TPTP Problem File: SLH0988^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : FSM_Tests/0077_FSM_Code_Datatype/prob_02069_100944__21264622_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1841 ( 756 unt; 564 typ;   0 def)
%            Number of atoms       : 3397 (1556 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11965 ( 313   ~;   2   |; 254   &;10176   @)
%                                         (   0 <=>;1220  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Number of types       :  129 ( 128 usr)
%            Number of type conns  : 2419 (2419   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  439 ( 436 usr;  32 con; 0-4 aty)
%            Number of variables   : 4677 ( 994   ^;3437   !; 246   ?;4677   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 11:31:00.634
%------------------------------------------------------------------------------
% Could-be-implicit typings (128)
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    member4039470153312254544od_c_d: produc2268670659274586663od_c_d > set_Pr137242208728970247od_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J,type,
    member5502040268208861279od_c_a: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J,type,
    member1160383752307813046od_c_a: produc2133359284458116493od_c_a > set_Pr272002107461315949od_c_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J,type,
    member1373485872446851001od_c_d: produc2346461404597154448od_c_d > set_Pr1931820589448984816od_c_d > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__d_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J,type,
    member4231278564905875615od_c_d: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__d_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J_J_J,type,
    member4652649041329350768od_c_d: set_Pr2898135061646675015od_c_d > set_se7324893142979633703od_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J,type,
    member9073573073229046463od_c_a: set_Pr8357624050790843030od_c_a > set_se312849170327590134od_c_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__d_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J_J,type,
    member939977866535389951od_c_d: set_Pr224028844097186518od_c_d > set_se1756390050576932150od_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    member_set_set_o: set_set_o > set_set_set_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_Mt__Product____Type__Oprod_Itf__d_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J_J_J_J,type,
    member1954756648797149776od_c_d: set_se7324893142979633703od_c_d > set_se3083084249005913095od_c_d > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__a_J_J_J_J_J,type,
    member6456832832760271135od_c_a: set_se312849170327590134od_c_a > set_se6970335898041508694od_c_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__d_Mt__Product____Type__Oprod_Itf__b_Mt__Product____Type__Oprod_Itf__c_Mtf__d_J_J_J_J_J,type,
    member7900373713009613151od_c_d: set_se1756390050576932150od_c_d > set_se3236662274807499670od_c_d > $o ).

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

thf(sy_c_member_001tf__d,type,
    member_d: d > set_d > $o ).

thf(sy_v_A____,type,
    a2: fSM_Co5485227252817617415_a_b_c ).

thf(sy_v_B____,type,
    b2: fSM_Co9191560239686800906_d_b_c ).

% Relevant facts (1276)
thf(fact_0_a1,axiom,
    fSM_Co7818001176552552063_a_b_c @ a2 ).

% a1
thf(fact_1_a2,axiom,
    fSM_Co6159912948407392514_d_b_c @ b2 ).

% a2
thf(fact_2_h__obs__impl_Ocases,axiom,
    ! [X: produc7413521230612682178od_c_d] :
      ~ ! [H: mappin3266157088286559569od_c_a,Q: mappin2657711750964880076od_c_a,X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
          ( X
         != ( produc180570349367543930od_c_d @ H @ ( produc5940215602367092546od_c_d @ Q @ ( produc2733640664908071511od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs_impl.cases
thf(fact_3_h__obs__impl_Ocases,axiom,
    ! [X: produc1071504100252382367od_c_d] :
      ~ ! [H: mappin600351445987705420_c_d_d,Q: d,X2: b,Y: product_prod_c_d] :
          ( X
         != ( produc3409962326170770897od_c_d @ H @ ( produc3045323491883059558od_c_d @ Q @ ( produc3440865926910568064od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs_impl.cases
thf(fact_4_h__obs__impl_Ocases,axiom,
    ! [X: produc3360334827542460560od_c_d] :
      ~ ! [H: mappin2657711750964880076od_c_a,Q: produc2669932201312201782od_c_a,X2: d,Y: produc2346461404597154448od_c_d] :
          ( X
         != ( produc5940215602367092546od_c_d @ H @ ( produc2733640664908071511od_c_d @ Q @ ( produc3045323491883059558od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs_impl.cases
thf(fact_5_UN__I,axiom,
    ! [A: $o,A2: set_o,B: $o,B2: $o > set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( member_o @ B @ ( B2 @ A ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_6_UN__I,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: $o,B2: produc2669932201312201782od_c_a > set_o] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ( member_o @ B @ ( B2 @ A ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_7_UN__I,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: $o,B2: produc1399170498009216118od_c_d > set_o] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member_o @ B @ ( B2 @ A ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_8_UN__I,axiom,
    ! [A: $o,A2: set_o,B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ A @ A2 )
     => ( ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_9_UN__I,axiom,
    ! [A: $o,A2: set_o,B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d] :
      ( ( member_o @ A @ A2 )
     => ( ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_10_UN__I,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_11_UN__I,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: produc1399170498009216118od_c_d,B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_12_UN__I,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: produc2669932201312201782od_c_a,B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_13_UN__I,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: produc1399170498009216118od_c_d,B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_14_UN__I,axiom,
    ! [A: $o,A2: set_o,B: produc2736867335369716327od_c_d,B2: $o > set_Pr2898135061646675015od_c_d] :
      ( ( member_o @ A @ A2 )
     => ( ( member4507666829407384208od_c_d @ B @ ( B2 @ A ) )
       => ( member4507666829407384208od_c_d @ B @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_I
thf(fact_15_UN__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_16_UN__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_17_UN__iff,axiom,
    ! [B: $o,B2: $o > set_o,A2: set_o] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member_o @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_18_UN__iff,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member4231278564905875615od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_19_UN__iff,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member5502040268208861279od_c_a @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_20_UN__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% UN_iff
thf(fact_21_SUP__identity__eq,axiom,
    ! [A2: set_set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_set_o_set_o
          @ ^ [X3: set_o] : X3
          @ A2 ) )
      = ( comple90263536869209701_set_o @ A2 ) ) ).

% SUP_identity_eq
thf(fact_22_SUP__identity__eq,axiom,
    ! [A2: set_se1756390050576932150od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [X3: set_Pr224028844097186518od_c_d] : X3
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ A2 ) ) ).

% SUP_identity_eq
thf(fact_23_SUP__identity__eq,axiom,
    ! [A2: set_se312849170327590134od_c_a] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_3355043664154344069od_c_a
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : X3
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ A2 ) ) ).

% SUP_identity_eq
thf(fact_24_SUP__identity__eq,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : X3
          @ A2 ) )
      = ( comple3966289243721909940od_c_d @ A2 ) ) ).

% SUP_identity_eq
thf(fact_25_SUP__identity__eq,axiom,
    ! [A2: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [X3: $o] : X3
          @ A2 ) )
      = ( complete_Sup_Sup_o @ A2 ) ) ).

% SUP_identity_eq
thf(fact_26_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_27_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_28_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: $o > set_o,A2: set_o,P: $o > $o] :
      ( ( ? [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: $o] :
                ( ( member_o @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_29_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: produc1399170498009216118od_c_d > $o] :
      ( ( ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_30_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_31_UN__ball__bex__simps_I4_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_32_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_33_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_34_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: $o > set_o,A2: set_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: $o] :
                ( ( member_o @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_35_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: produc1399170498009216118od_c_d > $o] :
      ( ( ! [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_36_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: produc2669932201312201782od_c_a > $o] :
      ( ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_37_UN__ball__bex__simps_I2_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_38_bex__UN,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_39_bex__UN,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_40_bex__UN,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_41_bex__UN,axiom,
    ! [B2: $o > set_o,A2: set_o,P: $o > $o] :
      ( ( ? [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: $o] :
                ( ( member_o @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_42_bex__UN,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: produc1399170498009216118od_c_d > $o] :
      ( ( ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_43_bex__UN,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ? [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_44_ball__UN,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_45_ball__UN,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_46_ball__UN,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_47_ball__UN,axiom,
    ! [B2: $o > set_o,A2: set_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: $o] :
                ( ( member_o @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_48_ball__UN,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: produc1399170498009216118od_c_d > $o] :
      ( ( ! [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_49_ball__UN,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: produc2669932201312201782od_c_a > $o] :
      ( ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ! [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_50_SUP__UNION,axiom,
    ! [F: $o > $o,G: $o > set_o,A2: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_51_SUP__UNION,axiom,
    ! [F: $o > set_o,G: $o > set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_52_SUP__UNION,axiom,
    ! [F: produc1399170498009216118od_c_d > $o,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ G @ A2 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_53_SUP__UNION,axiom,
    ! [F: produc2669932201312201782od_c_a > $o,G: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ F @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ G @ A2 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [Y2: $o] : ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_54_SUP__UNION,axiom,
    ! [F: produc1399170498009216118od_c_d > set_o,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ G @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_55_SUP__UNION,axiom,
    ! [F: produc2669932201312201782od_c_a > set_o,G: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ F @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ G @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_56_SUP__UNION,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,G: $o > set_o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_57_SUP__UNION,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,G: $o > set_o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ A2 ) ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_58_SUP__UNION,axiom,
    ! [F: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ G @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_59_SUP__UNION,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,G: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ F @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ G @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ F @ ( G @ Y2 ) ) )
          @ A2 ) ) ) ).

% SUP_UNION
thf(fact_60_prod__cases5,axiom,
    ! [Y3: produc2736867335369716327od_c_d] :
      ~ ! [A3: produc2669932201312201782od_c_a,B3: d,C: b,D: c,E: d] :
          ( Y3
         != ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ ( produc3440865926910568064od_c_d @ C @ ( product_Pair_c_d @ D @ E ) ) ) ) ) ).

% prod_cases5
thf(fact_61_prod__cases5,axiom,
    ! [Y3: produc3360334827542460560od_c_d] :
      ~ ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: b,E: product_prod_c_d] :
          ( Y3
         != ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ ( produc3440865926910568064od_c_d @ D @ E ) ) ) ) ) ).

% prod_cases5
thf(fact_62_prod__cases6,axiom,
    ! [Y3: produc3360334827542460560od_c_d] :
      ~ ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: b,E: c,F2: d] :
          ( Y3
         != ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ ( produc3440865926910568064od_c_d @ D @ ( product_Pair_c_d @ E @ F2 ) ) ) ) ) ) ).

% prod_cases6
thf(fact_63_old_Oprod_Oinject,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d,A4: produc2669932201312201782od_c_a,B4: produc1399170498009216118od_c_d] :
      ( ( ( produc2733640664908071511od_c_d @ A @ B )
        = ( produc2733640664908071511od_c_d @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_64_old_Oprod_Oinject,axiom,
    ! [A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d,A4: mappin2657711750964880076od_c_a,B4: produc2736867335369716327od_c_d] :
      ( ( ( produc5940215602367092546od_c_d @ A @ B )
        = ( produc5940215602367092546od_c_d @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_65_old_Oprod_Oinject,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,A4: d,B4: produc2346461404597154448od_c_d] :
      ( ( ( produc3045323491883059558od_c_d @ A @ B )
        = ( produc3045323491883059558od_c_d @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_66_prod_Oinject,axiom,
    ! [X1: produc2669932201312201782od_c_a,X22: produc1399170498009216118od_c_d,Y1: produc2669932201312201782od_c_a,Y22: produc1399170498009216118od_c_d] :
      ( ( ( produc2733640664908071511od_c_d @ X1 @ X22 )
        = ( produc2733640664908071511od_c_d @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_67_prod_Oinject,axiom,
    ! [X1: mappin2657711750964880076od_c_a,X22: produc2736867335369716327od_c_d,Y1: mappin2657711750964880076od_c_a,Y22: produc2736867335369716327od_c_d] :
      ( ( ( produc5940215602367092546od_c_d @ X1 @ X22 )
        = ( produc5940215602367092546od_c_d @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_68_prod_Oinject,axiom,
    ! [X1: d,X22: produc2346461404597154448od_c_d,Y1: d,Y22: produc2346461404597154448od_c_d] :
      ( ( ( produc3045323491883059558od_c_d @ X1 @ X22 )
        = ( produc3045323491883059558od_c_d @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_69_Union__iff,axiom,
    ! [A2: produc2736867335369716327od_c_d,C2: set_se7324893142979633703od_c_d] :
      ( ( member4507666829407384208od_c_d @ A2 @ ( comple3966289243721909940od_c_d @ C2 ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ C2 )
            & ( member4507666829407384208od_c_d @ A2 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_70_Union__iff,axiom,
    ! [A2: $o,C2: set_set_o] :
      ( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
      = ( ? [X3: set_o] :
            ( ( member_set_o @ X3 @ C2 )
            & ( member_o @ A2 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_71_Union__iff,axiom,
    ! [A2: produc1399170498009216118od_c_d,C2: set_se1756390050576932150od_c_d] :
      ( ( member4231278564905875615od_c_d @ A2 @ ( comple7812139117791525187od_c_d @ C2 ) )
      = ( ? [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ C2 )
            & ( member4231278564905875615od_c_d @ A2 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_72_Union__iff,axiom,
    ! [A2: produc2669932201312201782od_c_a,C2: set_se312849170327590134od_c_a] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( comple6722362287630405891od_c_a @ C2 ) )
      = ( ? [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ C2 )
            & ( member5502040268208861279od_c_a @ A2 @ X3 ) ) ) ) ).

% Union_iff
thf(fact_73_UnionI,axiom,
    ! [X4: set_Pr2898135061646675015od_c_d,C2: set_se7324893142979633703od_c_d,A2: produc2736867335369716327od_c_d] :
      ( ( member4652649041329350768od_c_d @ X4 @ C2 )
     => ( ( member4507666829407384208od_c_d @ A2 @ X4 )
       => ( member4507666829407384208od_c_d @ A2 @ ( comple3966289243721909940od_c_d @ C2 ) ) ) ) ).

% UnionI
thf(fact_74_UnionI,axiom,
    ! [X4: set_o,C2: set_set_o,A2: $o] :
      ( ( member_set_o @ X4 @ C2 )
     => ( ( member_o @ A2 @ X4 )
       => ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).

% UnionI
thf(fact_75_UnionI,axiom,
    ! [X4: set_Pr224028844097186518od_c_d,C2: set_se1756390050576932150od_c_d,A2: produc1399170498009216118od_c_d] :
      ( ( member939977866535389951od_c_d @ X4 @ C2 )
     => ( ( member4231278564905875615od_c_d @ A2 @ X4 )
       => ( member4231278564905875615od_c_d @ A2 @ ( comple7812139117791525187od_c_d @ C2 ) ) ) ) ).

% UnionI
thf(fact_76_UnionI,axiom,
    ! [X4: set_Pr8357624050790843030od_c_a,C2: set_se312849170327590134od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ( member9073573073229046463od_c_a @ X4 @ C2 )
     => ( ( member5502040268208861279od_c_a @ A2 @ X4 )
       => ( member5502040268208861279od_c_a @ A2 @ ( comple6722362287630405891od_c_a @ C2 ) ) ) ) ).

% UnionI
thf(fact_77_UN__ball__bex__simps_I1_J,axiom,
    ! [A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ A2 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ! [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ X3 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_78_UN__ball__bex__simps_I1_J,axiom,
    ! [A2: set_set_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ A2 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
           => ! [Y2: $o] :
                ( ( member_o @ Y2 @ X3 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_79_UN__ball__bex__simps_I1_J,axiom,
    ! [A2: set_se1756390050576932150od_c_d,P: produc1399170498009216118od_c_d > $o] :
      ( ( ! [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ A2 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
           => ! [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ X3 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_80_UN__ball__bex__simps_I1_J,axiom,
    ! [A2: set_se312849170327590134od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ A2 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
           => ! [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ X3 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_81_UN__ball__bex__simps_I3_J,axiom,
    ! [A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( ? [X3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ A2 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ? [Y2: produc2736867335369716327od_c_d] :
                ( ( member4507666829407384208od_c_d @ Y2 @ X3 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_82_UN__ball__bex__simps_I3_J,axiom,
    ! [A2: set_set_o,P: $o > $o] :
      ( ( ? [X3: $o] :
            ( ( member_o @ X3 @ ( comple90263536869209701_set_o @ A2 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
            & ? [Y2: $o] :
                ( ( member_o @ Y2 @ X3 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_83_UN__ball__bex__simps_I3_J,axiom,
    ! [A2: set_se1756390050576932150od_c_d,P: produc1399170498009216118od_c_d > $o] :
      ( ( ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ A2 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
            & ? [Y2: produc1399170498009216118od_c_d] :
                ( ( member4231278564905875615od_c_d @ Y2 @ X3 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_84_UN__ball__bex__simps_I3_J,axiom,
    ! [A2: set_se312849170327590134od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ A2 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
            & ? [Y2: produc2669932201312201782od_c_a] :
                ( ( member5502040268208861279od_c_a @ Y2 @ X3 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_85_Sup__set__def,axiom,
    ( comple3966289243721909940od_c_d
    = ( ^ [A5: set_se7324893142979633703od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [X3: produc2736867335369716327od_c_d] : ( complete_Sup_Sup_o @ ( image_2471105756073848854_c_d_o @ ( member4507666829407384208od_c_d @ X3 ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_86_Sup__set__def,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A5: set_set_o] :
          ( collect_o
          @ ^ [X3: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X3 ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_87_Sup__set__def,axiom,
    ( comple7812139117791525187od_c_d
    = ( ^ [A5: set_se1756390050576932150od_c_d] :
          ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] : ( complete_Sup_Sup_o @ ( image_1321941650430489543_c_d_o @ ( member4231278564905875615od_c_d @ X3 ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_88_Sup__set__def,axiom,
    ( comple6722362287630405891od_c_a
    = ( ^ [A5: set_se312849170327590134od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] : ( complete_Sup_Sup_o @ ( image_3105936341704502279_c_a_o @ ( member5502040268208861279od_c_a @ X3 ) @ A5 ) ) ) ) ) ).

