_order__less__subst2,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( ord_le1188267648640031866_ereal @ ( F @ B3 ) @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_1047_order__less__subst1,axiom,
    ! [A5: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1048_order__less__subst1,axiom,
    ! [A5: nat,F: extended_ereal > nat,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_less_nat @ A5 @ ( F @ B3 ) )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1049_order__less__subst1,axiom,
    ! [A5: extended_ereal,F: nat > extended_ereal,B3: nat,C2: nat] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1050_order__less__subst1,axiom,
    ! [A5: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ ( F @ B3 ) )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_1051_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1052_order__less__irrefl,axiom,
    ! [X3: extended_ereal] :
      ~ ( ord_le1188267648640031866_ereal @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1053_ord__less__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1054_ord__less__eq__subst,axiom,
    ! [A5: nat,B3: nat,F: nat > extended_ereal,C2: extended_ereal] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1055_ord__less__eq__subst,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > nat,C2: nat] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1056_ord__less__eq__subst,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ ( F @ A5 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1057_ord__eq__less__subst,axiom,
    ! [A5: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1058_ord__eq__less__subst,axiom,
    ! [A5: extended_ereal,F: nat > extended_ereal,B3: nat,C2: nat] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1059_ord__eq__less__subst,axiom,
    ! [A5: nat,F: extended_ereal > nat,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1060_ord__eq__less__subst,axiom,
    ! [A5: extended_ereal,F: extended_ereal > extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5
        = ( F @ B3 ) )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ! [X2: extended_ereal,Y2: extended_ereal] :
              ( ( ord_le1188267648640031866_ereal @ X2 @ Y2 )
             => ( ord_le1188267648640031866_ereal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_le1188267648640031866_ereal @ A5 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1061_order__less__trans,axiom,
    ! [X3: nat,Y4: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ( ( ord_less_nat @ Y4 @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_1062_order__less__trans,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal,Z3: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( ( ord_le1188267648640031866_ereal @ Y4 @ Z3 )
       => ( ord_le1188267648640031866_ereal @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_1063_order__less__asym_H,axiom,
    ! [A5: nat,B3: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A5 ) ) ).

% order_less_asym'
thf(fact_1064_order__less__asym_H,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ~ ( ord_le1188267648640031866_ereal @ B3 @ A5 ) ) ).

% order_less_asym'
thf(fact_1065_linorder__neq__iff,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( X3 != Y4 )
      = ( ( ord_less_nat @ X3 @ Y4 )
        | ( ord_less_nat @ Y4 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_1066_linorder__neq__iff,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( X3 != Y4 )
      = ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
        | ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_1067_order__less__asym,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ~ ( ord_less_nat @ Y4 @ X3 ) ) ).

% order_less_asym
thf(fact_1068_order__less__asym,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ~ ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ).

% order_less_asym
thf(fact_1069_linorder__neqE,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( X3 != Y4 )
     => ( ~ ( ord_less_nat @ X3 @ Y4 )
       => ( ord_less_nat @ Y4 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_1070_linorder__neqE,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( X3 != Y4 )
     => ( ~ ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
       => ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_1071_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: nat,A5: nat] :
      ( ( ord_less_nat @ B3 @ A5 )
     => ( A5 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1072_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: extended_ereal,A5: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ B3 @ A5 )
     => ( A5 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_1073_order_Ostrict__implies__not__eq,axiom,
    ! [A5: nat,B3: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( A5 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_1074_order_Ostrict__implies__not__eq,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( A5 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_1075_dual__order_Ostrict__trans,axiom,
    ! [B3: nat,A5: nat,C2: nat] :
      ( ( ord_less_nat @ B3 @ A5 )
     => ( ( ord_less_nat @ C2 @ B3 )
       => ( ord_less_nat @ C2 @ A5 ) ) ) ).

% dual_order.strict_trans
thf(fact_1076_dual__order_Ostrict__trans,axiom,
    ! [B3: extended_ereal,A5: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ B3 @ A5 )
     => ( ( ord_le1188267648640031866_ereal @ C2 @ B3 )
       => ( ord_le1188267648640031866_ereal @ C2 @ A5 ) ) ) ).

% dual_order.strict_trans
thf(fact_1077_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
      = ( ( ord_less_nat @ Y4 @ X3 )
        | ( X3 = Y4 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1078_not__less__iff__gr__or__eq,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ~ ( ord_le1188267648640031866_ereal @ X3 @ Y4 ) )
      = ( ( ord_le1188267648640031866_ereal @ Y4 @ X3 )
        | ( X3 = Y4 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1079_order_Ostrict__trans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ A5 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1080_order_Ostrict__trans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ord_le1188267648640031866_ereal @ A5 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_1081_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A5: nat,B3: nat] :
      ( ! [A2: nat,B5: nat] :
          ( ( ord_less_nat @ A2 @ B5 )
         => ( P @ A2 @ B5 ) )
     => ( ! [A2: nat] : ( P @ A2 @ A2 )
       => ( ! [A2: nat,B5: nat] :
              ( ( P @ B5 @ A2 )
             => ( P @ A2 @ B5 ) )
         => ( P @ A5 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1082_linorder__less__wlog,axiom,
    ! [P: extended_ereal > extended_ereal > $o,A5: extended_ereal,B3: extended_ereal] :
      ( ! [A2: extended_ereal,B5: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ A2 @ B5 )
         => ( P @ A2 @ B5 ) )
     => ( ! [A2: extended_ereal] : ( P @ A2 @ A2 )
       => ( ! [A2: extended_ereal,B5: extended_ereal] :
              ( ( P @ B5 @ A2 )
             => ( P @ A2 @ B5 ) )
         => ( P @ A5 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_1083_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P2: nat > $o] :
        ? [N2: nat] :
          ( ( P2 @ N2 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N2 )
             => ~ ( P2 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1084_dual__order_Oirrefl,axiom,
    ! [A5: nat] :
      ~ ( ord_less_nat @ A5 @ A5 ) ).

% dual_order.irrefl
thf(fact_1085_dual__order_Oirrefl,axiom,
    ! [A5: extended_ereal] :
      ~ ( ord_le1188267648640031866_ereal @ A5 @ A5 ) ).

% dual_order.irrefl
thf(fact_1086_dual__order_Oasym,axiom,
    ! [B3: nat,A5: nat] :
      ( ( ord_less_nat @ B3 @ A5 )
     => ~ ( ord_less_nat @ A5 @ B3 ) ) ).

% dual_order.asym
thf(fact_1087_dual__order_Oasym,axiom,
    ! [B3: extended_ereal,A5: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ B3 @ A5 )
     => ~ ( ord_le1188267648640031866_ereal @ A5 @ B3 ) ) ).

% dual_order.asym
thf(fact_1088_linorder__cases,axiom,
    ! [X3: nat,Y4: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y4 )
     => ( ( X3 != Y4 )
       => ( ord_less_nat @ Y4 @ X3 ) ) ) ).

% linorder_cases
thf(fact_1089_linorder__cases,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ~ ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( ( X3 != Y4 )
       => ( ord_le1188267648640031866_ereal @ Y4 @ X3 ) ) ) ).