% Sup_set_def
thf(fact_89_Pair__inject,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d,A4: produc2669932201312201782od_c_a,B4: produc1399170498009216118od_c_d] :
      ( ( ( produc2733640664908071511od_c_d @ A @ B )
        = ( produc2733640664908071511od_c_d @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_90_Pair__inject,axiom,
    ! [A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d,A4: mappin2657711750964880076od_c_a,B4: produc2736867335369716327od_c_d] :
      ( ( ( produc5940215602367092546od_c_d @ A @ B )
        = ( produc5940215602367092546od_c_d @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_91_Pair__inject,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,A4: d,B4: produc2346461404597154448od_c_d] :
      ( ( ( produc3045323491883059558od_c_d @ A @ B )
        = ( produc3045323491883059558od_c_d @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_92_prod__cases,axiom,
    ! [P: produc2736867335369716327od_c_d > $o,P2: produc2736867335369716327od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] : ( P @ ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_93_prod__cases,axiom,
    ! [P: produc3360334827542460560od_c_d > $o,P2: produc3360334827542460560od_c_d] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2736867335369716327od_c_d] : ( P @ ( produc5940215602367092546od_c_d @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_94_prod__cases,axiom,
    ! [P: produc1399170498009216118od_c_d > $o,P2: produc1399170498009216118od_c_d] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] : ( P @ ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_95_surj__pair,axiom,
    ! [P2: produc2736867335369716327od_c_d] :
    ? [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
      ( P2
      = ( produc2733640664908071511od_c_d @ X2 @ Y ) ) ).

% surj_pair
thf(fact_96_surj__pair,axiom,
    ! [P2: produc3360334827542460560od_c_d] :
    ? [X2: mappin2657711750964880076od_c_a,Y: produc2736867335369716327od_c_d] :
      ( P2
      = ( produc5940215602367092546od_c_d @ X2 @ Y ) ) ).

% surj_pair
thf(fact_97_surj__pair,axiom,
    ! [P2: produc1399170498009216118od_c_d] :
    ? [X2: d,Y: produc2346461404597154448od_c_d] :
      ( P2
      = ( produc3045323491883059558od_c_d @ X2 @ Y ) ) ).

% surj_pair
thf(fact_98_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2736867335369716327od_c_d] :
      ~ ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( Y3
         != ( produc2733640664908071511od_c_d @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_99_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3360334827542460560od_c_d] :
      ~ ! [A3: mappin2657711750964880076od_c_a,B3: produc2736867335369716327od_c_d] :
          ( Y3
         != ( produc5940215602367092546od_c_d @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_100_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1399170498009216118od_c_d] :
      ~ ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( Y3
         != ( produc3045323491883059558od_c_d @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_101_Sup_OSUP__cong,axiom,
    ! [A2: set_se7324893142979633703od_c_d,B2: set_se7324893142979633703od_c_d,C2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,D2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,Sup: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_242321854666033093od_c_d @ C2 @ A2 ) )
          = ( Sup @ ( image_242321854666033093od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_102_Sup_OSUP__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,D2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,Sup: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_8721290246026763926od_c_d @ C2 @ A2 ) )
          = ( Sup @ ( image_8721290246026763926od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_103_Sup_OSUP__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,D2: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,Sup: set_Pr2898135061646675015od_c_d > produc2736867335369716327od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_3436769108421412982od_c_d @ C2 @ A2 ) )
          = ( Sup @ ( image_3436769108421412982od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_104_Sup_OSUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr2898135061646675015od_c_d,D2: $o > set_Pr2898135061646675015od_c_d,Sup: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_8975626525747676852od_c_d @ C2 @ A2 ) )
          = ( Sup @ ( image_8975626525747676852od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_105_Sup_OSUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr224028844097186518od_c_d,D2: $o > set_Pr224028844097186518od_c_d,Sup: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_728402167444666691od_c_d @ C2 @ A2 ) )
          = ( Sup @ ( image_728402167444666691od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_106_Sup_OSUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr8357624050790843030od_c_a,D2: $o > set_Pr8357624050790843030od_c_a,Sup: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_8861997374138323203od_c_a @ C2 @ A2 ) )
          = ( Sup @ ( image_8861997374138323203od_c_a @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_107_Sup_OSUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_o,D2: $o > set_o,Sup: set_set_o > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_o_set_o @ C2 @ A2 ) )
          = ( Sup @ ( image_o_set_o @ D2 @ B2 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_108_Inf_OINF__cong,axiom,
    ! [A2: set_se7324893142979633703od_c_d,B2: set_se7324893142979633703od_c_d,C2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,D2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,Inf: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_242321854666033093od_c_d @ C2 @ A2 ) )
          = ( Inf @ ( image_242321854666033093od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_109_Inf_OINF__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,D2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,Inf: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_8721290246026763926od_c_d @ C2 @ A2 ) )
          = ( Inf @ ( image_8721290246026763926od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_110_Inf_OINF__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,D2: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,Inf: set_Pr2898135061646675015od_c_d > produc2736867335369716327od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_3436769108421412982od_c_d @ C2 @ A2 ) )
          = ( Inf @ ( image_3436769108421412982od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_111_Inf_OINF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr2898135061646675015od_c_d,D2: $o > set_Pr2898135061646675015od_c_d,Inf: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_8975626525747676852od_c_d @ C2 @ A2 ) )
          = ( Inf @ ( image_8975626525747676852od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_112_Inf_OINF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr224028844097186518od_c_d,D2: $o > set_Pr224028844097186518od_c_d,Inf: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_728402167444666691od_c_d @ C2 @ A2 ) )
          = ( Inf @ ( image_728402167444666691od_c_d @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_113_Inf_OINF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr8357624050790843030od_c_a,D2: $o > set_Pr8357624050790843030od_c_a,Inf: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_8861997374138323203od_c_a @ C2 @ A2 ) )
          = ( Inf @ ( image_8861997374138323203od_c_a @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_114_Inf_OINF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_o,D2: $o > set_o,Inf: set_set_o > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_o_set_o @ C2 @ A2 ) )
          = ( Inf @ ( image_o_set_o @ D2 @ B2 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_115_UnionE,axiom,
    ! [A2: produc2736867335369716327od_c_d,C2: set_se7324893142979633703od_c_d] :
      ( ( member4507666829407384208od_c_d @ A2 @ ( comple3966289243721909940od_c_d @ C2 ) )
     => ~ ! [X5: set_Pr2898135061646675015od_c_d] :
            ( ( member4507666829407384208od_c_d @ A2 @ X5 )
           => ~ ( member4652649041329350768od_c_d @ X5 @ C2 ) ) ) ).

% UnionE
thf(fact_116_UnionE,axiom,
    ! [A2: $o,C2: set_set_o] :
      ( ( member_o @ A2 @ ( comple90263536869209701_set_o @ C2 ) )
     => ~ ! [X5: set_o] :
            ( ( member_o @ A2 @ X5 )
           => ~ ( member_set_o @ X5 @ C2 ) ) ) ).

% UnionE
thf(fact_117_UnionE,axiom,
    ! [A2: produc1399170498009216118od_c_d,C2: set_se1756390050576932150od_c_d] :
      ( ( member4231278564905875615od_c_d @ A2 @ ( comple7812139117791525187od_c_d @ C2 ) )
     => ~ ! [X5: set_Pr224028844097186518od_c_d] :
            ( ( member4231278564905875615od_c_d @ A2 @ X5 )
           => ~ ( member939977866535389951od_c_d @ X5 @ C2 ) ) ) ).

% UnionE
thf(fact_118_UnionE,axiom,
    ! [A2: produc2669932201312201782od_c_a,C2: set_se312849170327590134od_c_a] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( comple6722362287630405891od_c_a @ C2 ) )
     => ~ ! [X5: set_Pr8357624050790843030od_c_a] :
            ( ( member5502040268208861279od_c_a @ A2 @ X5 )
           => ~ ( member9073573073229046463od_c_a @ X5 @ C2 ) ) ) ).

% UnionE
thf(fact_119_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( Sup
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : X3
          @ A2 ) )
      = ( Sup @ A2 ) ) ).

% Sup.SUP_identity_eq
thf(fact_120_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( Inf
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : X3
          @ A2 ) )
      = ( Inf @ A2 ) ) ).

% Inf.INF_identity_eq
thf(fact_121_prod__induct3,axiom,
    ! [P: produc2736867335369716327od_c_d > $o,X: produc2736867335369716327od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: d,C: produc2346461404597154448od_c_d] : ( P @ ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_122_prod__induct3,axiom,
    ! [P: produc3360334827542460560od_c_d > $o,X: produc3360334827542460560od_c_d] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: produc1399170498009216118od_c_d] : ( P @ ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_123_prod__induct3,axiom,
    ! [P: produc1399170498009216118od_c_d > $o,X: produc1399170498009216118od_c_d] :
      ( ! [A3: d,B3: b,C: product_prod_c_d] : ( P @ ( produc3045323491883059558od_c_d @ A3 @ ( produc3440865926910568064od_c_d @ B3 @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_124_prod__cases3,axiom,
    ! [Y3: produc2736867335369716327od_c_d] :
      ~ ! [A3: produc2669932201312201782od_c_a,B3: d,C: produc2346461404597154448od_c_d] :
          ( Y3
         != ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_125_prod__cases3,axiom,
    ! [Y3: produc3360334827542460560od_c_d] :
      ~ ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: produc1399170498009216118od_c_d] :
          ( Y3
         != ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_126_prod__cases3,axiom,
    ! [Y3: produc1399170498009216118od_c_d] :
      ~ ! [A3: d,B3: b,C: product_prod_c_d] :
          ( Y3
         != ( produc3045323491883059558od_c_d @ A3 @ ( produc3440865926910568064od_c_d @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_127_SUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > $o,D2: $o > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ C2 @ A2 ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_128_SUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_o,D2: $o > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C2 @ A2 ) )
          = ( comple90263536869209701_set_o @ ( image_o_set_o @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_129_SUP__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > $o,D2: produc2669932201312201782od_c_a > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ C2 @ A2 ) )
          = ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_130_SUP__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > $o,D2: produc1399170498009216118od_c_d > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ C2 @ A2 ) )
          = ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_131_SUP__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > set_o,D2: produc2669932201312201782od_c_a > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ C2 @ A2 ) )
          = ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_132_SUP__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > set_o,D2: produc1399170498009216118od_c_d > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ C2 @ A2 ) )
          = ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_133_SUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr224028844097186518od_c_d,D2: $o > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ C2 @ A2 ) )
          = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_134_SUP__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr8357624050790843030od_c_a,D2: $o > set_Pr8357624050790843030od_c_a] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ A2 ) )
          = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_135_SUP__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,D2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ A2 ) )
          = ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_136_SUP__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,D2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ A2 ) )
          = ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ D2 @ B2 ) ) ) ) ) ).

% SUP_cong
thf(fact_137_SUP__commute,axiom,
    ! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [I: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [J: $o] :
              ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_138_SUP__commute,axiom,
    ! [F: $o > $o > set_Pr224028844097186518od_c_d,B2: set_o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [I: $o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [J: $o] :
              ( comple7812139117791525187od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_139_SUP__commute,axiom,
    ! [F: $o > $o > set_Pr8357624050790843030od_c_a,B2: set_o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [I: $o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [J: $o] :
              ( comple6722362287630405891od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_140_SUP__commute,axiom,
    ! [F: $o > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple3966289243721909940od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_141_SUP__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple3966289243721909940od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_142_SUP__commute,axiom,
    ! [F: $o > produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,B2: set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [J: produc2669932201312201782od_c_a] :
              ( comple3966289243721909940od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_143_SUP__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,B2: set_Pr8357624050790843030od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [J: produc2669932201312201782od_c_a] :
              ( comple3966289243721909940od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_144_SUP__commute,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,B2: set_se7324893142979633703od_c_d,A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [J: set_Pr2898135061646675015od_c_d] :
              ( comple3966289243721909940od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_145_SUP__commute,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_se7324893142979633703od_c_d] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [I: set_Pr2898135061646675015od_c_d] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple3966289243721909940od_c_d
              @ ( image_242321854666033093od_c_d
                @ ^ [I: set_Pr2898135061646675015od_c_d] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_146_SUP__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,B2: set_se7324893142979633703od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [J: set_Pr2898135061646675015od_c_d] :
              ( comple3966289243721909940od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% SUP_commute
thf(fact_147_image__Union,axiom,
    ! [F: $o > $o,S: set_set_o] :
      ( ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple90263536869209701_set_o @ ( image_set_o_set_o @ ( image_o_o2 @ F ) @ S ) ) ) ).

% image_Union
thf(fact_148_image__Union,axiom,
    ! [F: $o > set_o,S: set_set_o] :
      ( ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple4436988014476444997_set_o @ ( image_5023573440332574309_set_o @ ( image_o_set_o @ F ) @ S ) ) ) ).

% image_Union
thf(fact_149_image__Union,axiom,
    ! [F: $o > produc1399170498009216118od_c_d,S: set_set_o] :
      ( ( image_3657811533049837283od_c_d @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple7812139117791525187od_c_d @ ( image_564716939699073635od_c_d @ ( image_3657811533049837283od_c_d @ F ) @ S ) ) ) ).

% image_Union
thf(fact_150_image__Union,axiom,
    ! [F: $o > produc2669932201312201782od_c_a,S: set_set_o] :
      ( ( image_4928573236352822947od_c_a @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple6722362287630405891od_c_a @ ( image_8698312146392730147od_c_a @ ( image_4928573236352822947od_c_a @ F ) @ S ) ) ) ).

% image_Union
thf(fact_151_image__Union,axiom,
    ! [F: produc1399170498009216118od_c_d > $o,S: set_se1756390050576932150od_c_d] :
      ( ( image_3816251319797307943_c_d_o @ F @ ( comple7812139117791525187od_c_d @ S ) )
      = ( comple90263536869209701_set_o @ ( image_2583072847733522087_set_o @ ( image_3816251319797307943_c_d_o @ F ) @ S ) ) ) ).

% image_Union
thf(fact_152_image__Union,axiom,
    ! [F: produc2669932201312201782od_c_a > $o,S: set_se312849170327590134od_c_a] :
      ( ( image_2427823168983691879_c_a_o @ F @ ( comple6722362287630405891od_c_a @ S ) )
      = ( comple90263536869209701_set_o @ ( image_1108906476932298471_set_o @ ( image_2427823168983691879_c_a_o @ F ) @ S ) ) ) ).

% image_Union
thf(fact_153_image__Union,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,S: set_set_o] :
      ( ( image_728402167444666691od_c_d @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple606099721242408867od_c_d @ ( image_9043331034324431043od_c_d @ ( image_728402167444666691od_c_d @ F ) @ S ) ) ) ).

% image_Union
thf(fact_154_image__Union,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,S: set_set_o] :
      ( ( image_8861997374138323203od_c_a @ F @ ( comple90263536869209701_set_o @ S ) )
      = ( comple8385930877847842659od_c_a @ ( image_7599790154075089027od_c_a @ ( image_8861997374138323203od_c_a @ F ) @ S ) ) ) ).

% image_Union
thf(fact_155_image__Union,axiom,
    ! [F: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d,S: set_se1756390050576932150od_c_d] :
      ( ( image_430859899312465541od_c_d @ F @ ( comple7812139117791525187od_c_d @ S ) )
      = ( comple7812139117791525187od_c_d @ ( image_563476247564946053od_c_d @ ( image_430859899312465541od_c_d @ F ) @ S ) ) ) ).

% image_Union
thf(fact_156_image__Union,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2669932201312201782od_c_a,S: set_se1756390050576932150od_c_d] :
      ( ( image_1701621602615451205od_c_a @ F @ ( comple7812139117791525187od_c_d @ S ) )
      = ( comple6722362287630405891od_c_a @ ( image_8697071454258602565od_c_a @ ( image_1701621602615451205od_c_a @ F ) @ S ) ) ) ).

% image_Union
thf(fact_157_UN__UN__flatten,axiom,
    ! [C2: $o > set_o,B2: $o > set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_158_UN__UN__flatten,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_o,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_159_UN__UN__flatten,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_o,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_160_UN__UN__flatten,axiom,
    ! [C2: $o > set_Pr224028844097186518od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_161_UN__UN__flatten,axiom,
    ! [C2: $o > set_Pr8357624050790843030od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_162_UN__UN__flatten,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_163_UN__UN__flatten,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_164_UN__UN__flatten,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] : ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_165_UN__UN__flatten,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] : ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_166_UN__UN__flatten,axiom,
    ! [C2: $o > set_Pr2898135061646675015od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [Y2: $o] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ C2 @ ( B2 @ Y2 ) ) )
          @ A2 ) ) ) ).

% UN_UN_flatten
thf(fact_167_UN__E,axiom,
    ! [B: $o,B2: $o > set_o,A2: set_o] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_168_UN__E,axiom,
    ! [B: $o,B2: produc2669932201312201782od_c_a > set_o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ B2 @ A2 ) ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_169_UN__E,axiom,
    ! [B: $o,B2: produc1399170498009216118od_c_d > set_o,A2: set_Pr224028844097186518od_c_d] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ B2 @ A2 ) ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ~ ( member_o @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_170_UN__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_171_UN__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_172_UN__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ A2 ) ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_173_UN__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ A2 ) ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_174_UN__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ A2 ) ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_175_UN__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,A2: set_Pr224028844097186518od_c_d] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ A2 ) ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_176_UN__E,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
     => ~ ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ~ ( member4507666829407384208od_c_d @ B @ ( B2 @ X2 ) ) ) ) ).