% linorder_cases
thf(fact_1090_antisym__conv3,axiom,
    ! [Y4: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y4 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y4 ) )
        = ( X3 = Y4 ) ) ) ).

% antisym_conv3
thf(fact_1091_antisym__conv3,axiom,
    ! [Y4: extended_ereal,X3: extended_ereal] :
      ( ~ ( ord_le1188267648640031866_ereal @ Y4 @ X3 )
     => ( ( ~ ( ord_le1188267648640031866_ereal @ X3 @ Y4 ) )
        = ( X3 = Y4 ) ) ) ).

% antisym_conv3
thf(fact_1092_less__induct,axiom,
    ! [P: nat > $o,A5: nat] :
      ( ! [X2: nat] :
          ( ! [Y3: nat] :
              ( ( ord_less_nat @ Y3 @ X2 )
             => ( P @ Y3 ) )
         => ( P @ X2 ) )
     => ( P @ A5 ) ) ).

% less_induct
thf(fact_1093_ord__less__eq__trans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_nat @ A5 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1094_ord__less__eq__trans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_le1188267648640031866_ereal @ A5 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_1095_ord__eq__less__trans,axiom,
    ! [A5: nat,B3: nat,C2: nat] :
      ( ( A5 = B3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ A5 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1096_ord__eq__less__trans,axiom,
    ! [A5: extended_ereal,B3: extended_ereal,C2: extended_ereal] :
      ( ( A5 = B3 )
     => ( ( ord_le1188267648640031866_ereal @ B3 @ C2 )
       => ( ord_le1188267648640031866_ereal @ A5 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_1097_order_Oasym,axiom,
    ! [A5: nat,B3: nat] :
      ( ( ord_less_nat @ A5 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A5 ) ) ).

% order.asym
thf(fact_1098_order_Oasym,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ A5 @ B3 )
     => ~ ( ord_le1188267648640031866_ereal @ B3 @ A5 ) ) ).

% order.asym
thf(fact_1099_less__imp__neq,axiom,
    ! [X3: nat,Y4: nat] :
      ( ( ord_less_nat @ X3 @ Y4 )
     => ( X3 != Y4 ) ) ).

% less_imp_neq
thf(fact_1100_less__imp__neq,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ( X3 != Y4 ) ) ).

% less_imp_neq
thf(fact_1101_dense,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( ord_le1188267648640031866_ereal @ X3 @ Y4 )
     => ? [Z2: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ X3 @ Z2 )
          & ( ord_le1188267648640031866_ereal @ Z2 @ Y4 ) ) ) ).

% dense
thf(fact_1102_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_1103_Sigma__Un__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_b] :
      ( ( product_Sigma_b_b @ ( sup_sup_set_b @ I @ J ) @ C )
      = ( sup_su2483643821041016987od_b_b @ ( product_Sigma_b_b @ I @ C ) @ ( product_Sigma_b_b @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1104_Sigma__Un__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_Product_prod_b_b] :
      ( ( produc8027630620858748621od_b_b @ ( sup_sup_set_nat @ I @ J ) @ C )
      = ( sup_su7306626712404911598od_b_b @ ( produc8027630620858748621od_b_b @ I @ C ) @ ( produc8027630620858748621od_b_b @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1105_Sigma__Un__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ I @ J ) @ C )
      = ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I @ C ) @ ( produc457027306803732586at_nat @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1106_Sigma__Un__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_b] :
      ( ( product_Sigma_nat_b @ ( sup_sup_set_nat @ I @ J ) @ C )
      = ( sup_su9013224398775143174_nat_b @ ( product_Sigma_nat_b @ I @ C ) @ ( product_Sigma_nat_b @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1107_Sigma__Un__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ ( sup_sup_set_b @ I @ J ) @ C )
      = ( sup_su492463831250443907od_b_b @ ( produc2915425143180021232od_b_b @ I @ C ) @ ( produc2915425143180021232od_b_b @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1108_Sigma__Un__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_nat] :
      ( ( product_Sigma_b_nat @ ( sup_sup_set_b @ I @ J ) @ C )
      = ( sup_su6056130500584125792_b_nat @ ( product_Sigma_b_nat @ I @ C ) @ ( product_Sigma_b_nat @ J @ C ) ) ) ).

% Sigma_Un_distrib1
thf(fact_1109_Sigma__Int__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_b] :
      ( ( product_Sigma_b_b @ ( inf_inf_set_b @ I @ J ) @ C )
      = ( inf_in8340392639285749429od_b_b @ ( product_Sigma_b_b @ I @ C ) @ ( product_Sigma_b_b @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1110_Sigma__Int__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_Product_prod_b_b] :
      ( ( produc8027630620858748621od_b_b @ ( inf_inf_set_nat @ I @ J ) @ C )
      = ( inf_in5735908020845025108od_b_b @ ( produc8027630620858748621od_b_b @ I @ C ) @ ( produc8027630620858748621od_b_b @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1111_Sigma__Int__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ ( inf_inf_set_nat @ I @ J ) @ C )
      = ( inf_in2572325071724192079at_nat @ ( produc457027306803732586at_nat @ I @ C ) @ ( produc457027306803732586at_nat @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1112_Sigma__Int__distrib1,axiom,
    ! [I: set_nat,J: set_nat,C: nat > set_b] :
      ( ( product_Sigma_nat_b @ ( inf_inf_set_nat @ I @ J ) @ C )
      = ( inf_in3142974464590631532_nat_b @ ( product_Sigma_nat_b @ I @ C ) @ ( product_Sigma_nat_b @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1113_Sigma__Int__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ ( inf_inf_set_b @ I @ J ) @ C )
      = ( inf_in5360946474388847261od_b_b @ ( produc2915425143180021232od_b_b @ I @ C ) @ ( produc2915425143180021232od_b_b @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1114_Sigma__Int__distrib1,axiom,
    ! [I: set_b,J: set_b,C: b > set_nat] :
      ( ( product_Sigma_b_nat @ ( inf_inf_set_b @ I @ J ) @ C )
      = ( inf_in185880566399614150_b_nat @ ( product_Sigma_b_nat @ I @ C ) @ ( product_Sigma_b_nat @ J @ C ) ) ) ).