% UN_E
thf(fact_177_UN__extend__simps_I8_J,axiom,
    ! [B2: $o > set_o,A2: set_set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_set_o_set_o
          @ ^ [Y2: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_178_UN__extend__simps_I8_J,axiom,
    ! [B2: produc1399170498009216118od_c_d > set_o,A2: set_se1756390050576932150od_c_d] :
      ( ( comple90263536869209701_set_o
        @ ( image_2583072847733522087_set_o
          @ ^ [Y2: set_Pr224028844097186518od_c_d] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ B2 @ ( comple7812139117791525187od_c_d @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_179_UN__extend__simps_I8_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_o,A2: set_se312849170327590134od_c_a] :
      ( ( comple90263536869209701_set_o
        @ ( image_1108906476932298471_set_o
          @ ^ [Y2: set_Pr8357624050790843030od_c_a] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ B2 @ ( comple6722362287630405891od_c_a @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_180_UN__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [Y2: set_o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_181_UN__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [Y2: set_o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_182_UN__extend__simps_I8_J,axiom,
    ! [B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_se1756390050576932150od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [Y2: set_Pr224028844097186518od_c_d] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ ( comple7812139117791525187od_c_d @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_183_UN__extend__simps_I8_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_se312849170327590134od_c_a] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_4444820494315463365od_c_d
          @ ^ [Y2: set_Pr8357624050790843030od_c_a] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ ( comple6722362287630405891od_c_a @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_184_UN__extend__simps_I8_J,axiom,
    ! [B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,A2: set_se1756390050576932150od_c_d] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8697071454258602565od_c_a
          @ ^ [Y2: set_Pr224028844097186518od_c_d] : ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ ( comple7812139117791525187od_c_d @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_185_UN__extend__simps_I8_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,A2: set_se312849170327590134od_c_a] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_3355043664154344069od_c_a
          @ ^ [Y2: set_Pr8357624050790843030od_c_a] : ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ ( comple6722362287630405891od_c_a @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_186_UN__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_5784685344308445140od_c_d
          @ ^ [Y2: set_o] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_187_UN__extend__simps_I9_J,axiom,
    ! [C2: $o > set_o,B2: $o > set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_o_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_188_UN__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_o,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_189_UN__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_o,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_190_UN__extend__simps_I9_J,axiom,
    ! [C2: $o > set_Pr224028844097186518od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_191_UN__extend__simps_I9_J,axiom,
    ! [C2: $o > set_Pr8357624050790843030od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_192_UN__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_193_UN__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_194_UN__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_195_UN__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_196_UN__extend__simps_I9_J,axiom,
    ! [C2: $o > set_Pr2898135061646675015od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [X3: $o] : ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% UN_extend_simps(9)
thf(fact_197_mem__Collect__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ A @ ( collec8405073525749625505od_c_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_198_mem__Collect__eq,axiom,
    ! [A: produc1399170498009216118od_c_d,P: produc1399170498009216118od_c_d > $o] :
      ( ( member4231278564905875615od_c_d @ A @ ( collec7134311822446639841od_c_d @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_199_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_200_mem__Collect__eq,axiom,
    ! [A: produc2736867335369716327od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( member4507666829407384208od_c_d @ A @ ( collec2789978228259155154od_c_d @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_201_Collect__mem__eq,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
    ! [A2: set_Pr224028844097186518od_c_d] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
    ! [A2: set_o] :
      ( ( collect_o
        @ ^ [X3: $o] : ( member_o @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_205_Collect__cong,axiom,
    ! [P: produc2736867335369716327od_c_d > $o,Q2: produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( P @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec2789978228259155154od_c_d @ P )
        = ( collec2789978228259155154od_c_d @ Q2 ) ) ) ).

% Collect_cong
thf(fact_206_prod__induct4,axiom,
    ! [P: produc2736867335369716327od_c_d > $o,X: produc2736867335369716327od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: d,C: b,D: product_prod_c_d] : ( P @ ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ ( produc3440865926910568064od_c_d @ C @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_207_prod__induct4,axiom,
    ! [P: produc3360334827542460560od_c_d > $o,X: produc3360334827542460560od_c_d] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: produc2346461404597154448od_c_d] : ( P @ ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_208_prod__induct4,axiom,
    ! [P: produc1399170498009216118od_c_d > $o,X: produc1399170498009216118od_c_d] :
      ( ! [A3: d,B3: b,C: c,D: d] : ( P @ ( produc3045323491883059558od_c_d @ A3 @ ( produc3440865926910568064od_c_d @ B3 @ ( product_Pair_c_d @ C @ D ) ) ) )
     => ( P @ X ) ) ).

% prod_induct4
thf(fact_209_prod__cases4,axiom,
    ! [Y3: produc2736867335369716327od_c_d] :
      ~ ! [A3: produc2669932201312201782od_c_a,B3: d,C: b,D: product_prod_c_d] :
          ( Y3
         != ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ ( produc3440865926910568064od_c_d @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_210_prod__cases4,axiom,
    ! [Y3: produc3360334827542460560od_c_d] :
      ~ ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: produc2346461404597154448od_c_d] :
          ( Y3
         != ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_211_prod__cases4,axiom,
    ! [Y3: produc1399170498009216118od_c_d] :
      ~ ! [A3: d,B3: b,C: c,D: d] :
          ( Y3
         != ( produc3045323491883059558od_c_d @ A3 @ ( produc3440865926910568064od_c_d @ B3 @ ( product_Pair_c_d @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_212_image__UN,axiom,
    ! [F: $o > $o,B2: $o > set_o,A2: set_o] :
      ( ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_o_o2 @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_213_image__UN,axiom,
    ! [F: $o > set_o,B2: $o > set_o,A2: set_o] :
      ( ( image_o_set_o @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple4436988014476444997_set_o
        @ ( image_o_set_set_o
          @ ^ [X3: $o] : ( image_o_set_o @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_214_image__UN,axiom,
    ! [F: $o > produc1399170498009216118od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( image_3657811533049837283od_c_d @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( image_3657811533049837283od_c_d @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_215_image__UN,axiom,
    ! [F: $o > produc2669932201312201782od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( image_4928573236352822947od_c_a @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( image_4928573236352822947od_c_a @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_216_image__UN,axiom,
    ! [F: produc1399170498009216118od_c_d > $o,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_3816251319797307943_c_d_o @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_3816251319797307943_c_d_o @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_217_image__UN,axiom,
    ! [F: produc2669932201312201782od_c_a > $o,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( image_2427823168983691879_c_a_o @ F @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_2427823168983691879_c_a_o @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_218_image__UN,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( image_728402167444666691od_c_d @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple606099721242408867od_c_d
        @ ( image_4184728393606117283od_c_d
          @ ^ [X3: $o] : ( image_728402167444666691od_c_d @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_219_image__UN,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( image_8861997374138323203od_c_a @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple8385930877847842659od_c_a
        @ ( image_2741187513356775267od_c_a
          @ ^ [X3: $o] : ( image_8861997374138323203od_c_a @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_220_image__UN,axiom,
    ! [F: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_430859899312465541od_c_d @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( image_430859899312465541od_c_d @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_221_image__UN,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2669932201312201782od_c_a,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_1701621602615451205od_c_a @ F @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( image_1701621602615451205od_c_a @ F @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% image_UN
thf(fact_222_UN__extend__simps_I10_J,axiom,
    ! [B2: $o > set_o,F: $o > $o,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_223_UN__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_o,F: $o > set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_set_o_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_224_UN__extend__simps_I10_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,F: $o > $o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_225_UN__extend__simps_I10_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,F: $o > $o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_226_UN__extend__simps_I10_J,axiom,
    ! [B2: set_Pr224028844097186518od_c_d > set_o,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_2583072847733522087_set_o @ B2 @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_227_UN__extend__simps_I10_J,axiom,
    ! [B2: set_Pr8357624050790843030od_c_a > set_o,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o @ ( image_1108906476932298471_set_o @ B2 @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_228_UN__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_Pr224028844097186518od_c_d,F: $o > set_o,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_564716939699073635od_c_d @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_229_UN__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_Pr8357624050790843030od_c_a,F: $o > set_o,A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a @ ( image_8698312146392730147od_c_a @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_230_UN__extend__simps_I10_J,axiom,
    ! [B2: set_Pr224028844097186518od_c_d > set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_563476247564946053od_c_d @ B2 @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_231_UN__extend__simps_I10_J,axiom,
    ! [B2: set_Pr8357624050790843030od_c_a > set_Pr224028844097186518od_c_d,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d @ ( image_4444820494315463365od_c_d @ B2 @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_232_prod__induct6,axiom,
    ! [P: produc3360334827542460560od_c_d > $o,X: produc3360334827542460560od_c_d] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: b,E: c,F2: d] : ( P @ ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ ( produc3440865926910568064od_c_d @ D @ ( product_Pair_c_d @ E @ F2 ) ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct6
thf(fact_233_prod__induct5,axiom,
    ! [P: produc2736867335369716327od_c_d > $o,X: produc2736867335369716327od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: d,C: b,D: c,E: d] : ( P @ ( produc2733640664908071511od_c_d @ A3 @ ( produc3045323491883059558od_c_d @ B3 @ ( produc3440865926910568064od_c_d @ C @ ( product_Pair_c_d @ D @ E ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct5
thf(fact_234_prod__induct5,axiom,
    ! [P: produc3360334827542460560od_c_d > $o,X: produc3360334827542460560od_c_d] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2669932201312201782od_c_a,C: d,D: b,E: product_prod_c_d] : ( P @ ( produc5940215602367092546od_c_d @ A3 @ ( produc2733640664908071511od_c_d @ B3 @ ( produc3045323491883059558od_c_d @ C @ ( produc3440865926910568064od_c_d @ D @ E ) ) ) ) )
     => ( P @ X ) ) ).

% prod_induct5
thf(fact_235_SUP__UN__eq2,axiom,
    ! [R: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,S: set_Pr8357624050790843030od_c_a] :
      ( ( comple2351186111055541915_c_d_o
        @ ( image_393729682460826237_c_d_o
          @ ^ [I: produc2669932201312201782od_c_a,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq2
thf(fact_236_SUP__UN__eq2,axiom,
    ! [R: $o > set_Pr2898135061646675015od_c_d,S: set_o] :
      ( ( comple2351186111055541915_c_d_o
        @ ( image_4475099463610824347_c_d_o
          @ ^ [I: $o,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq2
thf(fact_237_SUP__UN__eq2,axiom,
    ! [R: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,S: set_se7324893142979633703od_c_d] :
      ( ( comple2351186111055541915_c_d_o
        @ ( image_5533172864875290028_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq2
thf(fact_238_SUP__UN__eq2,axiom,
    ! [R: $o > set_Pr224028844097186518od_c_d,S: set_o] :
      ( ( comple4510354444638677324_c_d_o
        @ ( image_8048435621306247500_c_d_o
          @ ^ [I: $o,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq2
thf(fact_239_SUP__UN__eq2,axiom,
    ! [R: $o > set_Pr8357624050790843030od_c_a,S: set_o] :
      ( ( comple1107991413401925132_c_a_o
        @ ( image_4646072590069495308_c_a_o
          @ ^ [I: $o,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ R @ S ) ) ) ) ) ).

% SUP_UN_eq2
thf(fact_240_image__ident,axiom,
    ! [Y4: set_se7324893142979633703od_c_d] :
      ( ( image_242321854666033093od_c_d
        @ ^ [X3: set_Pr2898135061646675015od_c_d] : X3
        @ Y4 )
      = Y4 ) ).

% image_ident
thf(fact_241_SUP__Sup__eq2,axiom,
    ! [S: set_se7355440763634802940od_c_d] :
      ( ( comple916854894399990804_c_d_o
        @ ( image_4928181473391346958_c_d_o
          @ ^ [I: set_Pr5052118491147477062od_c_d,X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ ( comple5928930310332571033od_c_d @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_242_SUP__Sup__eq2,axiom,
    ! [S: set_se7324893142979633703od_c_d] :
      ( ( comple2351186111055541915_c_d_o
        @ ( image_5533172864875290028_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple3966289243721909940od_c_d @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_243_SUP__Sup__eq2,axiom,
    ! [S: set_se1756390050576932150od_c_d] :
      ( ( comple4510354444638677324_c_d_o
        @ ( image_1218452421501306510_c_d_o
          @ ^ [I: set_Pr224028844097186518od_c_d,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple7812139117791525187od_c_d @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_244_SUP__Sup__eq2,axiom,
    ! [S: set_se312849170327590134od_c_a] :
      ( ( comple1107991413401925132_c_a_o
        @ ( image_34615054399089550_c_a_o
          @ ^ [I: set_Pr8357624050790843030od_c_a,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple6722362287630405891od_c_a @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_245_image__eqI,axiom,
    ! [B: $o,F: $o > $o,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member_o @ B @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_246_image__eqI,axiom,
    ! [B: set_o,F: $o > set_o,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_247_image__eqI,axiom,
    ! [B: $o,F: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A2 )
       => ( member_o @ B @ ( image_2427823168983691879_c_a_o @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_248_image__eqI,axiom,
    ! [B: $o,F: produc1399170498009216118od_c_d > $o,X: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( B
        = ( F @ X ) )
     => ( ( member4231278564905875615od_c_d @ X @ A2 )
       => ( member_o @ B @ ( image_3816251319797307943_c_d_o @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_249_image__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: $o > produc2669932201312201782od_c_a,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( image_4928573236352822947od_c_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_250_image__eqI,axiom,
    ! [B: produc1399170498009216118od_c_d,F: $o > produc1399170498009216118od_c_d,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( image_3657811533049837283od_c_d @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_251_image__eqI,axiom,
    ! [B: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member939977866535389951od_c_d @ B @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_252_image__eqI,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a,X: $o,A2: set_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_o @ X @ A2 )
       => ( member9073573073229046463od_c_a @ B @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_253_image__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_254_image__eqI,axiom,
    ! [B: produc1399170498009216118od_c_d,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d,X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5502040268208861279od_c_a @ X @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( image_899056575407595205od_c_d @ F @ A2 ) ) ) ) ).

% image_eqI
thf(fact_255_SUP__UN__eq,axiom,
    ! [R: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,S: set_Pr8357624050790843030od_c_a] :
      ( ( comple4886903611889951529_c_d_o
        @ ( image_4757571491694354887_c_d_o
          @ ^ [I: produc2669932201312201782od_c_a,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_256_SUP__UN__eq,axiom,
    ! [R: $o > set_Pr2898135061646675015od_c_d,S: set_o] :
      ( ( comple4886903611889951529_c_d_o
        @ ( image_5121210478446633769_c_d_o
          @ ^ [I: $o,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_257_SUP__UN__eq,axiom,
    ! [R: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,S: set_se7324893142979633703od_c_d] :
      ( ( comple4886903611889951529_c_d_o
        @ ( image_6411191663572841816_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_258_SUP__UN__eq,axiom,
    ! [R: $o > set_o,S: set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_o_o_o
          @ ^ [I: $o,X3: $o] : ( member_o @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: $o] : ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_259_SUP__UN__eq,axiom,
    ! [R: $o > set_Pr224028844097186518od_c_d,S: set_o] :
      ( ( comple3325465037385151322_c_d_o
        @ ( image_6856080854014128986_c_d_o
          @ ^ [I: $o,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_260_SUP__UN__eq,axiom,
    ! [R: $o > set_Pr8357624050790843030od_c_a,S: set_o] :
      ( ( comple1937036886571535258_c_a_o
        @ ( image_5467652703200512922_c_a_o
          @ ^ [I: $o,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ R @ S ) ) ) ) ) ).

% SUP_UN_eq
thf(fact_261_UN__constant__eq,axiom,
    ! [A: $o,A2: set_o,F: $o > set_o,C3: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_262_UN__constant__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_o,C3: set_o] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_263_UN__constant__eq,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > set_o,C3: set_o] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_264_UN__constant__eq,axiom,
    ! [A: $o,A2: set_o,F: $o > set_Pr224028844097186518od_c_d,C3: set_Pr224028844097186518od_c_d] :
      ( ( member_o @ A @ A2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_265_UN__constant__eq,axiom,
    ! [A: $o,A2: set_o,F: $o > set_Pr8357624050790843030od_c_a,C3: set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ A @ A2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_266_UN__constant__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,C3: set_Pr224028844097186518od_c_d] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_267_UN__constant__eq,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,C3: set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_268_UN__constant__eq,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,C3: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple6722362287630405891od_c_a @ ( image_4003539448451559653od_c_a @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_269_UN__constant__eq,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,C3: set_Pr8357624050790843030od_c_a] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple6722362287630405891od_c_a @ ( image_2410488862838995109od_c_a @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_270_UN__constant__eq,axiom,
    ! [A: $o,A2: set_o,F: $o > set_Pr2898135061646675015od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( member_o @ A @ A2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ A2 )
           => ( ( F @ X2 )
              = C3 ) )
       => ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ F @ A2 ) )
          = C3 ) ) ) ).

% UN_constant_eq
thf(fact_271_setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,P: produc2669932201312201782od_c_a > $o] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_8721290246026763926od_c_d @ F @ ( collec8405073525749625505od_c_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_272_setcompr__eq__image,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,P: $o > $o] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_8975626525747676852od_c_d @ F @ ( collect_o @ P ) ) ) ).

% setcompr_eq_image
thf(fact_273_setcompr__eq__image,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,P: $o > $o] :
      ( ( collec4526355225760826689od_c_d
        @ ^ [Uu: set_Pr224028844097186518od_c_d] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_728402167444666691od_c_d @ F @ ( collect_o @ P ) ) ) ).