% Sigma_Int_distrib1
thf(fact_1115_rev__image__eqI,axiom,
    ! [X3: extended_ereal,A: set_Extended_ereal,B3: extended_ereal,F: extended_ereal > extended_ereal] :
      ( ( member2350847679896131959_ereal @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member2350847679896131959_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1116_rev__image__eqI,axiom,
    ! [X3: nat,A: set_nat,B3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal] :
      ( ( member_nat @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member7908768830364227535nnreal @ B3 @ ( image_8459861568512453903nnreal @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1117_rev__image__eqI,axiom,
    ! [X3: nat,A: set_nat,B3: extended_ereal,F: nat > extended_ereal] :
      ( ( member_nat @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member2350847679896131959_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1118_rev__image__eqI,axiom,
    ! [X3: nat,A: set_nat,B3: nat,F: nat > nat] :
      ( ( member_nat @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1119_rev__image__eqI,axiom,
    ! [X3: nat,A: set_nat,B3: b,F: nat > b] :
      ( ( member_nat @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member_b @ B3 @ ( image_nat_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1120_rev__image__eqI,axiom,
    ! [X3: b,A: set_b,B3: nat,F: b > nat] :
      ( ( member_b @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member_nat @ B3 @ ( image_b_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1121_rev__image__eqI,axiom,
    ! [X3: b,A: set_b,B3: b,F: b > b] :
      ( ( member_b @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member_b @ B3 @ ( image_b_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1122_rev__image__eqI,axiom,
    ! [X3: nat,A: set_nat,B3: product_prod_b_b,F: nat > product_prod_b_b] :
      ( ( member_nat @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member7862447936710763792od_b_b @ B3 @ ( image_6808858347418066896od_b_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1123_rev__image__eqI,axiom,
    ! [X3: b,A: set_b,B3: product_prod_b_b,F: b > product_prod_b_b] :
      ( ( member_b @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member7862447936710763792od_b_b @ B3 @ ( image_3973729904588732333od_b_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1124_rev__image__eqI,axiom,
    ! [X3: product_prod_b_b,A: set_Product_prod_b_b,B3: nat,F: product_prod_b_b > nat] :
      ( ( member7862447936710763792od_b_b @ X3 @ A )
     => ( ( B3
          = ( F @ X3 ) )
       => ( member_nat @ B3 @ ( image_6770982514055950706_b_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_1125_ball__imageD,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
      ( ! [X2: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X2 @ ( image_6042159593519690757_ereal @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X5: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X5 @ A )
         => ( P @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_1126_ball__imageD,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
      ( ! [X2: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ X2 @ ( image_8459861568512453903nnreal @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A )
         => ( P @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_1127_ball__imageD,axiom,
    ! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
      ( ! [X2: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X2 @ ( image_4309273772856505399_ereal @ F @ A ) )
         => ( P @ X2 ) )
     => ! [X5: nat] :
          ( ( member_nat @ X5 @ A )
         => ( P @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_1128_image__cong,axiom,
    ! [M2: set_Extended_ereal,N: set_Extended_ereal,F: extended_ereal > extended_ereal,G2: extended_ereal > extended_ereal] :
      ( ( M2 = N )
     => ( ! [X2: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X2 @ N )
           => ( ( F @ X2 )
              = ( G2 @ X2 ) ) )
       => ( ( image_6042159593519690757_ereal @ F @ M2 )
          = ( image_6042159593519690757_ereal @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_1129_image__cong,axiom,
    ! [M2: set_nat,N: set_nat,F: nat > extend8495563244428889912nnreal,G2: nat > extend8495563244428889912nnreal] :
      ( ( M2 = N )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ N )
           => ( ( F @ X2 )
              = ( G2 @ X2 ) ) )
       => ( ( image_8459861568512453903nnreal @ F @ M2 )
          = ( image_8459861568512453903nnreal @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_1130_image__cong,axiom,
    ! [M2: set_nat,N: set_nat,F: nat > extended_ereal,G2: nat > extended_ereal] :
      ( ( M2 = N )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ N )
           => ( ( F @ X2 )
              = ( G2 @ X2 ) ) )
       => ( ( image_4309273772856505399_ereal @ F @ M2 )
          = ( image_4309273772856505399_ereal @ G2 @ N ) ) ) ) ).

% image_cong
thf(fact_1131_bex__imageD,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
      ( ? [X5: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X5 @ ( image_6042159593519690757_ereal @ F @ A ) )
          & ( P @ X5 ) )
     => ? [X2: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_1132_bex__imageD,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
      ( ? [X5: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ X5 @ ( image_8459861568512453903nnreal @ F @ A ) )
          & ( P @ X5 ) )
     => ? [X2: nat] :
          ( ( member_nat @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_1133_bex__imageD,axiom,
    ! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
      ( ? [X5: extended_ereal] :
          ( ( member2350847679896131959_ereal @ X5 @ ( image_4309273772856505399_ereal @ F @ A ) )
          & ( P @ X5 ) )
     => ? [X2: nat] :
          ( ( member_nat @ X2 @ A )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_1134_image__iff,axiom,
    ! [Z3: extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
      ( ( member2350847679896131959_ereal @ Z3 @ ( image_6042159593519690757_ereal @ F @ A ) )
      = ( ? [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ A )
            & ( Z3
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_1135_image__iff,axiom,
    ! [Z3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( member7908768830364227535nnreal @ Z3 @ ( image_8459861568512453903nnreal @ F @ A ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ( Z3
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_1136_image__iff,axiom,
    ! [Z3: extended_ereal,F: nat > extended_ereal,A: set_nat] :
      ( ( member2350847679896131959_ereal @ Z3 @ ( image_4309273772856505399_ereal @ F @ A ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ( Z3
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_1137_image__Un,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,B: set_Extended_ereal] :
      ( ( image_6042159593519690757_ereal @ F @ ( sup_su2680283192902082946_ereal @ A @ B ) )
      = ( sup_su2680283192902082946_ereal @ ( image_6042159593519690757_ereal @ F @ A ) @ ( image_6042159593519690757_ereal @ F @ B ) ) ) ).

% image_Un
thf(fact_1138_image__Un,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat,B: set_nat] :
      ( ( image_8459861568512453903nnreal @ F @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_su1870425459839775770nnreal @ ( image_8459861568512453903nnreal @ F @ A ) @ ( image_8459861568512453903nnreal @ F @ B ) ) ) ).

% image_Un
thf(fact_1139_image__Un,axiom,
    ! [F: nat > extended_ereal,A: set_nat,B: set_nat] :
      ( ( image_4309273772856505399_ereal @ F @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_su2680283192902082946_ereal @ ( image_4309273772856505399_ereal @ F @ A ) @ ( image_4309273772856505399_ereal @ F @ B ) ) ) ).

% image_Un
thf(fact_1140_imageI,axiom,
    ! [X3: extended_ereal,A: set_Extended_ereal,F: extended_ereal > extended_ereal] :
      ( ( member2350847679896131959_ereal @ X3 @ A )
     => ( member2350847679896131959_ereal @ ( F @ X3 ) @ ( image_6042159593519690757_ereal @ F @ A ) ) ) ).

% imageI
thf(fact_1141_imageI,axiom,
    ! [X3: nat,A: set_nat,F: nat > extend8495563244428889912nnreal] :
      ( ( member_nat @ X3 @ A )
     => ( member7908768830364227535nnreal @ ( F @ X3 ) @ ( image_8459861568512453903nnreal @ F @ A ) ) ) ).

% imageI
thf(fact_1142_imageI,axiom,
    ! [X3: nat,A: set_nat,F: nat > extended_ereal] :
      ( ( member_nat @ X3 @ A )
     => ( member2350847679896131959_ereal @ ( F @ X3 ) @ ( image_4309273772856505399_ereal @ F @ A ) ) ) ).

% imageI
thf(fact_1143_imageI,axiom,
    ! [X3: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat @ X3 @ A )
     => ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1144_imageI,axiom,
    ! [X3: nat,A: set_nat,F: nat > b] :
      ( ( member_nat @ X3 @ A )
     => ( member_b @ ( F @ X3 ) @ ( image_nat_b @ F @ A ) ) ) ).

% imageI
thf(fact_1145_imageI,axiom,
    ! [X3: b,A: set_b,F: b > nat] :
      ( ( member_b @ X3 @ A )
     => ( member_nat @ ( F @ X3 ) @ ( image_b_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1146_imageI,axiom,
    ! [X3: b,A: set_b,F: b > b] :
      ( ( member_b @ X3 @ A )
     => ( member_b @ ( F @ X3 ) @ ( image_b_b @ F @ A ) ) ) ).

% imageI
thf(fact_1147_imageI,axiom,
    ! [X3: nat,A: set_nat,F: nat > product_prod_b_b] :
      ( ( member_nat @ X3 @ A )
     => ( member7862447936710763792od_b_b @ ( F @ X3 ) @ ( image_6808858347418066896od_b_b @ F @ A ) ) ) ).

% imageI
thf(fact_1148_imageI,axiom,
    ! [X3: b,A: set_b,F: b > product_prod_b_b] :
      ( ( member_b @ X3 @ A )
     => ( member7862447936710763792od_b_b @ ( F @ X3 ) @ ( image_3973729904588732333od_b_b @ F @ A ) ) ) ).

% imageI
thf(fact_1149_imageI,axiom,
    ! [X3: product_prod_b_b,A: set_Product_prod_b_b,F: product_prod_b_b > nat] :
      ( ( member7862447936710763792od_b_b @ X3 @ A )
     => ( member_nat @ ( F @ X3 ) @ ( image_6770982514055950706_b_nat @ F @ A ) ) ) ).

% imageI
thf(fact_1150_Compr__image__eq,axiom,
    ! [F: extended_ereal > extended_ereal,A: set_Extended_ereal,P: extended_ereal > $o] :
      ( ( collec5835592288176408249_ereal
        @ ^ [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_6042159593519690757_ereal @ F
        @ ( collec5835592288176408249_ereal
          @ ^ [X: extended_ereal] :
              ( ( member2350847679896131959_ereal @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1151_Compr__image__eq,axiom,
    ! [F: nat > extend8495563244428889912nnreal,A: set_nat,P: extend8495563244428889912nnreal > $o] :
      ( ( collec6648975593938027277nnreal
        @ ^ [X: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ ( image_8459861568512453903nnreal @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_8459861568512453903nnreal @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1152_Compr__image__eq,axiom,
    ! [F: nat > extended_ereal,A: set_nat,P: extended_ereal > $o] :
      ( ( collec5835592288176408249_ereal
        @ ^ [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ ( image_4309273772856505399_ereal @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_4309273772856505399_ereal @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1153_Compr__image__eq,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1154_Compr__image__eq,axiom,
    ! [F: b > nat,A: set_b,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_b_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_b_nat @ F
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1155_Compr__image__eq,axiom,
    ! [F: nat > b,A: set_nat,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_nat_b @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_b @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1156_Compr__image__eq,axiom,
    ! [F: b > b,A: set_b,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_b_b @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_b_b @ F
        @ ( collect_b
          @ ^ [X: b] :
              ( ( member_b @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1157_Compr__image__eq,axiom,
    ! [F: product_prod_b_b > nat,A: set_Product_prod_b_b,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_6770982514055950706_b_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_6770982514055950706_b_nat @ F
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1158_Compr__image__eq,axiom,
    ! [F: product_prod_b_b > b,A: set_Product_prod_b_b,P: b > $o] :
      ( ( collect_b
        @ ^ [X: b] :
            ( ( member_b @ X @ ( image_8398514867482601949_b_b_b @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_8398514867482601949_b_b_b @ F
        @ ( collec548942219715005266od_b_b
          @ ^ [X: product_prod_b_b] :
              ( ( member7862447936710763792od_b_b @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1159_Compr__image__eq,axiom,
    ! [F: nat > product_prod_b_b,A: set_nat,P: product_prod_b_b > $o] :
      ( ( collec548942219715005266od_b_b
        @ ^ [X: product_prod_b_b] :
            ( ( member7862447936710763792od_b_b @ X @ ( image_6808858347418066896od_b_b @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_6808858347418066896od_b_b @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1160_image__image,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G2: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( image_8394674774369097847nnreal @ F @ ( image_8459861568512453903nnreal @ G2 @ A ) )
      = ( image_8459861568512453903nnreal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1161_image__image,axiom,
    ! [F: extend8495563244428889912nnreal > extended_ereal,G2: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( image_6393943237584228047_ereal @ F @ ( image_8459861568512453903nnreal @ G2 @ A ) )
      = ( image_4309273772856505399_ereal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1162_image__image,axiom,
    ! [F: extended_ereal > extend8495563244428889912nnreal,G2: nat > extended_ereal,A: set_nat] :
      ( ( image_8614087454967683265nnreal @ F @ ( image_4309273772856505399_ereal @ G2 @ A ) )
      = ( image_8459861568512453903nnreal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1163_image__image,axiom,
    ! [F: extended_ereal > extended_ereal,G2: extended_ereal > extended_ereal,A: set_Extended_ereal] :
      ( ( image_6042159593519690757_ereal @ F @ ( image_6042159593519690757_ereal @ G2 @ A ) )
      = ( image_6042159593519690757_ereal
        @ ^ [X: extended_ereal] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1164_image__image,axiom,
    ! [F: extended_ereal > extended_ereal,G2: nat > extended_ereal,A: set_nat] :
      ( ( image_6042159593519690757_ereal @ F @ ( image_4309273772856505399_ereal @ G2 @ A ) )
      = ( image_4309273772856505399_ereal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1165_image__image,axiom,
    ! [F: nat > extend8495563244428889912nnreal,G2: nat > nat,A: set_nat] :
      ( ( image_8459861568512453903nnreal @ F @ ( image_nat_nat @ G2 @ A ) )
      = ( image_8459861568512453903nnreal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1166_image__image,axiom,
    ! [F: nat > extended_ereal,G2: extended_ereal > nat,A: set_Extended_ereal] :
      ( ( image_4309273772856505399_ereal @ F @ ( image_7659842161140344153al_nat @ G2 @ A ) )
      = ( image_6042159593519690757_ereal
        @ ^ [X: extended_ereal] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1167_image__image,axiom,
    ! [F: nat > extended_ereal,G2: nat > nat,A: set_nat] :
      ( ( image_4309273772856505399_ereal @ F @ ( image_nat_nat @ G2 @ A ) )
      = ( image_4309273772856505399_ereal
        @ ^ [X: nat] : ( F @ ( G2 @ X ) )
        @ A ) ) ).

% image_image
thf(fact_1168_imageE,axiom,
    ! [B3: extended_ereal,F: extended_ereal > extended_ereal,A: set_Extended_ereal] :
      ( ( member2350847679896131959_ereal @ B3 @ ( image_6042159593519690757_ereal @ F @ A ) )
     => ~ ! [X2: extended_ereal] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member2350847679896131959_ereal @ X2 @ A ) ) ) ).

% imageE
thf(fact_1169_imageE,axiom,
    ! [B3: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,A: set_nat] :
      ( ( member7908768830364227535nnreal @ B3 @ ( image_8459861568512453903nnreal @ F @ A ) )
     => ~ ! [X2: nat] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ).