% setcompr_eq_image
thf(fact_274_setcompr__eq__image,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,P: $o > $o] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_8861997374138323203od_c_a @ F @ ( collect_o @ P ) ) ) ).

% setcompr_eq_image
thf(fact_275_setcompr__eq__image,axiom,
    ! [F: $o > set_o,P: $o > $o] :
      ( ( collect_set_o
        @ ^ [Uu: set_o] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_o_set_o @ F @ ( collect_o @ P ) ) ) ).

% setcompr_eq_image
thf(fact_276_setcompr__eq__image,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_242321854666033093od_c_d @ F @ ( collec6732257886551399090od_c_d @ P ) ) ) ).

% setcompr_eq_image
thf(fact_277_setcompr__eq__image,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,P: produc1399170498009216118od_c_d > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: produc1399170498009216118od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_3436769108421412982od_c_d @ F @ ( collec7134311822446639841od_c_d @ P ) ) ) ).

% setcompr_eq_image
thf(fact_278_setcompr__eq__image,axiom,
    ! [F: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: produc2736867335369716327od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( P @ X3 ) ) )
      = ( image_645533970575654853od_c_d @ F @ ( collec2789978228259155154od_c_d @ P ) ) ) ).

% setcompr_eq_image
thf(fact_279_Setcompr__eq__image,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member4652649041329350768od_c_d @ X3 @ A2 ) ) )
      = ( image_242321854666033093od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_280_Setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member5502040268208861279od_c_a @ X3 @ A2 ) ) )
      = ( image_8721290246026763926od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_281_Setcompr__eq__image,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [Uu: set_Pr2898135061646675015od_c_d] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_o @ X3 @ A2 ) ) )
      = ( image_8975626525747676852od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_282_Setcompr__eq__image,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( collec4526355225760826689od_c_d
        @ ^ [Uu: set_Pr224028844097186518od_c_d] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_o @ X3 @ A2 ) ) )
      = ( image_728402167444666691od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_283_Setcompr__eq__image,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_o @ X3 @ A2 ) ) )
      = ( image_8861997374138323203od_c_a @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_284_Setcompr__eq__image,axiom,
    ! [F: $o > set_o,A2: set_o] :
      ( ( collect_set_o
        @ ^ [Uu: set_o] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_o @ X3 @ A2 ) ) )
      = ( image_o_set_o @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_285_Setcompr__eq__image,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2736867335369716327od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member5502040268208861279od_c_a @ X3 @ A2 ) ) )
      = ( image_3416532124207988918od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_286_Setcompr__eq__image,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: produc1399170498009216118od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member4231278564905875615od_c_d @ X3 @ A2 ) ) )
      = ( image_3436769108421412982od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_287_Setcompr__eq__image,axiom,
    ! [F: $o > produc2736867335369716327od_c_d,A2: set_o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: $o] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member_o @ X3 @ A2 ) ) )
      = ( image_8029560678096478420od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_288_Setcompr__eq__image,axiom,
    ! [F: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d,A2: set_Pr2898135061646675015od_c_d] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [Uu: produc2736867335369716327od_c_d] :
          ? [X3: produc2736867335369716327od_c_d] :
            ( ( Uu
              = ( F @ X3 ) )
            & ( member4507666829407384208od_c_d @ X3 @ A2 ) ) )
      = ( image_645533970575654853od_c_d @ F @ A2 ) ) ).

% Setcompr_eq_image
thf(fact_289_h__obs_Ocases,axiom,
    ! [X: produc1664448241625060997od_c_d] :
      ~ ! [M: fSM_fs8514149560489614281od_c_d,Q: produc2669932201312201782od_c_a,X2: d,Y: produc2346461404597154448od_c_d] :
          ( X
         != ( produc2816823678332465269od_c_d @ M @ ( produc2733640664908071511od_c_d @ Q @ ( produc3045323491883059558od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs.cases
thf(fact_290_h__obs_Ocases,axiom,
    ! [X: produc1158241047046068175od_c_d] :
      ~ ! [M: fSM_fs7627742230287599454od_c_d,Q: mappin2657711750964880076od_c_a,X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
          ( X
         != ( produc8875723801892655751od_c_d @ M @ ( produc5940215602367092546od_c_d @ Q @ ( produc2733640664908071511od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs.cases
thf(fact_291_h__obs_Ocases,axiom,
    ! [X: produc1978195128095926127od_c_d] :
      ~ ! [M: fSM_fs256625017975418798od_c_d,Q: d,X2: b,Y: product_prod_c_d] :
          ( X
         != ( produc856611777525334239od_c_d @ M @ ( produc3045323491883059558od_c_d @ Q @ ( produc3440865926910568064od_c_d @ X2 @ Y ) ) ) ) ).

% h_obs.cases
thf(fact_292_SUP__Sup__eq,axiom,
    ! [S: set_se7324893142979633703od_c_d] :
      ( ( comple4886903611889951529_c_d_o
        @ ( image_6411191663572841816_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_293_SUP__Sup__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_set_o_o_o
          @ ^ [I: set_o,X3: $o] : ( member_o @ X3 @ I )
          @ S ) )
      = ( ^ [X3: $o] : ( member_o @ X3 @ ( comple90263536869209701_set_o @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_294_SUP__Sup__eq,axiom,
    ! [S: set_se1756390050576932150od_c_d] :
      ( ( comple3325465037385151322_c_d_o
        @ ( image_1484836191878656152_c_d_o
          @ ^ [I: set_Pr224028844097186518od_c_d,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_295_SUP__Sup__eq,axiom,
    ! [S: set_se312849170327590134od_c_a] :
      ( ( comple1937036886571535258_c_a_o
        @ ( image_7664211964329984152_c_a_o
          @ ^ [I: set_Pr8357624050790843030od_c_a,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_296_Compr__image__eq,axiom,
    ! [F: $o > $o,A2: set_o,P: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ ( image_o_o2 @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_o_o2 @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_297_Compr__image__eq,axiom,
    ! [F: $o > set_o,A2: set_o,P: set_o > $o] :
      ( ( collect_set_o
        @ ^ [X3: set_o] :
            ( ( member_set_o @ X3 @ ( image_o_set_o @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_o_set_o @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_298_Compr__image__eq,axiom,
    ! [F: $o > produc2669932201312201782od_c_a,A2: set_o,P: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( image_4928573236352822947od_c_a @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_4928573236352822947od_c_a @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_299_Compr__image__eq,axiom,
    ! [F: $o > produc1399170498009216118od_c_d,A2: set_o,P: produc1399170498009216118od_c_d > $o] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ ( image_3657811533049837283od_c_d @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_3657811533049837283od_c_d @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_300_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > $o,A2: set_Pr8357624050790843030od_c_a,P: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ ( image_2427823168983691879_c_a_o @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_2427823168983691879_c_a_o @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_301_Compr__image__eq,axiom,
    ! [F: produc1399170498009216118od_c_d > $o,A2: set_Pr224028844097186518od_c_d,P: $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
            ( ( member_o @ X3 @ ( image_3816251319797307943_c_d_o @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_3816251319797307943_c_d_o @ F
        @ ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_302_Compr__image__eq,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: set_Pr224028844097186518od_c_d > $o] :
      ( ( collec4526355225760826689od_c_d
        @ ^ [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ ( image_728402167444666691od_c_d @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_728402167444666691od_c_d @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_303_Compr__image__eq,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: set_Pr8357624050790843030od_c_a > $o] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_8861997374138323203od_c_a @ F
        @ ( collect_o
          @ ^ [X3: $o] :
              ( ( member_o @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_304_Compr__image__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( image_2169818278710580869od_c_a @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_2169818278710580869od_c_a @ F
        @ ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_305_Compr__image__eq,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2669932201312201782od_c_a,A2: set_Pr224028844097186518od_c_d,P: produc2669932201312201782od_c_a > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ ( image_1701621602615451205od_c_a @ F @ A2 ) )
            & ( P @ X3 ) ) )
      = ( image_1701621602615451205od_c_a @ F
        @ ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ X3 @ A2 )
              & ( P @ ( F @ X3 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_306_Sup__bool__def,axiom,
    ( complete_Sup_Sup_o
    = ( member_o @ $true ) ) ).

% Sup_bool_def
thf(fact_307_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > $o] :
      ( ( member_o @ X @ A2 )
     => ( member_o @ ( F @ X ) @ ( image_o_o2 @ F @ A2 ) ) ) ).

% imageI
thf(fact_308_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > set_o] :
      ( ( member_o @ X @ A2 )
     => ( member_set_o @ ( F @ X ) @ ( image_o_set_o @ F @ A2 ) ) ) ).

% imageI
thf(fact_309_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( member_o @ ( F @ X ) @ ( image_2427823168983691879_c_a_o @ F @ A2 ) ) ) ).

% imageI
thf(fact_310_imageI,axiom,
    ! [X: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > $o] :
      ( ( member4231278564905875615od_c_d @ X @ A2 )
     => ( member_o @ ( F @ X ) @ ( image_3816251319797307943_c_d_o @ F @ A2 ) ) ) ).

% imageI
thf(fact_311_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > produc2669932201312201782od_c_a] :
      ( ( member_o @ X @ A2 )
     => ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_4928573236352822947od_c_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_312_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > produc1399170498009216118od_c_d] :
      ( ( member_o @ X @ A2 )
     => ( member4231278564905875615od_c_d @ ( F @ X ) @ ( image_3657811533049837283od_c_d @ F @ A2 ) ) ) ).

% imageI
thf(fact_313_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > set_Pr224028844097186518od_c_d] :
      ( ( member_o @ X @ A2 )
     => ( member939977866535389951od_c_d @ ( F @ X ) @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ).

% imageI
thf(fact_314_imageI,axiom,
    ! [X: $o,A2: set_o,F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ X @ A2 )
     => ( member9073573073229046463od_c_a @ ( F @ X ) @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_315_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_2169818278710580869od_c_a @ F @ A2 ) ) ) ).

% imageI
thf(fact_316_imageI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( member4231278564905875615od_c_d @ ( F @ X ) @ ( image_899056575407595205od_c_d @ F @ A2 ) ) ) ).

% imageI
thf(fact_317_image__iff,axiom,
    ! [Z: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member4652649041329350768od_c_d @ Z @ ( image_8721290246026763926od_c_d @ F @ A2 ) )
      = ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_318_image__iff,axiom,
    ! [Z: set_Pr2898135061646675015od_c_d,F: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( member4652649041329350768od_c_d @ Z @ ( image_8975626525747676852od_c_d @ F @ A2 ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_319_image__iff,axiom,
    ! [Z: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( member939977866535389951od_c_d @ Z @ ( image_728402167444666691od_c_d @ F @ A2 ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_320_image__iff,axiom,
    ! [Z: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( member9073573073229046463od_c_a @ Z @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_321_image__iff,axiom,
    ! [Z: set_o,F: $o > set_o,A2: set_o] :
      ( ( member_set_o @ Z @ ( image_o_set_o @ F @ A2 ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_322_image__iff,axiom,
    ! [Z: set_Pr2898135061646675015od_c_d,F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( member4652649041329350768od_c_d @ Z @ ( image_242321854666033093od_c_d @ F @ A2 ) )
      = ( ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_323_image__iff,axiom,
    ! [Z: produc2736867335369716327od_c_d,F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( member4507666829407384208od_c_d @ Z @ ( image_3436769108421412982od_c_d @ F @ A2 ) )
      = ( ? [X3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X3 @ A2 )
            & ( Z
              = ( F @ X3 ) ) ) ) ) ).

% image_iff
thf(fact_324_bex__imageD,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ? [X6: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X6 @ ( image_8721290246026763926od_c_d @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_325_bex__imageD,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ? [X6: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X6 @ ( image_3436769108421412982od_c_d @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_326_bex__imageD,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ? [X6: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X6 @ ( image_8975626525747676852od_c_d @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: $o] :
          ( ( member_o @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_327_bex__imageD,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: set_Pr224028844097186518od_c_d > $o] :
      ( ? [X6: set_Pr224028844097186518od_c_d] :
          ( ( member939977866535389951od_c_d @ X6 @ ( image_728402167444666691od_c_d @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: $o] :
          ( ( member_o @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_328_bex__imageD,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: set_Pr8357624050790843030od_c_a > $o] :
      ( ? [X6: set_Pr8357624050790843030od_c_a] :
          ( ( member9073573073229046463od_c_a @ X6 @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: $o] :
          ( ( member_o @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_329_bex__imageD,axiom,
    ! [F: $o > set_o,A2: set_o,P: set_o > $o] :
      ( ? [X6: set_o] :
          ( ( member_set_o @ X6 @ ( image_o_set_o @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: $o] :
          ( ( member_o @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_330_bex__imageD,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ? [X6: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X6 @ ( image_242321854666033093od_c_d @ F @ A2 ) )
          & ( P @ X6 ) )
     => ? [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ A2 )
          & ( P @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_331_image__cong,axiom,
    ! [M2: set_se7324893142979633703od_c_d,N: set_se7324893142979633703od_c_d,F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,G: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( M2 = N )
     => ( ! [X2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_242321854666033093od_c_d @ F @ M2 )
          = ( image_242321854666033093od_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_332_image__cong,axiom,
    ! [M2: set_Pr8357624050790843030od_c_a,N: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,G: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( M2 = N )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_8721290246026763926od_c_d @ F @ M2 )
          = ( image_8721290246026763926od_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_333_image__cong,axiom,
    ! [M2: set_Pr224028844097186518od_c_d,N: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,G: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d] :
      ( ( M2 = N )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_3436769108421412982od_c_d @ F @ M2 )
          = ( image_3436769108421412982od_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_334_image__cong,axiom,
    ! [M2: set_o,N: set_o,F: $o > set_Pr2898135061646675015od_c_d,G: $o > set_Pr2898135061646675015od_c_d] :
      ( ( M2 = N )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_8975626525747676852od_c_d @ F @ M2 )
          = ( image_8975626525747676852od_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_335_image__cong,axiom,
    ! [M2: set_o,N: set_o,F: $o > set_Pr224028844097186518od_c_d,G: $o > set_Pr224028844097186518od_c_d] :
      ( ( M2 = N )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_728402167444666691od_c_d @ F @ M2 )
          = ( image_728402167444666691od_c_d @ G @ N ) ) ) ) ).

% image_cong
thf(fact_336_image__cong,axiom,
    ! [M2: set_o,N: set_o,F: $o > set_Pr8357624050790843030od_c_a,G: $o > set_Pr8357624050790843030od_c_a] :
      ( ( M2 = N )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_8861997374138323203od_c_a @ F @ M2 )
          = ( image_8861997374138323203od_c_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_337_image__cong,axiom,
    ! [M2: set_o,N: set_o,F: $o > set_o,G: $o > set_o] :
      ( ( M2 = N )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ N )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_o_set_o @ F @ M2 )
          = ( image_o_set_o @ G @ N ) ) ) ) ).

% image_cong
thf(fact_338_ball__imageD,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ ( image_8721290246026763926od_c_d @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_339_ball__imageD,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d,P: produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X2 @ ( image_3436769108421412982od_c_d @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_340_ball__imageD,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,A2: set_o,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ ( image_8975626525747676852od_c_d @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: $o] :
          ( ( member_o @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_341_ball__imageD,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o,P: set_Pr224028844097186518od_c_d > $o] :
      ( ! [X2: set_Pr224028844097186518od_c_d] :
          ( ( member939977866535389951od_c_d @ X2 @ ( image_728402167444666691od_c_d @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: $o] :
          ( ( member_o @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_342_ball__imageD,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o,P: set_Pr8357624050790843030od_c_a > $o] :
      ( ! [X2: set_Pr8357624050790843030od_c_a] :
          ( ( member9073573073229046463od_c_a @ X2 @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: $o] :
          ( ( member_o @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_343_ball__imageD,axiom,
    ! [F: $o > set_o,A2: set_o,P: set_o > $o] :
      ( ! [X2: set_o] :
          ( ( member_set_o @ X2 @ ( image_o_set_o @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: $o] :
          ( ( member_o @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_344_ball__imageD,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d,P: set_Pr2898135061646675015od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ ( image_242321854666033093od_c_d @ F @ A2 ) )
         => ( P @ X2 ) )
     => ! [X6: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X6 @ A2 )
         => ( P @ ( F @ X6 ) ) ) ) ).

% ball_imageD
thf(fact_345_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: $o,F: $o > $o] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_346_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: set_o,F: $o > set_o] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_347_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: $o,F: produc2669932201312201782od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_2427823168983691879_c_a_o @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_348_rev__image__eqI,axiom,
    ! [X: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: $o,F: produc1399170498009216118od_c_d > $o] :
      ( ( member4231278564905875615od_c_d @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member_o @ B @ ( image_3816251319797307943_c_d_o @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_349_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: produc2669932201312201782od_c_a,F: $o > produc2669932201312201782od_c_a] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member5502040268208861279od_c_a @ B @ ( image_4928573236352822947od_c_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_350_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: produc1399170498009216118od_c_d,F: $o > produc1399170498009216118od_c_d] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member4231278564905875615od_c_d @ B @ ( image_3657811533049837283od_c_d @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_351_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member939977866535389951od_c_d @ B @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_352_rev__image__eqI,axiom,
    ! [X: $o,A2: set_o,B: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member9073573073229046463od_c_a @ B @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_353_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_354_rev__image__eqI,axiom,
    ! [X: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: produc1399170498009216118od_c_d,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ X @ A2 )
     => ( ( B
          = ( F @ X ) )
       => ( member4231278564905875615od_c_d @ B @ ( image_899056575407595205od_c_d @ F @ A2 ) ) ) ) ).