% imageE
thf(fact_1170_imageE,axiom,
    ! [B3: extended_ereal,F: nat > extended_ereal,A: set_nat] :
      ( ( member2350847679896131959_ereal @ B3 @ ( image_4309273772856505399_ereal @ F @ A ) )
     => ~ ! [X2: nat] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ).

% imageE
thf(fact_1171_imageE,axiom,
    ! [B3: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) )
     => ~ ! [X2: nat] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ).

% imageE
thf(fact_1172_imageE,axiom,
    ! [B3: nat,F: b > nat,A: set_b] :
      ( ( member_nat @ B3 @ ( image_b_nat @ F @ A ) )
     => ~ ! [X2: b] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_b @ X2 @ A ) ) ) ).

% imageE
thf(fact_1173_imageE,axiom,
    ! [B3: b,F: nat > b,A: set_nat] :
      ( ( member_b @ B3 @ ( image_nat_b @ F @ A ) )
     => ~ ! [X2: nat] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ).

% imageE
thf(fact_1174_imageE,axiom,
    ! [B3: b,F: b > b,A: set_b] :
      ( ( member_b @ B3 @ ( image_b_b @ F @ A ) )
     => ~ ! [X2: b] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_b @ X2 @ A ) ) ) ).

% imageE
thf(fact_1175_imageE,axiom,
    ! [B3: nat,F: product_prod_b_b > nat,A: set_Product_prod_b_b] :
      ( ( member_nat @ B3 @ ( image_6770982514055950706_b_nat @ F @ A ) )
     => ~ ! [X2: product_prod_b_b] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member7862447936710763792od_b_b @ X2 @ A ) ) ) ).

% imageE
thf(fact_1176_imageE,axiom,
    ! [B3: b,F: product_prod_b_b > b,A: set_Product_prod_b_b] :
      ( ( member_b @ B3 @ ( image_8398514867482601949_b_b_b @ F @ A ) )
     => ~ ! [X2: product_prod_b_b] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member7862447936710763792od_b_b @ X2 @ A ) ) ) ).

% imageE
thf(fact_1177_imageE,axiom,
    ! [B3: product_prod_b_b,F: nat > product_prod_b_b,A: set_nat] :
      ( ( member7862447936710763792od_b_b @ B3 @ ( image_6808858347418066896od_b_b @ F @ A ) )
     => ~ ! [X2: nat] :
            ( ( B3
              = ( F @ X2 ) )
           => ~ ( member_nat @ X2 @ A ) ) ) ).

% imageE
thf(fact_1178_finite__conv__nat__seg__image,axiom,
    ( finite3782138982310603983nnreal
    = ( ^ [A4: set_Ex3793607809372303086nnreal] :
        ? [N2: nat,F3: nat > extend8495563244428889912nnreal] :
          ( A4
          = ( image_8459861568512453903nnreal @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1179_finite__conv__nat__seg__image,axiom,
    ( finite7198162374296863863_ereal
    = ( ^ [A4: set_Extended_ereal] :
        ? [N2: nat,F3: nat > extended_ereal] :
          ( A4
          = ( image_4309273772856505399_ereal @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1180_finite__conv__nat__seg__image,axiom,
    ( finite3757003017338540048od_b_b
    = ( ^ [A4: set_Product_prod_b_b] :
        ? [N2: nat,F3: nat > product_prod_b_b] :
          ( A4
          = ( image_6808858347418066896od_b_b @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1181_finite__conv__nat__seg__image,axiom,
    ( finite_finite_b
    = ( ^ [A4: set_b] :
        ? [N2: nat,F3: nat > b] :
          ( A4
          = ( image_nat_b @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1182_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
        ? [N2: nat,F3: nat > nat] :
          ( A4
          = ( image_nat_nat @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1183_finite__conv__nat__seg__image,axiom,
    ( finite7768965217515309219od_b_b
    = ( ^ [A4: set_Pr5139338970096277698od_b_b] :
        ? [N2: nat,F3: nat > produc1536031394801701132od_b_b] :
          ( A4
          = ( image_5000473625437735139od_b_b @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1184_finite__conv__nat__seg__image,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat] :
        ? [N2: nat,F3: nat > product_prod_nat_nat] :
          ( A4
          = ( image_5846123807819985514at_nat @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1185_finite__conv__nat__seg__image,axiom,
    ( finite659689794318260667_nat_b
    = ( ^ [A4: set_Pr4264375888882495962_nat_b] :
        ? [N2: nat,F3: nat > product_prod_nat_b] :
          ( A4
          = ( image_8668673924000913915_nat_b @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1186_finite__conv__nat__seg__image,axiom,
    ( finite4644902770518909432od_b_b
    = ( ^ [A4: set_Pr4323519195528460463od_b_b] :
        ? [N2: nat,F3: nat > produc2840042325109449167od_b_b] :
          ( A4
          = ( image_2386253535795868088od_b_b @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1187_finite__conv__nat__seg__image,axiom,
    ( finite7880342692102525205_b_nat
    = ( ^ [A4: set_Pr1307281990691478580_b_nat] :
        ? [N2: nat,F3: nat > product_prod_b_nat] :
          ( A4
          = ( image_6665954784930402645_b_nat @ F3
            @ ( collect_nat
              @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_1188_nat__seg__image__imp__finite,axiom,
    ! [A: set_Ex3793607809372303086nnreal,F: nat > extend8495563244428889912nnreal,N3: nat] :
      ( ( A
        = ( image_8459861568512453903nnreal @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite3782138982310603983nnreal @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1189_nat__seg__image__imp__finite,axiom,
    ! [A: set_Extended_ereal,F: nat > extended_ereal,N3: nat] :
      ( ( A
        = ( image_4309273772856505399_ereal @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite7198162374296863863_ereal @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1190_nat__seg__image__imp__finite,axiom,
    ! [A: set_Product_prod_b_b,F: nat > product_prod_b_b,N3: nat] :
      ( ( A
        = ( image_6808858347418066896od_b_b @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite3757003017338540048od_b_b @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1191_nat__seg__image__imp__finite,axiom,
    ! [A: set_b,F: nat > b,N3: nat] :
      ( ( A
        = ( image_nat_b @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite_finite_b @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1192_nat__seg__image__imp__finite,axiom,
    ! [A: set_nat,F: nat > nat,N3: nat] :
      ( ( A
        = ( image_nat_nat @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite_finite_nat @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1193_nat__seg__image__imp__finite,axiom,
    ! [A: set_Pr5139338970096277698od_b_b,F: nat > produc1536031394801701132od_b_b,N3: nat] :
      ( ( A
        = ( image_5000473625437735139od_b_b @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite7768965217515309219od_b_b @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1194_nat__seg__image__imp__finite,axiom,
    ! [A: set_Pr1261947904930325089at_nat,F: nat > product_prod_nat_nat,N3: nat] :
      ( ( A
        = ( image_5846123807819985514at_nat @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite6177210948735845034at_nat @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1195_nat__seg__image__imp__finite,axiom,
    ! [A: set_Pr4264375888882495962_nat_b,F: nat > product_prod_nat_b,N3: nat] :
      ( ( A
        = ( image_8668673924000913915_nat_b @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite659689794318260667_nat_b @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1196_nat__seg__image__imp__finite,axiom,
    ! [A: set_Pr4323519195528460463od_b_b,F: nat > produc2840042325109449167od_b_b,N3: nat] :
      ( ( A
        = ( image_2386253535795868088od_b_b @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite4644902770518909432od_b_b @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1197_nat__seg__image__imp__finite,axiom,
    ! [A: set_Pr1307281990691478580_b_nat,F: nat > product_prod_b_nat,N3: nat] :
      ( ( A
        = ( image_6665954784930402645_b_nat @ F
          @ ( collect_nat
            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) ) )
     => ( finite7880342692102525205_b_nat @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_1198_Times__Un__distrib1,axiom,
    ! [A: set_b,B: set_b,C: set_b] :
      ( ( product_Sigma_b_b @ ( sup_sup_set_b @ A @ B )
        @ ^ [Uu: b] : C )
      = ( sup_su2483643821041016987od_b_b
        @ ( product_Sigma_b_b @ A
          @ ^ [Uu: b] : C )
        @ ( product_Sigma_b_b @ B
          @ ^ [Uu: b] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1199_Times__Un__distrib1,axiom,
    ! [A: set_nat,B: set_nat,C: set_Product_prod_b_b] :
      ( ( produc8027630620858748621od_b_b @ ( sup_sup_set_nat @ A @ B )
        @ ^ [Uu: nat] : C )
      = ( sup_su7306626712404911598od_b_b
        @ ( produc8027630620858748621od_b_b @ A
          @ ^ [Uu: nat] : C )
        @ ( produc8027630620858748621od_b_b @ B
          @ ^ [Uu: nat] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1200_Times__Un__distrib1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( produc457027306803732586at_nat @ ( sup_sup_set_nat @ A @ B )
        @ ^ [Uu: nat] : C )
      = ( sup_su6327502436637775413at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : C )
        @ ( produc457027306803732586at_nat @ B
          @ ^ [Uu: nat] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1201_Times__Un__distrib1,axiom,
    ! [A: set_nat,B: set_nat,C: set_b] :
      ( ( product_Sigma_nat_b @ ( sup_sup_set_nat @ A @ B )
        @ ^ [Uu: nat] : C )
      = ( sup_su9013224398775143174_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : C )
        @ ( product_Sigma_nat_b @ B
          @ ^ [Uu: nat] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1202_Times__Un__distrib1,axiom,
    ! [A: set_b,B: set_b,C: set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ ( sup_sup_set_b @ A @ B )
        @ ^ [Uu: b] : C )
      = ( sup_su492463831250443907od_b_b
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : C )
        @ ( produc2915425143180021232od_b_b @ B
          @ ^ [Uu: b] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1203_Times__Un__distrib1,axiom,
    ! [A: set_b,B: set_b,C: set_nat] :
      ( ( product_Sigma_b_nat @ ( sup_sup_set_b @ A @ B )
        @ ^ [Uu: b] : C )
      = ( sup_su6056130500584125792_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : C )
        @ ( product_Sigma_b_nat @ B
          @ ^ [Uu: b] : C ) ) ) ).