% rev_image_eqI
thf(fact_355_Sup__SUP__eq,axiom,
    ( comple4886903611889951529_c_d_o
    = ( ^ [S2: set_Pr4683839544804131468_c_d_o,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple3966289243721909940od_c_d @ ( image_7682309925354437162od_c_d @ collec2789978228259155154od_c_d @ S2 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_356_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_o_o
    = ( ^ [S2: set_o_o,X3: $o] : ( member_o @ X3 @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_357_Sup__SUP__eq,axiom,
    ( comple3325465037385151322_c_d_o
    = ( ^ [S2: set_Pr5476883483191388733_c_d_o,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple7812139117791525187od_c_d @ ( image_1266220945562961352od_c_d @ collec7134311822446639841od_c_d @ S2 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_358_Sup__SUP__eq,axiom,
    ( comple1937036886571535258_c_a_o
    = ( ^ [S2: set_Pr7260878174465401469_c_a_o,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple6722362287630405891od_c_a @ ( image_1542064655996835144od_c_a @ collec8405073525749625505od_c_a @ S2 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_359_pred__equals__eq2,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ R2 ) )
        = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_360_pred__equals__eq2,axiom,
    ! [R2: set_Pr2898135061646675015od_c_d,S: set_Pr2898135061646675015od_c_d] :
      ( ( ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ R2 ) )
        = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_361_pred__equals__eq2,axiom,
    ! [R2: set_Pr5052118491147477062od_c_d,S: set_Pr5052118491147477062od_c_d] :
      ( ( ( ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ R2 ) )
        = ( ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_362_pred__equals__eq2,axiom,
    ! [R2: set_Pr224028844097186518od_c_d,S: set_Pr224028844097186518od_c_d] :
      ( ( ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ R2 ) )
        = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ S ) ) )
      = ( R2 = S ) ) ).

% pred_equals_eq2
thf(fact_363_imageE,axiom,
    ! [B: $o,F: $o > $o,A2: set_o] :
      ( ( member_o @ B @ ( image_o_o2 @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_364_imageE,axiom,
    ! [B: set_o,F: $o > set_o,A2: set_o] :
      ( ( member_set_o @ B @ ( image_o_set_o @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_365_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: $o > produc2669932201312201782od_c_a,A2: set_o] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_4928573236352822947od_c_a @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_366_imageE,axiom,
    ! [B: produc1399170498009216118od_c_d,F: $o > produc1399170498009216118od_c_d,A2: set_o] :
      ( ( member4231278564905875615od_c_d @ B @ ( image_3657811533049837283od_c_d @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_367_imageE,axiom,
    ! [B: $o,F: produc2669932201312201782od_c_a > $o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ B @ ( image_2427823168983691879_c_a_o @ F @ A2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member5502040268208861279od_c_a @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_368_imageE,axiom,
    ! [B: $o,F: produc1399170498009216118od_c_d > $o,A2: set_Pr224028844097186518od_c_d] :
      ( ( member_o @ B @ ( image_3816251319797307943_c_d_o @ F @ A2 ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member4231278564905875615od_c_d @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_369_imageE,axiom,
    ! [B: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( member939977866535389951od_c_d @ B @ ( image_728402167444666691od_c_d @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_370_imageE,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( member9073573073229046463od_c_a @ B @ ( image_8861997374138323203od_c_a @ F @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_371_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_2169818278710580869od_c_a @ F @ A2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member5502040268208861279od_c_a @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_372_imageE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: produc1399170498009216118od_c_d > produc2669932201312201782od_c_a,A2: set_Pr224028844097186518od_c_d] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_1701621602615451205od_c_a @ F @ A2 ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( B
              = ( F @ X2 ) )
           => ~ ( member4231278564905875615od_c_d @ X2 @ A2 ) ) ) ).

% imageE
thf(fact_373_image__image,axiom,
    ! [F: $o > set_o,G: $o > $o,A2: set_o] :
      ( ( image_o_set_o @ F @ ( image_o_o2 @ G @ A2 ) )
      = ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_374_image__image,axiom,
    ! [F: set_o > set_o,G: $o > set_o,A2: set_o] :
      ( ( image_set_o_set_o @ F @ ( image_o_set_o @ G @ A2 ) )
      = ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_375_image__image,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,G: $o > $o,A2: set_o] :
      ( ( image_728402167444666691od_c_d @ F @ ( image_o_o2 @ G @ A2 ) )
      = ( image_728402167444666691od_c_d
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_376_image__image,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,G: $o > $o,A2: set_o] :
      ( ( image_8861997374138323203od_c_a @ F @ ( image_o_o2 @ G @ A2 ) )
      = ( image_8861997374138323203od_c_a
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_377_image__image,axiom,
    ! [F: set_Pr224028844097186518od_c_d > set_o,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_2583072847733522087_set_o @ F @ ( image_728402167444666691od_c_d @ G @ A2 ) )
      = ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_378_image__image,axiom,
    ! [F: set_Pr8357624050790843030od_c_a > set_o,G: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( image_1108906476932298471_set_o @ F @ ( image_8861997374138323203od_c_a @ G @ A2 ) )
      = ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_379_image__image,axiom,
    ! [F: set_o > set_Pr224028844097186518od_c_d,G: $o > set_o,A2: set_o] :
      ( ( image_564716939699073635od_c_d @ F @ ( image_o_set_o @ G @ A2 ) )
      = ( image_728402167444666691od_c_d
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_380_image__image,axiom,
    ! [F: set_o > set_Pr8357624050790843030od_c_a,G: $o > set_o,A2: set_o] :
      ( ( image_8698312146392730147od_c_a @ F @ ( image_o_set_o @ G @ A2 ) )
      = ( image_8861997374138323203od_c_a
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_381_image__image,axiom,
    ! [F: set_Pr224028844097186518od_c_d > set_Pr224028844097186518od_c_d,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_563476247564946053od_c_d @ F @ ( image_728402167444666691od_c_d @ G @ A2 ) )
      = ( image_728402167444666691od_c_d
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_382_image__image,axiom,
    ! [F: set_Pr224028844097186518od_c_d > set_Pr8357624050790843030od_c_a,G: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_8697071454258602565od_c_a @ F @ ( image_728402167444666691od_c_d @ G @ A2 ) )
      = ( image_8861997374138323203od_c_a
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ A2 ) ) ).

% image_image
thf(fact_383_refl__on__UNION,axiom,
    ! [S: set_Pr8357624050790843030od_c_a,A2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,R: produc2669932201312201782od_c_a > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ A2 @ S ) ) @ ( comple7203151847829604148od_c_d @ ( image_5364838190352781078od_c_d @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_384_refl__on__UNION,axiom,
    ! [S: set_o,A2: $o > set_Pr2898135061646675015od_c_d,R: $o > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ A2 @ S ) ) @ ( comple7203151847829604148od_c_d @ ( image_5794988778541631284od_c_d @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_385_refl__on__UNION,axiom,
    ! [S: set_se7324893142979633703od_c_d,A2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,R: set_Pr2898135061646675015od_c_d > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ A2 @ S ) ) @ ( comple7203151847829604148od_c_d @ ( image_914099667705184325od_c_d @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_386_refl__on__UNION,axiom,
    ! [S: set_o,A2: $o > set_o,R: $o > set_Product_prod_o_o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_on_o @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_on_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ S ) ) @ ( comple2567785453016644468od_o_o @ ( image_5520585345982086004od_o_o @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_387_refl__on__UNION,axiom,
    ! [S: set_o,A2: $o > set_Pr224028844097186518od_c_d,R: $o > set_Pr137242208728970247od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o899880292924752245od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o899880292924752245od_c_d @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ A2 @ S ) ) @ ( comple1205396390804205172od_c_d @ ( image_6214733672829972084od_c_d @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_388_refl__on__UNION,axiom,
    ! [S: set_o,A2: $o > set_Pr8357624050790843030od_c_a,R: $o > set_Pr1454594181397332999od_c_a] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o2170641996227737909od_c_a @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2170641996227737909od_c_a @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ A2 @ S ) ) @ ( comple2522748363472567924od_c_a @ ( image_7532085645498334836od_c_a @ R @ S ) ) ) ) ).

% refl_on_UNION
thf(fact_389_pair__in__swap__image,axiom,
    ! [Y3: produc2133359284458116493od_c_a,X: a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member5164570601847750273_c_a_a @ ( produc2999150402704957000_c_a_a @ Y3 @ X ) @ ( image_1832348612349469863_c_a_a @ produc3568985373970804038od_c_a @ A2 ) )
      = ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_390_pair__in__swap__image,axiom,
    ! [Y3: a,X: produc2133359284458116493od_c_a,A2: set_Pr2582991818569342904_c_a_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ Y3 @ X ) @ ( image_6327630916522945251od_c_a @ produc805334219641084904_c_a_a @ A2 ) )
      = ( member5164570601847750273_c_a_a @ ( produc2999150402704957000_c_a_a @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_391_pair__in__swap__image,axiom,
    ! [Y3: produc1399170498009216118od_c_d,X: produc2669932201312201782od_c_a,A2: set_Pr2898135061646675015od_c_d] :
      ( ( member2949693323151135248od_c_a @ ( produc3536205692115927511od_c_a @ Y3 @ X ) @ ( image_8310932501174181701od_c_a @ produc4204837798695895031od_c_d @ A2 ) )
      = ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_392_pair__in__swap__image,axiom,
    ! [Y3: produc2736867335369716327od_c_d,X: mappin2657711750964880076od_c_a,A2: set_Pr5052118491147477062od_c_d] :
      ( ( member5758217800826081913od_c_a @ ( produc2046884284851656156od_c_a @ Y3 @ X ) @ ( image_5887946549550171257od_c_a @ produc6894619464842006946od_c_d @ A2 ) )
      = ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_393_pair__in__swap__image,axiom,
    ! [Y3: produc2346461404597154448od_c_d,X: d,A2: set_Pr224028844097186518od_c_d] :
      ( ( member3893209869398447809_c_d_d @ ( produc1727789657345968136_c_d_d @ Y3 @ X ) @ ( image_92791203805037735_c_d_d @ produc851507308819187462od_c_d @ A2 ) )
      = ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_394_pair__in__swap__image,axiom,
    ! [Y3: produc2669932201312201782od_c_a,X: produc1399170498009216118od_c_d,A2: set_Pr7917073365334404039od_c_a] :
      ( ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ Y3 @ X ) @ ( image_7658187320581095493od_c_d @ produc5007402825903751031od_c_a @ A2 ) )
      = ( member2949693323151135248od_c_a @ ( produc3536205692115927511od_c_a @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_395_pair__in__swap__image,axiom,
    ! [Y3: mappin2657711750964880076od_c_a,X: produc2736867335369716327od_c_d,A2: set_Pr2358889278880801944od_c_a] :
      ( ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ Y3 @ X ) @ ( image_4775091365484192889od_c_d @ produc3001288147326570556od_c_a @ A2 ) )
      = ( member5758217800826081913od_c_a @ ( produc2046884284851656156od_c_a @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_396_pair__in__swap__image,axiom,
    ! [Y3: d,X: produc2346461404597154448od_c_d,A2: set_Pr1437931961497901048_c_d_d] :
      ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ Y3 @ X ) @ ( image_8220429349476133603od_c_d @ produc8757345511136871848_c_d_d @ A2 ) )
      = ( member3893209869398447809_c_d_d @ ( produc1727789657345968136_c_d_d @ X @ Y3 ) @ A2 ) ) ).

% pair_in_swap_image
thf(fact_397_UN__Image,axiom,
    ! [X4: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,I2: set_Pr8357624050790843030od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( image_5265990627690709651od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ X4 @ I2 ) ) @ S )
      = ( comple7812139117791525187od_c_d
        @ ( image_5093316278612678949od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( image_5265990627690709651od_c_d @ ( X4 @ I ) @ S )
          @ I2 ) ) ) ).

% UN_Image
thf(fact_398_UN__Image,axiom,
    ! [X4: $o > set_Pr2898135061646675015od_c_d,I2: set_o,S: set_Pr8357624050790843030od_c_a] :
      ( ( image_5265990627690709651od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ X4 @ I2 ) ) @ S )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [I: $o] : ( image_5265990627690709651od_c_d @ ( X4 @ I ) @ S )
          @ I2 ) ) ) ).

% UN_Image
thf(fact_399_UN__Image,axiom,
    ! [X4: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,I2: set_se7324893142979633703od_c_d,S: set_Pr8357624050790843030od_c_a] :
      ( ( image_5265990627690709651od_c_d @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ X4 @ I2 ) ) @ S )
      = ( comple7812139117791525187od_c_d
        @ ( image_2269266353923825236od_c_d
          @ ^ [I: set_Pr2898135061646675015od_c_d] : ( image_5265990627690709651od_c_d @ ( X4 @ I ) @ S )
          @ I2 ) ) ) ).

% UN_Image
thf(fact_400_UN__Image,axiom,
    ! [X4: $o > set_Pr224028844097186518od_c_d,I2: set_o,S: set_d] :
      ( ( image_4235397684752546210od_c_d @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ X4 @ I2 ) ) @ S )
      = ( comple8603459066105068637od_c_d
        @ ( image_721604527892118621od_c_d
          @ ^ [I: $o] : ( image_4235397684752546210od_c_d @ ( X4 @ I ) @ S )
          @ I2 ) ) ) ).

% UN_Image
thf(fact_401_UN__Image,axiom,
    ! [X4: $o > set_Pr8357624050790843030od_c_a,I2: set_o,S: set_a] :
      ( ( image_6952875749904162786od_c_a @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ X4 @ I2 ) ) @ S )
      = ( comple6943640584117399770od_c_a
        @ ( image_8285158082759225562od_c_a
          @ ^ [I: $o] : ( image_6952875749904162786od_c_a @ ( X4 @ I ) @ S )
          @ I2 ) ) ) ).

% UN_Image
thf(fact_402_reflp__on__Sup,axiom,
    ! [S: set_Pr8357624050790843030od_c_a,A2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,R2: produc2669932201312201782od_c_a > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ A2 @ S ) ) @ ( comple3993424857363069083_c_d_o @ ( image_8192407713068371069_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_403_reflp__on__Sup,axiom,
    ! [S: set_o,A2: $o > set_Pr2898135061646675015od_c_d,R2: $o > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ A2 @ S ) ) @ ( comple3993424857363069083_c_d_o @ ( image_8434443802585561243_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_404_reflp__on__Sup,axiom,
    ! [S: set_se7324893142979633703od_c_d,A2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,R2: set_Pr2898135061646675015od_c_d > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ A2 @ S ) ) @ ( comple3993424857363069083_c_d_o @ ( image_1525462150832592812_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_405_reflp__on__Sup,axiom,
    ! [S: set_o,A2: $o > set_o,R2: $o > $o > $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_on_o @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_on_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A2 @ S ) ) @ ( comple8004442457126985307_o_o_o @ ( image_o_o_o_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_406_reflp__on__Sup,axiom,
    ! [S: set_o,A2: $o > set_Pr224028844097186518od_c_d,R2: $o > produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_8343749969777128071od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_8343749969777128071od_c_d @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ A2 @ S ) ) @ ( comple5563752445805895387_c_d_o @ ( image_7687665798361177819_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_407_reflp__on__Sup,axiom,
    ! [S: set_o,A2: $o > set_Pr8357624050790843030od_c_a,R2: $o > produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_391139636225337927od_c_a @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_391139636225337927od_c_a @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ A2 @ S ) ) @ ( comple4135180802329554651_c_a_o @ ( image_6259094154884837083_c_a_o @ R2 @ S ) ) ) ) ).

% reflp_on_Sup
thf(fact_408_same__fstI,axiom,
    ! [P: produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a,Y5: produc1399170498009216118od_c_d,Y3: produc1399170498009216118od_c_d,R2: produc2669932201312201782od_c_a > set_Pr137242208728970247od_c_d] :
      ( ( P @ X )
     => ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ Y5 @ Y3 ) @ ( R2 @ X ) )
       => ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ ( produc2733640664908071511od_c_d @ X @ Y5 ) @ ( produc2733640664908071511od_c_d @ X @ Y3 ) ) @ ( same_f5452307918870061868od_c_d @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_409_same__fstI,axiom,
    ! [P: mappin2657711750964880076od_c_a > $o,X: mappin2657711750964880076od_c_a,Y5: produc2736867335369716327od_c_d,Y3: produc2736867335369716327od_c_d,R2: mappin2657711750964880076od_c_a > set_Pr8731163265463710919od_c_d] :
      ( ( P @ X )
     => ( ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ Y5 @ Y3 ) @ ( R2 @ X ) )
       => ( member784859968957666602od_c_d @ ( produc2276813182815557561od_c_d @ ( produc5940215602367092546od_c_d @ X @ Y5 ) @ ( produc5940215602367092546od_c_d @ X @ Y3 ) ) @ ( same_f139212207016127533od_c_d @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_410_same__fstI,axiom,
    ! [P: d > $o,X: d,Y5: produc2346461404597154448od_c_d,Y3: produc2346461404597154448od_c_d,R2: d > set_Pr5408497735240033927od_c_d] :
      ( ( P @ X )
     => ( ( member4193451924820284624od_c_d @ ( produc4475737221850782103od_c_d @ Y5 @ Y3 ) @ ( R2 @ X ) )
       => ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ ( produc3045323491883059558od_c_d @ X @ Y5 ) @ ( produc3045323491883059558od_c_d @ X @ Y3 ) ) @ ( same_f6781433740390309435od_c_d @ P @ R2 ) ) ) ) ).