% Times_Un_distrib1
thf(fact_1204_Sigma__Un__distrib2,axiom,
    ! [I: set_b,A: b > set_b,B: b > set_b] :
      ( ( product_Sigma_b_b @ I
        @ ^ [I2: b] : ( sup_sup_set_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su2483643821041016987od_b_b @ ( product_Sigma_b_b @ I @ A ) @ ( product_Sigma_b_b @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1205_Sigma__Un__distrib2,axiom,
    ! [I: set_nat,A: nat > set_Product_prod_b_b,B: nat > set_Product_prod_b_b] :
      ( ( produc8027630620858748621od_b_b @ I
        @ ^ [I2: nat] : ( sup_su2483643821041016987od_b_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su7306626712404911598od_b_b @ ( produc8027630620858748621od_b_b @ I @ A ) @ ( produc8027630620858748621od_b_b @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1206_Sigma__Un__distrib2,axiom,
    ! [I: set_nat,A: nat > set_nat,B: nat > set_nat] :
      ( ( produc457027306803732586at_nat @ I
        @ ^ [I2: nat] : ( sup_sup_set_nat @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( produc457027306803732586at_nat @ I @ A ) @ ( produc457027306803732586at_nat @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1207_Sigma__Un__distrib2,axiom,
    ! [I: set_nat,A: nat > set_b,B: nat > set_b] :
      ( ( product_Sigma_nat_b @ I
        @ ^ [I2: nat] : ( sup_sup_set_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su9013224398775143174_nat_b @ ( product_Sigma_nat_b @ I @ A ) @ ( product_Sigma_nat_b @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1208_Sigma__Un__distrib2,axiom,
    ! [I: set_b,A: b > set_Product_prod_b_b,B: b > set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ I
        @ ^ [I2: b] : ( sup_su2483643821041016987od_b_b @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su492463831250443907od_b_b @ ( produc2915425143180021232od_b_b @ I @ A ) @ ( produc2915425143180021232od_b_b @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1209_Sigma__Un__distrib2,axiom,
    ! [I: set_b,A: b > set_nat,B: b > set_nat] :
      ( ( product_Sigma_b_nat @ I
        @ ^ [I2: b] : ( sup_sup_set_nat @ ( A @ I2 ) @ ( B @ I2 ) ) )
      = ( sup_su6056130500584125792_b_nat @ ( product_Sigma_b_nat @ I @ A ) @ ( product_Sigma_b_nat @ I @ B ) ) ) ).

% Sigma_Un_distrib2
thf(fact_1210_Times__Int__distrib1,axiom,
    ! [A: set_nat,B: set_nat,C: set_b] :
      ( ( product_Sigma_nat_b @ ( inf_inf_set_nat @ A @ B )
        @ ^ [Uu: nat] : C )
      = ( inf_in3142974464590631532_nat_b
        @ ( product_Sigma_nat_b @ A
          @ ^ [Uu: nat] : C )
        @ ( product_Sigma_nat_b @ B
          @ ^ [Uu: nat] : C ) ) ) ).

% Times_Int_distrib1
thf(fact_1211_Times__Int__distrib1,axiom,
    ! [A: set_b,B: set_b,C: set_Product_prod_b_b] :
      ( ( produc2915425143180021232od_b_b @ ( inf_inf_set_b @ A @ B )
        @ ^ [Uu: b] : C )
      = ( inf_in5360946474388847261od_b_b
        @ ( produc2915425143180021232od_b_b @ A
          @ ^ [Uu: b] : C )
        @ ( produc2915425143180021232od_b_b @ B
          @ ^ [Uu: b] : C ) ) ) ).

% Times_Int_distrib1
thf(fact_1212_Times__Int__distrib1,axiom,
    ! [A: set_b,B: set_b,C: set_nat] :
      ( ( product_Sigma_b_nat @ ( inf_inf_set_b @ A @ B )
        @ ^ [Uu: b] : C )
      = ( inf_in185880566399614150_b_nat
        @ ( product_Sigma_b_nat @ A
          @ ^ [Uu: b] : C )
        @ ( product_Sigma_b_nat @ B
          @ ^ [Uu: b] : C ) ) ) ).

% Times_Int_distrib1
thf(fact_1213_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_1214_nat__descend__induct,axiom,
    ! [N3: nat,P: nat > $o,M3: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N3 @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N3 )
           => ( ! [I3: nat] :
                  ( ( ord_less_nat @ K @ I3 )
                 => ( P @ I3 ) )
             => ( P @ K ) ) )
       => ( P @ M3 ) ) ) ).

% nat_descend_induct
thf(fact_1215_unbounded__k__infinite,axiom,
    ! [K2: nat,S: set_nat] :
      ( ! [M: nat] :
          ( ( ord_less_nat @ K2 @ M )
         => ? [N4: nat] :
              ( ( ord_less_nat @ M @ N4 )
              & ( member_nat @ N4 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_1216_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M4: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M4 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1217_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_1218_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M4: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M4 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1219_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1220_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_1221_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I4: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I4 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_1222_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F @ N5 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_1223_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M2: nat] :
      ( ( P @ X3 )
     => ( ! [X2: nat] :
            ( ( P @ X2 )
           => ( ord_less_eq_nat @ X2 @ M2 ) )
       => ~ ! [M: nat] :
              ( ( P @ M )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1224_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M4: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N6 )
         => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1225_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I4: nat,J2: nat] :
      ( ! [I5: nat,J3: nat] :
          ( ( ord_less_nat @ I5 @ J3 )
         => ( ord_less_nat @ ( F @ I5 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I4 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I4 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1226_le__neq__implies__less,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ( M3 != N3 )
       => ( ord_less_nat @ M3 @ N3 ) ) ) ).

% le_neq_implies_less
thf(fact_1227_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B3: nat] :
      ( ( P @ K2 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ? [X2: nat] :
            ( ( P @ X2 )
            & ! [Y3: nat] :
                ( ( P @ Y3 )
               => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1228_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B3: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_1229_Greatest__le__nat,axiom,
    ! [P: nat > $o,K2: nat,B3: nat] :
      ( ( P @ K2 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_1230_nat__le__linear,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
      | ( ord_less_eq_nat @ N3 @ M3 ) ) ).

% nat_le_linear
thf(fact_1231_GreatestI__nat,axiom,
    ! [P: nat > $o,K2: nat,B3: nat] :
      ( ( P @ K2 )
     => ( ! [Y2: nat] :
            ( ( P @ Y2 )
           => ( ord_less_eq_nat @ Y2 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_1232_le__antisym,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ M3 )
       => ( M3 = N3 ) ) ) ).

% le_antisym
thf(fact_1233_eq__imp__le,axiom,
    ! [M3: nat,N3: nat] :
      ( ( M3 = N3 )
     => ( ord_less_eq_nat @ M3 @ N3 ) ) ).

% eq_imp_le
thf(fact_1234_le__trans,axiom,
    ! [I4: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K2 )
       => ( ord_less_eq_nat @ I4 @ K2 ) ) ) ).

% le_trans
thf(fact_1235_le__refl,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).

% le_refl
thf(fact_1236_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M4 @ N2 )
          & ( M4 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_1237_less__imp__le__nat,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_nat @ M3 @ N3 )
     => ( ord_less_eq_nat @ M3 @ N3 ) ) ).

% less_imp_le_nat
thf(fact_1238_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N2: nat] :
          ( ( ord_less_nat @ M4 @ N2 )
          | ( M4 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1239_less__or__eq__imp__le,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ( ord_less_nat @ M3 @ N3 )
        | ( M3 = N3 ) )
     => ( ord_less_eq_nat @ M3 @ N3 ) ) ).

% less_or_eq_imp_le
thf(fact_1240_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N3: nat] :
      ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
     => ( ord_less_eq_nat @ N3 @ ( F @ N3 ) ) ) ).

% strict_mono_imp_increasing
thf(fact_1241_infinite__enumerate,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ? [R3: nat > nat] :
          ( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R3 )
          & ! [N4: nat] : ( member_nat @ ( R3 @ N4 ) @ S ) ) ) ).

% infinite_enumerate
thf(fact_1242_ereal__incseq__uminus,axiom,
    ! [F: nat > extended_ereal] :
      ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal
        @ ^ [X: nat] : ( uminus27091377158695749_ereal @ ( F @ X ) ) )
      = ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y5 @ X )
        @ F ) ) ).

% ereal_incseq_uminus
thf(fact_1243_ereal__range__uminus,axiom,
    ( ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ top_to5683747375963461374_ereal )
    = top_to5683747375963461374_ereal ) ).

% ereal_range_uminus
thf(fact_1244_ereal__minus__le__minus,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A5 ) @ ( uminus27091377158695749_ereal @ B3 ) )
      = ( ord_le1083603963089353582_ereal @ B3 @ A5 ) ) ).

% ereal_minus_le_minus
thf(fact_1245_ereal__decseq__uminus,axiom,
    ! [F: nat > extended_ereal] :
      ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
        @ ^ [X: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y5 @ X )
        @ ^ [X: nat] : ( uminus27091377158695749_ereal @ ( F @ X ) ) )
      = ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat @ ord_le1083603963089353582_ereal @ F ) ) ).

% ereal_decseq_uminus
thf(fact_1246_ereal__uminus__le__reorder,axiom,
    ! [A5: extended_ereal,B3: extended_ereal] :
      ( ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ A5 ) @ B3 )
      = ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ B3 ) @ A5 ) ) ).

% ereal_uminus_le_reorder
thf(fact_1247_ereal__complete__uminus__eq,axiom,
    ! [S: set_Extended_ereal,X3: extended_ereal] :
      ( ( ! [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
           => ( ord_le1083603963089353582_ereal @ X @ X3 ) )
        & ! [Z5: extended_ereal] :
            ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ ( image_6042159593519690757_ereal @ uminus27091377158695749_ereal @ S ) )
               => ( ord_le1083603963089353582_ereal @ X @ Z5 ) )
           => ( ord_le1083603963089353582_ereal @ X3 @ Z5 ) ) )
      = ( ! [X: extended_ereal] :
            ( ( member2350847679896131959_ereal @ X @ S )
           => ( ord_le1083603963089353582_ereal @ ( uminus27091377158695749_ereal @ X3 ) @ X ) )
        & ! [Z5: extended_ereal] :
            ( ! [X: extended_ereal] :
                ( ( member2350847679896131959_ereal @ X @ S )
               => ( ord_le1083603963089353582_ereal @ Z5 @ X ) )
           => ( ord_le1083603963089353582_ereal @ Z5 @ ( uminus27091377158695749_ereal @ X3 ) ) ) ) ) ).

% ereal_complete_uminus_eq
thf(fact_1248_less__eq__ereal__def,axiom,
    ( ord_le1083603963089353582_ereal
    = ( ^ [X: extended_ereal,Y5: extended_ereal] :
          ( ( ord_le1188267648640031866_ereal @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% less_eq_ereal_def
thf(fact_1249_ereal__complete__Sup,axiom,
    ! [S: set_Extended_ereal] :
    ? [X2: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member2350847679896131959_ereal @ Xa @ S )
         => ( ord_le1083603963089353582_ereal @ Xa @ X2 ) )
      & ! [Z6: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ Xa2 @ S )
             => ( ord_le1083603963089353582_ereal @ Xa2 @ Z6 ) )
         => ( ord_le1083603963089353582_ereal @ X2 @ Z6 ) ) ) ).