% same_fstI
thf(fact_411_Collect__ex__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > produc2669932201312201782od_c_a > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [X7: produc2669932201312201782od_c_a] : ( P @ X3 @ X7 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [Y2: produc2669932201312201782od_c_a] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_to7710394033828511718od_c_a ) ) ) ).

% Collect_ex_eq
thf(fact_412_Collect__ex__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > $o > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [X7: $o] : ( P @ X3 @ X7 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [Y2: $o] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_top_set_o ) ) ) ).

% Collect_ex_eq
thf(fact_413_Collect__ex__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > set_Pr2898135061646675015od_c_d > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [X7: set_Pr2898135061646675015od_c_d] : ( P @ X3 @ X7 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_to8663423184816037751od_c_d ) ) ) ).

% Collect_ex_eq
thf(fact_414_Collect__ex__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > nat > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [X7: nat] : ( P @ X3 @ X7 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_388539031256343344od_c_d
          @ ^ [Y2: nat] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_415_Collect__ex__eq,axiom,
    ! [P: $o > $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
          ? [X7: $o] : ( P @ X3 @ X7 ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( collect_o
              @ ^ [X3: $o] : ( P @ X3 @ Y2 ) )
          @ top_top_set_o ) ) ) ).

% Collect_ex_eq
thf(fact_416_Collect__ex__eq,axiom,
    ! [P: $o > nat > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
          ? [X7: nat] : ( P @ X3 @ X7 ) )
      = ( comple90263536869209701_set_o
        @ ( image_nat_set_o
          @ ^ [Y2: nat] :
              ( collect_o
              @ ^ [X3: $o] : ( P @ X3 @ Y2 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_417_Collect__ex__eq,axiom,
    ! [P: produc1399170498009216118od_c_d > $o > $o] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] :
          ? [X7: $o] : ( P @ X3 @ X7 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( collec7134311822446639841od_c_d
              @ ^ [X3: produc1399170498009216118od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_top_set_o ) ) ) ).

% Collect_ex_eq
thf(fact_418_Collect__ex__eq,axiom,
    ! [P: produc1399170498009216118od_c_d > nat > $o] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] :
          ? [X7: nat] : ( P @ X3 @ X7 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_8269028152250893759od_c_d
          @ ^ [Y2: nat] :
              ( collec7134311822446639841od_c_d
              @ ^ [X3: produc1399170498009216118od_c_d] : ( P @ X3 @ Y2 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_419_Collect__ex__eq,axiom,
    ! [P: produc2669932201312201782od_c_a > $o > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
          ? [X7: $o] : ( P @ X3 @ X7 ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] :
              ( collec8405073525749625505od_c_a
              @ ^ [X3: produc2669932201312201782od_c_a] : ( P @ X3 @ Y2 ) )
          @ top_top_set_o ) ) ) ).

% Collect_ex_eq
thf(fact_420_Collect__ex__eq,axiom,
    ! [P: produc2669932201312201782od_c_a > nat > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
          ? [X7: nat] : ( P @ X3 @ X7 ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_7179251322089774463od_c_a
          @ ^ [Y2: nat] :
              ( collec8405073525749625505od_c_a
              @ ^ [X3: produc2669932201312201782od_c_a] : ( P @ X3 @ Y2 ) )
          @ top_top_set_nat ) ) ) ).

% Collect_ex_eq
thf(fact_421_INT__extend__simps_I8_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se3083084249005913095od_c_d] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_1688364829223432677od_c_d
          @ ^ [Y2: set_se7324893142979633703od_c_d] : ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ ( comple6045575216621938836od_c_d @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_422_INT__extend__simps_I8_J,axiom,
    ! [B2: produc2736867335369716327od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] : ( comple5098958804508528846od_c_d @ ( image_217721511564321189od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d @ ( image_217721511564321189od_c_d @ B2 @ ( comple3966289243721909940od_c_d @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_423_INT__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_set_o] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_5784685344308445140od_c_d
          @ ^ [Y2: set_o] : ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_424_INT__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [Y2: set_o] : ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_425_INT__extend__simps_I8_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [Y2: set_o] : ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_426_INT__extend__simps_I8_J,axiom,
    ! [B2: $o > set_o,A2: set_set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_set_o_set_o
          @ ^ [Y2: set_o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ ( comple90263536869209701_set_o @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_427_INT__extend__simps_I8_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_se312849170327590134od_c_a] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_5083520090745792566od_c_d
          @ ^ [Y2: set_Pr8357624050790843030od_c_a] : ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ Y2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ ( comple6722362287630405891od_c_a @ A2 ) ) ) ) ).

% INT_extend_simps(8)
thf(fact_428_Sup__SUP__eq2,axiom,
    ( comple916854894399990804_c_d_o
    = ( ^ [S2: set_ma5789154285582676471_c_d_o,X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ ( comple5928930310332571033od_c_d @ ( image_7829844492532784832od_c_d @ collec1512216693319809125od_c_d @ ( image_2436371738898717431_c_d_o @ produc7497122261040117353_c_d_o @ S2 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_429_Sup__SUP__eq2,axiom,
    ( comple2351186111055541915_c_d_o
    = ( ^ [S2: set_Pr6674123571878231694_c_d_o,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple3966289243721909940od_c_d @ ( image_7682309925354437162od_c_d @ collec2789978228259155154od_c_d @ ( image_6408603934379594687_c_d_o @ produc1656800403805021740_c_d_o @ S2 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_430_Sup__SUP__eq2,axiom,
    ( comple4510354444638677324_c_d_o
    = ( ^ [S2: set_d_4343312612943956159_c_d_o,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple7812139117791525187od_c_d @ ( image_1266220945562961352od_c_d @ collec7134311822446639841od_c_d @ ( image_8921174598359947809_c_d_o @ produc2693131752807987549_c_d_o @ S2 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_431_Sup__SUP__eq2,axiom,
    ( comple1107991413401925132_c_a_o
    = ( ^ [S2: set_a_6354160464449729919_c_a_o,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple6722362287630405891od_c_a @ ( image_1542064655996835144od_c_a @ collec8405073525749625505od_c_a @ ( image_3449380194339102497_c_a_o @ produc1940289858450342685_c_a_o @ S2 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_432_UNIV__I,axiom,
    ! [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ top_to7710394033828511718od_c_a ) ).

% UNIV_I
thf(fact_433_UNIV__I,axiom,
    ! [X: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X @ top_to8800170863989631014od_c_d ) ).

% UNIV_I
thf(fact_434_UNIV__I,axiom,
    ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% UNIV_I
thf(fact_435_UNIV__I,axiom,
    ! [X: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X @ top_to3206388957835446679od_c_d ) ).

% UNIV_I
thf(fact_436_UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_437_Inter__iff,axiom,
    ! [A2: produc2669932201312201782od_c_a,C2: set_se312849170327590134od_c_a] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( comple2587467249947738909od_c_a @ C2 ) )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ C2 )
           => ( member5502040268208861279od_c_a @ A2 @ X3 ) ) ) ) ).

% Inter_iff
thf(fact_438_Inter__iff,axiom,
    ! [A2: produc1399170498009216118od_c_d,C2: set_se1756390050576932150od_c_d] :
      ( ( member4231278564905875615od_c_d @ A2 @ ( comple3677244080108858205od_c_d @ C2 ) )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ C2 )
           => ( member4231278564905875615od_c_d @ A2 @ X3 ) ) ) ) ).

% Inter_iff
thf(fact_439_Inter__iff,axiom,
    ! [A2: $o,C2: set_set_o] :
      ( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ C2 )
           => ( member_o @ A2 @ X3 ) ) ) ) ).

% Inter_iff
thf(fact_440_Inter__iff,axiom,
    ! [A2: produc2736867335369716327od_c_d,C2: set_se7324893142979633703od_c_d] :
      ( ( member4507666829407384208od_c_d @ A2 @ ( comple5098958804508528846od_c_d @ C2 ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ C2 )
           => ( member4507666829407384208od_c_d @ A2 @ X3 ) ) ) ) ).

% Inter_iff
thf(fact_441_InterI,axiom,
    ! [C2: set_se312849170327590134od_c_a,A2: produc2669932201312201782od_c_a] :
      ( ! [X5: set_Pr8357624050790843030od_c_a] :
          ( ( member9073573073229046463od_c_a @ X5 @ C2 )
         => ( member5502040268208861279od_c_a @ A2 @ X5 ) )
     => ( member5502040268208861279od_c_a @ A2 @ ( comple2587467249947738909od_c_a @ C2 ) ) ) ).

% InterI
thf(fact_442_InterI,axiom,
    ! [C2: set_se1756390050576932150od_c_d,A2: produc1399170498009216118od_c_d] :
      ( ! [X5: set_Pr224028844097186518od_c_d] :
          ( ( member939977866535389951od_c_d @ X5 @ C2 )
         => ( member4231278564905875615od_c_d @ A2 @ X5 ) )
     => ( member4231278564905875615od_c_d @ A2 @ ( comple3677244080108858205od_c_d @ C2 ) ) ) ).

% InterI
thf(fact_443_InterI,axiom,
    ! [C2: set_set_o,A2: $o] :
      ( ! [X5: set_o] :
          ( ( member_set_o @ X5 @ C2 )
         => ( member_o @ A2 @ X5 ) )
     => ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) ) ) ).

% InterI
thf(fact_444_InterI,axiom,
    ! [C2: set_se7324893142979633703od_c_d,A2: produc2736867335369716327od_c_d] :
      ( ! [X5: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X5 @ C2 )
         => ( member4507666829407384208od_c_d @ A2 @ X5 ) )
     => ( member4507666829407384208od_c_d @ A2 @ ( comple5098958804508528846od_c_d @ C2 ) ) ) ).

% InterI
thf(fact_445_Inf__top__conv_I1_J,axiom,
    ! [A2: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A2 )
        = top_top_set_nat )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
           => ( X3 = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_446_Inf__top__conv_I1_J,axiom,
    ! [A2: set_o] :
      ( ( ( complete_Inf_Inf_o @ A2 )
        = top_top_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( X3 = top_top_o ) ) ) ) ).

% Inf_top_conv(1)
thf(fact_447_Inf__top__conv_I2_J,axiom,
    ! [A2: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A2 ) )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
           => ( X3 = top_top_set_nat ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_448_Inf__top__conv_I2_J,axiom,
    ! [A2: set_o] :
      ( ( top_top_o
        = ( complete_Inf_Inf_o @ A2 ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( X3 = top_top_o ) ) ) ) ).

% Inf_top_conv(2)
thf(fact_449_Inter__UNIV__conv_I1_J,axiom,
    ! [A2: set_set_nat] :
      ( ( ( comple7806235888213564991et_nat @ A2 )
        = top_top_set_nat )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
           => ( X3 = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(1)
thf(fact_450_Inter__UNIV__conv_I2_J,axiom,
    ! [A2: set_set_nat] :
      ( ( top_top_set_nat
        = ( comple7806235888213564991et_nat @ A2 ) )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
           => ( X3 = top_top_set_nat ) ) ) ) ).

% Inter_UNIV_conv(2)
thf(fact_451_ImageI,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o,A2: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
     => ( ( member_o @ A @ A2 )
       => ( member_o @ B @ ( image_o_o @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_452_ImageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,A2: set_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ R )
     => ( ( member_a @ A @ A2 )
       => ( member1160383752307813046od_c_a @ B @ ( image_6952875749904162786od_c_a @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_453_ImageI,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,R: set_Pr224028844097186518od_c_d,A2: set_d] :
      ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ A @ B ) @ R )
     => ( ( member_d @ A @ A2 )
       => ( member1373485872446851001od_c_d @ B @ ( image_4235397684752546210od_c_d @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_454_ImageI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: $o,R: set_Pr3612504503696799721_c_a_o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member2207542994726438194_c_a_o @ ( produc4532043784946001145_c_a_o @ A @ B ) @ R )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member_o @ B @ ( image_7960254329004709685_c_a_o @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_455_ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,B: $o,R: set_Pr1828509812422786985_c_d_o,A2: set_Pr224028844097186518od_c_d] :
      ( ( member3595971145540054258_c_d_o @ ( produc5920471935759617209_c_d_o @ A @ B ) @ R )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member_o @ B @ ( image_125310442963549941_c_d_o @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_456_ImageI,axiom,
    ! [A: $o,B: produc2669932201312201782od_c_a,R: set_Pr9081864598239045925od_c_a,A2: set_o] :
      ( ( member1408854235948340334od_c_a @ ( produc7032793852315132213od_c_a @ A @ B ) @ R )
     => ( ( member_o @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( image_1237632359519064945od_c_a @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_457_ImageI,axiom,
    ! [A: $o,B: produc1399170498009216118od_c_d,R: set_Pr1302033441633612133od_c_d,A2: set_o] :
      ( ( member2498631066109459630od_c_d @ ( produc5762032149012146549od_c_d @ A @ B ) @ R )
     => ( ( member_o @ A @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( image_9190242693070855089od_c_d @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_458_ImageI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ R )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( image_6536752330993695315od_c_a @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_459_ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,B: produc2669932201312201782od_c_a,R: set_Pr7917073365334404039od_c_a,A2: set_Pr224028844097186518od_c_d] :
      ( ( member2949693323151135248od_c_a @ ( produc3536205692115927511od_c_a @ A @ B ) @ R )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( image_6068555654898565651od_c_a @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_460_ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,B: produc1399170498009216118od_c_d,R: set_Pr137242208728970247od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ A @ B ) @ R )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( image_4797793951595579987od_c_d @ R @ A2 ) ) ) ) ).

% ImageI
thf(fact_461_swap__simp,axiom,
    ! [X: produc1399170498009216118od_c_d,Y3: produc2669932201312201782od_c_a] :
      ( ( produc5007402825903751031od_c_a @ ( produc3536205692115927511od_c_a @ X @ Y3 ) )
      = ( produc2733640664908071511od_c_d @ Y3 @ X ) ) ).

% swap_simp
thf(fact_462_swap__simp,axiom,
    ! [X: produc2736867335369716327od_c_d,Y3: mappin2657711750964880076od_c_a] :
      ( ( produc3001288147326570556od_c_a @ ( produc2046884284851656156od_c_a @ X @ Y3 ) )
      = ( produc5940215602367092546od_c_d @ Y3 @ X ) ) ).

% swap_simp
thf(fact_463_swap__simp,axiom,
    ! [X: produc2346461404597154448od_c_d,Y3: d] :
      ( ( produc8757345511136871848_c_d_d @ ( produc1727789657345968136_c_d_d @ X @ Y3 ) )
      = ( produc3045323491883059558od_c_d @ Y3 @ X ) ) ).

% swap_simp
thf(fact_464_swap__simp,axiom,
    ! [X: produc2669932201312201782od_c_a,Y3: produc1399170498009216118od_c_d] :
      ( ( produc4204837798695895031od_c_d @ ( produc2733640664908071511od_c_d @ X @ Y3 ) )
      = ( produc3536205692115927511od_c_a @ Y3 @ X ) ) ).

% swap_simp
thf(fact_465_swap__simp,axiom,
    ! [X: mappin2657711750964880076od_c_a,Y3: produc2736867335369716327od_c_d] :
      ( ( produc6894619464842006946od_c_d @ ( produc5940215602367092546od_c_d @ X @ Y3 ) )
      = ( produc2046884284851656156od_c_a @ Y3 @ X ) ) ).

% swap_simp
thf(fact_466_swap__simp,axiom,
    ! [X: d,Y3: produc2346461404597154448od_c_d] :
      ( ( produc851507308819187462od_c_d @ ( produc3045323491883059558od_c_d @ X @ Y3 ) )
      = ( produc1727789657345968136_c_d_d @ Y3 @ X ) ) ).

% swap_simp
thf(fact_467_case__prodI2,axiom,
    ! [P2: produc2736867335369716327od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( ( P2
            = ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc1656800403805021740_c_d_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_468_case__prodI2,axiom,
    ! [P2: produc3360334827542460560od_c_d,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > $o] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2736867335369716327od_c_d] :
          ( ( P2
            = ( produc5940215602367092546od_c_d @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc7497122261040117353_c_d_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_469_case__prodI2,axiom,
    ! [P2: produc1399170498009216118od_c_d,C3: d > produc2346461404597154448od_c_d > $o] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( ( P2
            = ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
         => ( C3 @ A3 @ B3 ) )
     => ( produc2693131752807987549_c_d_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_470_case__prodI,axiom,
    ! [F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( F @ A @ B )
     => ( produc1656800403805021740_c_d_o @ F @ ( produc2733640664908071511od_c_d @ A @ B ) ) ) ).

% case_prodI
thf(fact_471_case__prodI,axiom,
    ! [F: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > $o,A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d] :
      ( ( F @ A @ B )
     => ( produc7497122261040117353_c_d_o @ F @ ( produc5940215602367092546od_c_d @ A @ B ) ) ) ).

% case_prodI
thf(fact_472_case__prodI,axiom,
    ! [F: d > produc2346461404597154448od_c_d > $o,A: d,B: produc2346461404597154448od_c_d] :
      ( ( F @ A @ B )
     => ( produc2693131752807987549_c_d_o @ F @ ( produc3045323491883059558od_c_d @ A @ B ) ) ) ).

% case_prodI
thf(fact_473_INF__identity__eq,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : X3
          @ A2 ) )
      = ( comple5098958804508528846od_c_d @ A2 ) ) ).

% INF_identity_eq
thf(fact_474_INF__identity__eq,axiom,
    ! [A2: set_o] :
      ( ( complete_Inf_Inf_o
        @ ( image_o_o2
          @ ^ [X3: $o] : X3
          @ A2 ) )
      = ( complete_Inf_Inf_o @ A2 ) ) ).