% ereal_complete_Sup
thf(fact_1250_ereal__complete__Inf,axiom,
    ! [S: set_Extended_ereal] :
    ? [X2: extended_ereal] :
      ( ! [Xa: extended_ereal] :
          ( ( member2350847679896131959_ereal @ Xa @ S )
         => ( ord_le1083603963089353582_ereal @ X2 @ Xa ) )
      & ! [Z6: extended_ereal] :
          ( ! [Xa2: extended_ereal] :
              ( ( member2350847679896131959_ereal @ Xa2 @ S )
             => ( ord_le1083603963089353582_ereal @ Z6 @ Xa2 ) )
         => ( ord_le1083603963089353582_ereal @ Z6 @ X2 ) ) ) ).

% ereal_complete_Inf
thf(fact_1251_Suc__le__mono,axiom,
    ! [N3: nat,M3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M3 ) )
      = ( ord_less_eq_nat @ N3 @ M3 ) ) ).

% Suc_le_mono
thf(fact_1252_transitive__stepwise__le,axiom,
    ! [M3: nat,N3: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ! [X2: nat] : ( R @ X2 @ X2 )
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z2 )
               => ( R @ X2 @ Z2 ) ) )
         => ( ! [N5: nat] : ( R @ N5 @ ( suc @ N5 ) )
           => ( R @ M3 @ N3 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1253_nat__induct__at__least,axiom,
    ! [M3: nat,N3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ( P @ M3 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ M3 @ N5 )
             => ( ( P @ N5 )
               => ( P @ ( suc @ N5 ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct_at_least
thf(fact_1254_full__nat__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N5: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N5 )
             => ( P @ M5 ) )
         => ( P @ N5 ) )
     => ( P @ N3 ) ) ).

% full_nat_induct
thf(fact_1255_not__less__eq__eq,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ~ ( ord_less_eq_nat @ M3 @ N3 ) )
      = ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 ) ) ).

% not_less_eq_eq
thf(fact_1256_Suc__n__not__le__n,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).

% Suc_n_not_le_n
thf(fact_1257_le__Suc__eq,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
      = ( ( ord_less_eq_nat @ M3 @ N3 )
        | ( M3
          = ( suc @ N3 ) ) ) ) ).

% le_Suc_eq
thf(fact_1258_Suc__le__D,axiom,
    ! [N3: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ M6 )
     => ? [M: nat] :
          ( M6
          = ( suc @ M ) ) ) ).

% Suc_le_D
thf(fact_1259_le__SucI,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) ) ) ).

% le_SucI
thf(fact_1260_le__SucE,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
     => ( ~ ( ord_less_eq_nat @ M3 @ N3 )
       => ( M3
          = ( suc @ N3 ) ) ) ) ).

% le_SucE
thf(fact_1261_Suc__leD,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
     => ( ord_less_eq_nat @ M3 @ N3 ) ) ).

% Suc_leD
thf(fact_1262_Suc__leI,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_nat @ M3 @ N3 )
     => ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 ) ) ).

% Suc_leI
thf(fact_1263_Suc__le__eq,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
      = ( ord_less_nat @ M3 @ N3 ) ) ).

% Suc_le_eq
thf(fact_1264_dec__induct,axiom,
    ! [I4: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( P @ I4 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I4 @ N5 )
             => ( ( ord_less_nat @ N5 @ J2 )
               => ( ( P @ N5 )
                 => ( P @ ( suc @ N5 ) ) ) ) )
         => ( P @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1265_inc__induct,axiom,
    ! [I4: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I4 @ J2 )
     => ( ( P @ J2 )
       => ( ! [N5: nat] :
              ( ( ord_less_eq_nat @ I4 @ N5 )
             => ( ( ord_less_nat @ N5 @ J2 )
               => ( ( P @ ( suc @ N5 ) )
                 => ( P @ N5 ) ) ) )
         => ( P @ I4 ) ) ) ) ).

% inc_induct
thf(fact_1266_Suc__le__lessD,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
     => ( ord_less_nat @ M3 @ N3 ) ) ).

% Suc_le_lessD
thf(fact_1267_le__less__Suc__eq,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
        = ( N3 = M3 ) ) ) ).

% le_less_Suc_eq
thf(fact_1268_less__Suc__eq__le,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
      = ( ord_less_eq_nat @ M3 @ N3 ) ) ).

% less_Suc_eq_le
thf(fact_1269_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1270_le__imp__less__Suc,axiom,
    ! [M3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M3 @ N3 )
     => ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).

% le_imp_less_Suc
thf(fact_1271_mono__Suc,axiom,
    monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).

% mono_Suc
thf(fact_1272_ennreal__Inf__countable__INF,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( A != bot_bo4854962954004695426nnreal )
     => ? [F4: nat > extend8495563244428889912nnreal] :
          ( ( monoto2291723841412853873nnreal @ top_top_set_nat @ ord_less_eq_nat
            @ ^ [X: extend8495563244428889912nnreal,Y5: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ Y5 @ X )
            @ F4 )
          & ( ord_le6787938422905777998nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) @ A )
          & ( ( comple7330758040695736817nnreal @ A )
            = ( comple7330758040695736817nnreal @ ( image_8459861568512453903nnreal @ F4 @ top_top_set_nat ) ) ) ) ) ).

% ennreal_Inf_countable_INF
thf(fact_1273_Inf__countable__INF,axiom,
    ! [A: set_Extended_ereal] :
      ( ( A != bot_bo8367695208629047834_ereal )
     => ? [F4: nat > extended_ereal] :
          ( ( monoto8452838292781035605_ereal @ top_top_set_nat @ ord_less_eq_nat
            @ ^ [X: extended_ereal,Y5: extended_ereal] : ( ord_le1083603963089353582_ereal @ Y5 @ X )
            @ F4 )
          & ( ord_le1644982726543182158_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) @ A )
          & ( ( comple3556804143462414037_ereal @ A )
            = ( comple3556804143462414037_ereal @ ( image_4309273772856505399_ereal @ F4 @ top_top_set_nat ) ) ) ) ) ).

% Inf_countable_INF

% Helper facts (5)
thf(help_If_2_1_If_001t__Extended____Real__Oereal_T,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( if_Extended_ereal @ $false @ X3 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Extended____Real__Oereal_T,axiom,
    ! [X3: extended_ereal,Y4: extended_ereal] :
      ( ( if_Extended_ereal @ $true @ X3 @ Y4 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( if_Ext9135588136721118450nnreal @ $false @ X3 @ Y4 )
      = Y4 ) ).

thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
    ! [X3: extend8495563244428889912nnreal,Y4: extend8495563244428889912nnreal] :
      ( ( if_Ext9135588136721118450nnreal @ $true @ X3 @ Y4 )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( finite3757003017338540048od_b_b
    @ ( product_Sigma_b_b @ i
      @ ^ [Uu: b] : i ) ) ).

%------------------------------------------------------------------------------