% INF_identity_eq
thf(fact_475_INT__iff,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member5502040268208861279od_c_a @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_476_INT__iff,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member4231278564905875615od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_477_INT__iff,axiom,
    ! [B: $o,B2: $o > set_o,A2: set_o] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member_o @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_478_INT__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_479_INT__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_480_INT__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( member4507666829407384208od_c_d @ B @ ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ B @ ( B2 @ X3 ) ) ) ) ) ).

% INT_iff
thf(fact_481_INT__I,axiom,
    ! [A2: set_o,B: $o,B2: $o > set_o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A2 )
         => ( member_o @ B @ ( B2 @ X2 ) ) )
     => ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_482_INT__I,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B: $o,B2: produc2669932201312201782od_c_a > set_o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ A2 )
         => ( member_o @ B @ ( B2 @ X2 ) ) )
     => ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_483_INT__I,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B: $o,B2: produc1399170498009216118od_c_d > set_o] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ A2 )
         => ( member_o @ B @ ( B2 @ X2 ) ) )
     => ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_484_INT__I,axiom,
    ! [A2: set_o,B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A2 )
         => ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) )
     => ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_485_INT__I,axiom,
    ! [A2: set_o,B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A2 )
         => ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) )
     => ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_486_INT__I,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ A2 )
         => ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) )
     => ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_487_INT__I,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B: produc1399170498009216118od_c_d,B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ A2 )
         => ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) )
     => ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_488_INT__I,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B: produc2669932201312201782od_c_a,B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ A2 )
         => ( member5502040268208861279od_c_a @ B @ ( B2 @ X2 ) ) )
     => ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_489_INT__I,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B: produc1399170498009216118od_c_d,B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ A2 )
         => ( member4231278564905875615od_c_d @ B @ ( B2 @ X2 ) ) )
     => ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_490_INT__I,axiom,
    ! [A2: set_o,B: produc2736867335369716327od_c_d,B2: $o > set_Pr2898135061646675015od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A2 )
         => ( member4507666829407384208od_c_d @ B @ ( B2 @ X2 ) ) )
     => ( member4507666829407384208od_c_d @ B @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) ) ) ).

% INT_I
thf(fact_491_Sup__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Sup_UNIV
thf(fact_492_Sup__UNIV,axiom,
    ( ( comple3966289243721909940od_c_d @ top_to8663423184816037751od_c_d )
    = top_to3206388957835446679od_c_d ) ).

% Sup_UNIV
thf(fact_493_Sup__UNIV,axiom,
    ( ( complete_Sup_Sup_o @ top_top_set_o )
    = top_top_o ) ).

% Sup_UNIV
thf(fact_494_Sup__UNIV,axiom,
    ( ( comple90263536869209701_set_o @ top_top_set_set_o )
    = top_top_set_o ) ).

% Sup_UNIV
thf(fact_495_Sup__UNIV,axiom,
    ( ( comple7812139117791525187od_c_d @ top_to2093899158997654150od_c_d )
    = top_to8800170863989631014od_c_d ) ).

% Sup_UNIV
thf(fact_496_Sup__UNIV,axiom,
    ( ( comple6722362287630405891od_c_a @ top_to650358278748312134od_c_a )
    = top_to7710394033828511718od_c_a ) ).

% Sup_UNIV
thf(fact_497_bex__UNIV,axiom,
    ! [P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ top_top_set_nat )
            & ( P @ X3 ) ) )
      = ( ? [X7: nat] : ( P @ X7 ) ) ) ).

% bex_UNIV
thf(fact_498_INF__top,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [X3: produc2669932201312201782od_c_a] : top_to3206388957835446679od_c_d
          @ A2 ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_top
thf(fact_499_INF__top,axiom,
    ! [A2: set_o] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [X3: $o] : top_to3206388957835446679od_c_d
          @ A2 ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_top
thf(fact_500_INF__top,axiom,
    ! [A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : top_to8800170863989631014od_c_d
          @ A2 ) )
      = top_to8800170863989631014od_c_d ) ).

% INF_top
thf(fact_501_INF__top,axiom,
    ! [A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : top_to7710394033828511718od_c_a
          @ A2 ) )
      = top_to7710394033828511718od_c_a ) ).

% INF_top
thf(fact_502_INF__top,axiom,
    ! [A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : top_top_set_o
          @ A2 ) )
      = top_top_set_o ) ).

% INF_top
thf(fact_503_INF__top,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : top_to3206388957835446679od_c_d
          @ A2 ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_top
thf(fact_504_INF__top__conv_I1_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(1)
thf(fact_505_INF__top__conv_I1_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(1)
thf(fact_506_INF__top__conv_I1_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
        = top_to8800170863989631014od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to8800170863989631014od_c_d ) ) ) ) ).

% INF_top_conv(1)
thf(fact_507_INF__top__conv_I1_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
        = top_to7710394033828511718od_c_a )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to7710394033828511718od_c_a ) ) ) ) ).

% INF_top_conv(1)
thf(fact_508_INF__top__conv_I1_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) )
        = top_top_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_top_set_o ) ) ) ) ).

% INF_top_conv(1)
thf(fact_509_INF__top__conv_I1_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(1)
thf(fact_510_INF__top__conv_I2_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(2)
thf(fact_511_INF__top__conv_I2_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(2)
thf(fact_512_INF__top__conv_I2_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( top_to8800170863989631014od_c_d
        = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to8800170863989631014od_c_d ) ) ) ) ).

% INF_top_conv(2)
thf(fact_513_INF__top__conv_I2_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( top_to7710394033828511718od_c_a
        = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to7710394033828511718od_c_a ) ) ) ) ).

% INF_top_conv(2)
thf(fact_514_INF__top__conv_I2_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( top_top_set_o
        = ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_top_set_o ) ) ) ) ).

% INF_top_conv(2)
thf(fact_515_INF__top__conv_I2_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INF_top_conv(2)
thf(fact_516_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > $o] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_2427823168983691879_c_a_o @ ( produc1940289858450342685_c_a_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_517_pair__imageI,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,A2: set_Pr224028844097186518od_c_d,F: d > produc2346461404597154448od_c_d > $o] :
      ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_3816251319797307943_c_d_o @ ( produc2693131752807987549_c_d_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_518_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A2 )
     => ( member5502040268208861279od_c_a @ ( F @ A @ B ) @ ( image_2169818278710580869od_c_a @ ( produc4268383296163969595od_c_a @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_519_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A2 )
     => ( member4231278564905875615od_c_d @ ( F @ A @ B ) @ ( image_899056575407595205od_c_d @ ( produc2997621592860983931od_c_d @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_520_pair__imageI,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,A2: set_Pr224028844097186518od_c_d,F: d > produc2346461404597154448od_c_d > produc2669932201312201782od_c_a] :
      ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ A @ B ) @ A2 )
     => ( member5502040268208861279od_c_a @ ( F @ A @ B ) @ ( image_1701621602615451205od_c_a @ ( produc228941528955932795od_c_a @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_521_pair__imageI,axiom,
    ! [A: d,B: produc2346461404597154448od_c_d,A2: set_Pr224028844097186518od_c_d,F: d > produc2346461404597154448od_c_d > produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ A @ B ) @ A2 )
     => ( member4231278564905875615od_c_d @ ( F @ A @ B ) @ ( image_430859899312465541od_c_d @ ( produc8181551862507722939od_c_d @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_522_pair__imageI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d,A2: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o] :
      ( ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ A @ B ) @ A2 )
     => ( member_o @ ( F @ A @ B ) @ ( image_1680233356757228022_c_d_o @ ( produc1656800403805021740_c_d_o @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_523_pair__imageI,axiom,
    ! [A: a,B: produc2133359284458116493od_c_a,A2: set_Pr8357624050790843030od_c_a,F: a > produc2133359284458116493od_c_a > produc2736867335369716327od_c_d] :
      ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ A2 )
     => ( member4507666829407384208od_c_d @ ( F @ A @ B ) @ ( image_3416532124207988918od_c_d @ ( produc9215568718939491436od_c_d @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_524_pair__imageI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d,A2: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > produc2669932201312201782od_c_a] :
      ( ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ A @ B ) @ A2 )
     => ( member5502040268208861279od_c_a @ ( F @ A @ B ) @ ( image_5572119458559193492od_c_a @ ( produc8148172013387288778od_c_a @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_525_pair__imageI,axiom,
    ! [A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d,A2: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > produc1399170498009216118od_c_d] :
      ( ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ A @ B ) @ A2 )
     => ( member4231278564905875615od_c_d @ ( F @ A @ B ) @ ( image_4301357755256207828od_c_d @ ( produc6877410310084303114od_c_d @ F ) @ A2 ) ) ) ).

% pair_imageI
thf(fact_526_Image__Collect__case__prod,axiom,
    ! [P: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( image_5265990627690709651od_c_d @ ( collec2789978228259155154od_c_d @ ( produc1656800403805021740_c_d_o @ P ) ) @ A2 )
      = ( collec7134311822446639841od_c_d
        @ ^ [Y2: produc1399170498009216118od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ( P @ X3 @ Y2 ) ) ) ) ).

% Image_Collect_case_prod
thf(fact_527_reflp__refl__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( reflp_on_nat @ top_top_set_nat
        @ ^ [X3: nat,Y2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R ) )
      = ( refl_on_nat @ top_top_set_nat @ R ) ) ).

% reflp_refl_eq
thf(fact_528_reflp__on__mono,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
      ( ( reflp_391139636225337927od_c_a @ A2 @ R2 )
     => ( ! [X2: produc2669932201312201782od_c_a,Y: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ A2 )
           => ( ( member5502040268208861279od_c_a @ Y @ A2 )
             => ( ( R2 @ X2 @ Y )
               => ( Q2 @ X2 @ Y ) ) ) )
       => ( reflp_391139636225337927od_c_a @ A2 @ Q2 ) ) ) ).

% reflp_on_mono
thf(fact_529_reflp__on__mono,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R2: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o,Q2: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o] :
      ( ( reflp_8343749969777128071od_c_d @ A2 @ R2 )
     => ( ! [X2: produc1399170498009216118od_c_d,Y: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ A2 )
           => ( ( member4231278564905875615od_c_d @ Y @ A2 )
             => ( ( R2 @ X2 @ Y )
               => ( Q2 @ X2 @ Y ) ) ) )
       => ( reflp_8343749969777128071od_c_d @ A2 @ Q2 ) ) ) ).

% reflp_on_mono
thf(fact_530_reflp__on__mono,axiom,
    ! [A2: set_o,R2: $o > $o > $o,Q2: $o > $o > $o] :
      ( ( reflp_on_o @ A2 @ R2 )
     => ( ! [X2: $o,Y: $o] :
            ( ( member_o @ X2 @ A2 )
           => ( ( member_o @ Y @ A2 )
             => ( ( R2 @ X2 @ Y )
               => ( Q2 @ X2 @ Y ) ) ) )
       => ( reflp_on_o @ A2 @ Q2 ) ) ) ).

% reflp_on_mono
thf(fact_531_reflp__on__mono,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R2: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o,Q2: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ( reflp_4191898174370630776od_c_d @ A2 @ R2 )
     => ( ! [X2: produc2736867335369716327od_c_d,Y: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X2 @ A2 )
           => ( ( member4507666829407384208od_c_d @ Y @ A2 )
             => ( ( R2 @ X2 @ Y )
               => ( Q2 @ X2 @ Y ) ) ) )
       => ( reflp_4191898174370630776od_c_d @ A2 @ Q2 ) ) ) ).

% reflp_on_mono
thf(fact_532_reflp__mono,axiom,
    ! [R2: nat > nat > $o,Q2: nat > nat > $o] :
      ( ( reflp_on_nat @ top_top_set_nat @ R2 )
     => ( ! [X2: nat,Y: nat] :
            ( ( R2 @ X2 @ Y )
           => ( Q2 @ X2 @ Y ) )
       => ( reflp_on_nat @ top_top_set_nat @ Q2 ) ) ) ).

% reflp_mono
thf(fact_533_reflp__onI,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ A2 )
         => ( R2 @ X2 @ X2 ) )
     => ( reflp_391139636225337927od_c_a @ A2 @ R2 ) ) ).

% reflp_onI
thf(fact_534_reflp__onI,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R2: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ A2 )
         => ( R2 @ X2 @ X2 ) )
     => ( reflp_8343749969777128071od_c_d @ A2 @ R2 ) ) ).

% reflp_onI
thf(fact_535_reflp__onI,axiom,
    ! [A2: set_o,R2: $o > $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A2 )
         => ( R2 @ X2 @ X2 ) )
     => ( reflp_on_o @ A2 @ R2 ) ) ).

% reflp_onI
thf(fact_536_reflp__onI,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R2: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X2 @ A2 )
         => ( R2 @ X2 @ X2 ) )
     => ( reflp_4191898174370630776od_c_d @ A2 @ R2 ) ) ).

% reflp_onI
thf(fact_537_reflp__onD,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R2: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o,X: produc2669932201312201782od_c_a] :
      ( ( reflp_391139636225337927od_c_a @ A2 @ R2 )
     => ( ( member5502040268208861279od_c_a @ X @ A2 )
       => ( R2 @ X @ X ) ) ) ).

% reflp_onD
thf(fact_538_reflp__onD,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R2: produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o,X: produc1399170498009216118od_c_d] :
      ( ( reflp_8343749969777128071od_c_d @ A2 @ R2 )
     => ( ( member4231278564905875615od_c_d @ X @ A2 )
       => ( R2 @ X @ X ) ) ) ).

% reflp_onD
thf(fact_539_reflp__onD,axiom,
    ! [A2: set_o,R2: $o > $o > $o,X: $o] :
      ( ( reflp_on_o @ A2 @ R2 )
     => ( ( member_o @ X @ A2 )
       => ( R2 @ X @ X ) ) ) ).

% reflp_onD
thf(fact_540_reflp__onD,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R2: produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o,X: produc2736867335369716327od_c_d] :
      ( ( reflp_4191898174370630776od_c_d @ A2 @ R2 )
     => ( ( member4507666829407384208od_c_d @ X @ A2 )
       => ( R2 @ X @ X ) ) ) ).

% reflp_onD
thf(fact_541_UNIV__witness,axiom,
    ? [X2: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X2 @ top_to7710394033828511718od_c_a ) ).

% UNIV_witness
thf(fact_542_UNIV__witness,axiom,
    ? [X2: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X2 @ top_to8800170863989631014od_c_d ) ).

% UNIV_witness
thf(fact_543_UNIV__witness,axiom,
    ? [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_544_UNIV__witness,axiom,
    ? [X2: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X2 @ top_to3206388957835446679od_c_d ) ).

% UNIV_witness
thf(fact_545_UNIV__witness,axiom,
    ? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_546_reflpI,axiom,
    ! [R2: nat > nat > $o] :
      ( ! [X2: nat] : ( R2 @ X2 @ X2 )
     => ( reflp_on_nat @ top_top_set_nat @ R2 ) ) ).

% reflpI
thf(fact_547_reflpE,axiom,
    ! [R: nat > nat > $o,X: nat] :
      ( ( reflp_on_nat @ top_top_set_nat @ R )
     => ( R @ X @ X ) ) ).

% reflpE
thf(fact_548_reflpD,axiom,
    ! [R2: nat > nat > $o,X: nat] :
      ( ( reflp_on_nat @ top_top_set_nat @ R2 )
     => ( R2 @ X @ X ) ) ).

% reflpD
thf(fact_549_UNIV__eq__I,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ! [X2: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X2 @ A2 )
     => ( top_to7710394033828511718od_c_a = A2 ) ) ).

% UNIV_eq_I
thf(fact_550_UNIV__eq__I,axiom,
    ! [A2: set_Pr224028844097186518od_c_d] :
      ( ! [X2: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X2 @ A2 )
     => ( top_to8800170863989631014od_c_d = A2 ) ) ).

% UNIV_eq_I
thf(fact_551_UNIV__eq__I,axiom,
    ! [A2: set_o] :
      ( ! [X2: $o] : ( member_o @ X2 @ A2 )
     => ( top_top_set_o = A2 ) ) ).

% UNIV_eq_I
thf(fact_552_UNIV__eq__I,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d] :
      ( ! [X2: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X2 @ A2 )
     => ( top_to3206388957835446679od_c_d = A2 ) ) ).

% UNIV_eq_I
thf(fact_553_UNIV__eq__I,axiom,
    ! [A2: set_nat] :
      ( ! [X2: nat] : ( member_nat @ X2 @ A2 )
     => ( top_top_set_nat = A2 ) ) ).

% UNIV_eq_I
thf(fact_554_Bex__def,axiom,
    ( bex_Pr8276177465575388054od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,P3: produc2669932201312201782od_c_a > $o] :
        ? [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_555_Bex__def,axiom,
    ( bex_Pr7005415762272402390od_c_d
    = ( ^ [A5: set_Pr224028844097186518od_c_d,P3: produc1399170498009216118od_c_d > $o] :
        ? [X3: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_556_Bex__def,axiom,
    ( bex_o
    = ( ^ [A5: set_o,P3: $o > $o] :
        ? [X3: $o] :
          ( ( member_o @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_557_Bex__def,axiom,
    ( bex_Pr679877479140882887od_c_d
    = ( ^ [A5: set_Pr2898135061646675015od_c_d,P3: produc2736867335369716327od_c_d > $o] :
        ? [X3: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_558_InterE,axiom,
    ! [A2: produc2669932201312201782od_c_a,C2: set_se312849170327590134od_c_a,X4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( comple2587467249947738909od_c_a @ C2 ) )
     => ( ( member9073573073229046463od_c_a @ X4 @ C2 )
       => ( member5502040268208861279od_c_a @ A2 @ X4 ) ) ) ).

% InterE
thf(fact_559_InterE,axiom,
    ! [A2: produc1399170498009216118od_c_d,C2: set_se1756390050576932150od_c_d,X4: set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ A2 @ ( comple3677244080108858205od_c_d @ C2 ) )
     => ( ( member939977866535389951od_c_d @ X4 @ C2 )
       => ( member4231278564905875615od_c_d @ A2 @ X4 ) ) ) ).

% InterE
thf(fact_560_InterE,axiom,
    ! [A2: $o,C2: set_set_o,X4: set_o] :
      ( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
     => ( ( member_set_o @ X4 @ C2 )
       => ( member_o @ A2 @ X4 ) ) ) ).

% InterE
thf(fact_561_InterE,axiom,
    ! [A2: produc2736867335369716327od_c_d,C2: set_se7324893142979633703od_c_d,X4: set_Pr2898135061646675015od_c_d] :
      ( ( member4507666829407384208od_c_d @ A2 @ ( comple5098958804508528846od_c_d @ C2 ) )
     => ( ( member4652649041329350768od_c_d @ X4 @ C2 )
       => ( member4507666829407384208od_c_d @ A2 @ X4 ) ) ) ).

% InterE
thf(fact_562_InterD,axiom,
    ! [A2: produc2669932201312201782od_c_a,C2: set_se312849170327590134od_c_a,X4: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ A2 @ ( comple2587467249947738909od_c_a @ C2 ) )
     => ( ( member9073573073229046463od_c_a @ X4 @ C2 )
       => ( member5502040268208861279od_c_a @ A2 @ X4 ) ) ) ).

% InterD
thf(fact_563_InterD,axiom,
    ! [A2: produc1399170498009216118od_c_d,C2: set_se1756390050576932150od_c_d,X4: set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ A2 @ ( comple3677244080108858205od_c_d @ C2 ) )
     => ( ( member939977866535389951od_c_d @ X4 @ C2 )
       => ( member4231278564905875615od_c_d @ A2 @ X4 ) ) ) ).

% InterD
thf(fact_564_InterD,axiom,
    ! [A2: $o,C2: set_set_o,X4: set_o] :
      ( ( member_o @ A2 @ ( comple3063163877087187839_set_o @ C2 ) )
     => ( ( member_set_o @ X4 @ C2 )
       => ( member_o @ A2 @ X4 ) ) ) ).

% InterD
thf(fact_565_InterD,axiom,
    ! [A2: produc2736867335369716327od_c_d,C2: set_se7324893142979633703od_c_d,X4: set_Pr2898135061646675015od_c_d] :
      ( ( member4507666829407384208od_c_d @ A2 @ ( comple5098958804508528846od_c_d @ C2 ) )
     => ( ( member4652649041329350768od_c_d @ X4 @ C2 )
       => ( member4507666829407384208od_c_d @ A2 @ X4 ) ) ) ).

% InterD
thf(fact_566_reflI,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
     => ( refl_on_nat @ top_top_set_nat @ R ) ) ).

% reflI
thf(fact_567_reflD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_568_Union__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Union_UNIV
thf(fact_569_Union__UNIV,axiom,
    ( ( comple3966289243721909940od_c_d @ top_to8663423184816037751od_c_d )
    = top_to3206388957835446679od_c_d ) ).

% Union_UNIV
thf(fact_570_Union__UNIV,axiom,
    ( ( comple90263536869209701_set_o @ top_top_set_set_o )
    = top_top_set_o ) ).

% Union_UNIV
thf(fact_571_Union__UNIV,axiom,
    ( ( comple7812139117791525187od_c_d @ top_to2093899158997654150od_c_d )
    = top_to8800170863989631014od_c_d ) ).

% Union_UNIV
thf(fact_572_Union__UNIV,axiom,
    ( ( comple6722362287630405891od_c_a @ top_to650358278748312134od_c_a )
    = top_to7710394033828511718od_c_a ) ).

% Union_UNIV
thf(fact_573_Image__def,axiom,
    ( image_6952875749904162786od_c_a
    = ( ^ [R3: set_Pr8357624050790843030od_c_a,S3: set_a] :
          ( collec3135042693873630968od_c_a
          @ ^ [Y2: produc2133359284458116493od_c_a] :
            ? [X3: a] :
              ( ( member_a @ X3 @ S3 )
              & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ R3 ) ) ) ) ) ).

% Image_def
thf(fact_574_Image__def,axiom,
    ( image_5265990627690709651od_c_d
    = ( ^ [R3: set_Pr2898135061646675015od_c_d,S3: set_Pr8357624050790843030od_c_a] :
          ( collec7134311822446639841od_c_d
          @ ^ [Y2: produc1399170498009216118od_c_d] :
            ? [X3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X3 @ S3 )
              & ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ R3 ) ) ) ) ) ).

% Image_def
thf(fact_575_Image__def,axiom,
    ( image_8957767082753035270od_c_d
    = ( ^ [R3: set_Pr5052118491147477062od_c_d,S3: set_ma4406981705840051330od_c_a] :
          ( collec2789978228259155154od_c_d
          @ ^ [Y2: produc2736867335369716327od_c_d] :
            ? [X3: mappin2657711750964880076od_c_a] :
              ( ( member7151184318602930019od_c_a @ X3 @ S3 )
              & ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ R3 ) ) ) ) ) ).

% Image_def
thf(fact_576_Image__def,axiom,
    ( image_4235397684752546210od_c_d
    = ( ^ [R3: set_Pr224028844097186518od_c_d,S3: set_d] :
          ( collec3348144814012668923od_c_d
          @ ^ [Y2: produc2346461404597154448od_c_d] :
            ? [X3: d] :
              ( ( member_d @ X3 @ S3 )
              & ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ R3 ) ) ) ) ) ).

% Image_def
thf(fact_577_UNIV__def,axiom,
    ( top_to3206388957835446679od_c_d
    = ( collec2789978228259155154od_c_d
      @ ^ [X3: produc2736867335369716327od_c_d] : $true ) ) ).

% UNIV_def
thf(fact_578_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X3: nat] : $true ) ) ).

% UNIV_def
thf(fact_579_refl__on__INTER,axiom,
    ! [S: set_Pr8357624050790843030od_c_a,A2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,R: produc2669932201312201782od_c_a > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ A2 @ S ) ) @ ( comple7123888634065404238od_c_d @ ( image_5364838190352781078od_c_d @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_580_refl__on__INTER,axiom,
    ! [S: set_o,A2: $o > set_Pr2898135061646675015od_c_d,R: $o > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ A2 @ S ) ) @ ( comple7123888634065404238od_c_d @ ( image_5794988778541631284od_c_d @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_581_refl__on__INTER,axiom,
    ! [S: set_o,A2: $o > set_Pr224028844097186518od_c_d,R: $o > set_Pr137242208728970247od_c_d] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o899880292924752245od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o899880292924752245od_c_d @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ A2 @ S ) ) @ ( comple2338065951590824078od_c_d @ ( image_6214733672829972084od_c_d @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_582_refl__on__INTER,axiom,
    ! [S: set_o,A2: $o > set_Pr8357624050790843030od_c_a,R: $o > set_Pr1454594181397332999od_c_a] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_o2170641996227737909od_c_a @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2170641996227737909od_c_a @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ A2 @ S ) ) @ ( comple3655417924259186830od_c_a @ ( image_7532085645498334836od_c_a @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_583_refl__on__INTER,axiom,
    ! [S: set_o,A2: $o > set_o,R: $o > set_Product_prod_o_o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( refl_on_o @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_on_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ S ) ) @ ( comple1697049258504608654od_o_o @ ( image_5520585345982086004od_o_o @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_584_refl__on__INTER,axiom,
    ! [S: set_se7324893142979633703od_c_d,A2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,R: set_Pr2898135061646675015od_c_d > set_Pr8731163265463710919od_c_d] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ S )
         => ( refl_o2973842278997123942od_c_d @ ( A2 @ X2 ) @ ( R @ X2 ) ) )
     => ( refl_o2973842278997123942od_c_d @ ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ A2 @ S ) ) @ ( comple7123888634065404238od_c_d @ ( image_914099667705184325od_c_d @ R @ S ) ) ) ) ).

% refl_on_INTER
thf(fact_585_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_586_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_587_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( top_to8800170863989631014od_c_d
        = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to8800170863989631014od_c_d ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_588_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( top_to7710394033828511718od_c_a
        = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to7710394033828511718od_c_a ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_589_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( top_top_set_o
        = ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_top_set_o ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_590_INTER__UNIV__conv_I1_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( top_to3206388957835446679od_c_d
        = ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(1)
thf(fact_591_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_592_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_593_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
        = top_to8800170863989631014od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to8800170863989631014od_c_d ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_594_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
        = top_to7710394033828511718od_c_a )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to7710394033828511718od_c_a ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_595_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) )
        = top_top_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_top_set_o ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_596_INTER__UNIV__conv_I2_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
        = top_to3206388957835446679od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = top_to3206388957835446679od_c_d ) ) ) ) ).

% INTER_UNIV_conv(2)
thf(fact_597_rev__ImageI,axiom,
    ! [A: $o,A2: set_o,B: $o,R: set_Product_prod_o_o] :
      ( ( member_o @ A @ A2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
       => ( member_o @ B @ ( image_o_o @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_598_rev__ImageI,axiom,
    ! [A: a,A2: set_a,B: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a] :
      ( ( member_a @ A @ A2 )
     => ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ A @ B ) @ R )
       => ( member1160383752307813046od_c_a @ B @ ( image_6952875749904162786od_c_a @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_599_rev__ImageI,axiom,
    ! [A: d,A2: set_d,B: produc2346461404597154448od_c_d,R: set_Pr224028844097186518od_c_d] :
      ( ( member_d @ A @ A2 )
     => ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ A @ B ) @ R )
       => ( member1373485872446851001od_c_d @ B @ ( image_4235397684752546210od_c_d @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_600_rev__ImageI,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: $o,R: set_Pr3612504503696799721_c_a_o] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ( member2207542994726438194_c_a_o @ ( produc4532043784946001145_c_a_o @ A @ B ) @ R )
       => ( member_o @ B @ ( image_7960254329004709685_c_a_o @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_601_rev__ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: $o,R: set_Pr1828509812422786985_c_d_o] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member3595971145540054258_c_d_o @ ( produc5920471935759617209_c_d_o @ A @ B ) @ R )
       => ( member_o @ B @ ( image_125310442963549941_c_d_o @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_602_rev__ImageI,axiom,
    ! [A: $o,A2: set_o,B: produc2669932201312201782od_c_a,R: set_Pr9081864598239045925od_c_a] :
      ( ( member_o @ A @ A2 )
     => ( ( member1408854235948340334od_c_a @ ( produc7032793852315132213od_c_a @ A @ B ) @ R )
       => ( member5502040268208861279od_c_a @ B @ ( image_1237632359519064945od_c_a @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_603_rev__ImageI,axiom,
    ! [A: $o,A2: set_o,B: produc1399170498009216118od_c_d,R: set_Pr1302033441633612133od_c_d] :
      ( ( member_o @ A @ A2 )
     => ( ( member2498631066109459630od_c_d @ ( produc5762032149012146549od_c_d @ A @ B ) @ R )
       => ( member4231278564905875615od_c_d @ B @ ( image_9190242693070855089od_c_d @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_604_rev__ImageI,axiom,
    ! [A: produc2669932201312201782od_c_a,A2: set_Pr8357624050790843030od_c_a,B: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a] :
      ( ( member5502040268208861279od_c_a @ A @ A2 )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ R )
       => ( member5502040268208861279od_c_a @ B @ ( image_6536752330993695315od_c_a @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_605_rev__ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: produc2669932201312201782od_c_a,R: set_Pr7917073365334404039od_c_a] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member2949693323151135248od_c_a @ ( produc3536205692115927511od_c_a @ A @ B ) @ R )
       => ( member5502040268208861279od_c_a @ B @ ( image_6068555654898565651od_c_a @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_606_rev__ImageI,axiom,
    ! [A: produc1399170498009216118od_c_d,A2: set_Pr224028844097186518od_c_d,B: produc1399170498009216118od_c_d,R: set_Pr137242208728970247od_c_d] :
      ( ( member4231278564905875615od_c_d @ A @ A2 )
     => ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ A @ B ) @ R )
       => ( member4231278564905875615od_c_d @ B @ ( image_4797793951595579987od_c_d @ R @ A2 ) ) ) ) ).

% rev_ImageI
thf(fact_607_Image__iff,axiom,
    ! [B: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,A2: set_a] :
      ( ( member1160383752307813046od_c_a @ B @ ( image_6952875749904162786od_c_a @ R @ A2 ) )
      = ( ? [X3: a] :
            ( ( member_a @ X3 @ A2 )
            & ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_608_Image__iff,axiom,
    ! [B: produc1399170498009216118od_c_d,R: set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member4231278564905875615od_c_d @ B @ ( image_5265990627690709651od_c_d @ R @ A2 ) )
      = ( ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_609_Image__iff,axiom,
    ! [B: produc2736867335369716327od_c_d,R: set_Pr5052118491147477062od_c_d,A2: set_ma4406981705840051330od_c_a] :
      ( ( member4507666829407384208od_c_d @ B @ ( image_8957767082753035270od_c_d @ R @ A2 ) )
      = ( ? [X3: mappin2657711750964880076od_c_a] :
            ( ( member7151184318602930019od_c_a @ X3 @ A2 )
            & ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_610_Image__iff,axiom,
    ! [B: produc2346461404597154448od_c_d,R: set_Pr224028844097186518od_c_d,A2: set_d] :
      ( ( member1373485872446851001od_c_d @ B @ ( image_4235397684752546210od_c_d @ R @ A2 ) )
      = ( ? [X3: d] :
            ( ( member_d @ X3 @ A2 )
            & ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ B ) @ R ) ) ) ) ).

% Image_iff
thf(fact_611_ImageE,axiom,
    ! [B: $o,R: set_Product_prod_o_o,A2: set_o] :
      ( ( member_o @ B @ ( image_o_o @ R @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ B ) @ R )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_612_ImageE,axiom,
    ! [B: produc2133359284458116493od_c_a,R: set_Pr8357624050790843030od_c_a,A2: set_a] :
      ( ( member1160383752307813046od_c_a @ B @ ( image_6952875749904162786od_c_a @ R @ A2 ) )
     => ~ ! [X2: a] :
            ( ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X2 @ B ) @ R )
           => ~ ( member_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_613_ImageE,axiom,
    ! [B: produc2346461404597154448od_c_d,R: set_Pr224028844097186518od_c_d,A2: set_d] :
      ( ( member1373485872446851001od_c_d @ B @ ( image_4235397684752546210od_c_d @ R @ A2 ) )
     => ~ ! [X2: d] :
            ( ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X2 @ B ) @ R )
           => ~ ( member_d @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_614_ImageE,axiom,
    ! [B: produc2669932201312201782od_c_a,R: set_Pr9081864598239045925od_c_a,A2: set_o] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_1237632359519064945od_c_a @ R @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( member1408854235948340334od_c_a @ ( produc7032793852315132213od_c_a @ X2 @ B ) @ R )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_615_ImageE,axiom,
    ! [B: produc1399170498009216118od_c_d,R: set_Pr1302033441633612133od_c_d,A2: set_o] :
      ( ( member4231278564905875615od_c_d @ B @ ( image_9190242693070855089od_c_d @ R @ A2 ) )
     => ~ ! [X2: $o] :
            ( ( member2498631066109459630od_c_d @ ( produc5762032149012146549od_c_d @ X2 @ B ) @ R )
           => ~ ( member_o @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_616_ImageE,axiom,
    ! [B: $o,R: set_Pr3612504503696799721_c_a_o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member_o @ B @ ( image_7960254329004709685_c_a_o @ R @ A2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( member2207542994726438194_c_a_o @ ( produc4532043784946001145_c_a_o @ X2 @ B ) @ R )
           => ~ ( member5502040268208861279od_c_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_617_ImageE,axiom,
    ! [B: $o,R: set_Pr1828509812422786985_c_d_o,A2: set_Pr224028844097186518od_c_d] :
      ( ( member_o @ B @ ( image_125310442963549941_c_d_o @ R @ A2 ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member3595971145540054258_c_d_o @ ( produc5920471935759617209_c_d_o @ X2 @ B ) @ R )
           => ~ ( member4231278564905875615od_c_d @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_618_ImageE,axiom,
    ! [B: produc2669932201312201782od_c_a,R: set_Pr1454594181397332999od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_6536752330993695315od_c_a @ R @ A2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X2 @ B ) @ R )
           => ~ ( member5502040268208861279od_c_a @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_619_ImageE,axiom,
    ! [B: produc2669932201312201782od_c_a,R: set_Pr7917073365334404039od_c_a,A2: set_Pr224028844097186518od_c_d] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_6068555654898565651od_c_a @ R @ A2 ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member2949693323151135248od_c_a @ ( produc3536205692115927511od_c_a @ X2 @ B ) @ R )
           => ~ ( member4231278564905875615od_c_d @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_620_ImageE,axiom,
    ! [B: produc1399170498009216118od_c_d,R: set_Pr137242208728970247od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( member4231278564905875615od_c_d @ B @ ( image_4797793951595579987od_c_d @ R @ A2 ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ X2 @ B ) @ R )
           => ~ ( member4231278564905875615od_c_d @ X2 @ A2 ) ) ) ).

% ImageE
thf(fact_621_INF__cong,axiom,
    ! [A2: set_se7324893142979633703od_c_d,B2: set_se7324893142979633703od_c_d,C2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,D2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ C2 @ A2 ) )
          = ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_622_INF__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,D2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ C2 @ A2 ) )
          = ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_623_INF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr2898135061646675015od_c_d,D2: $o > set_Pr2898135061646675015od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ C2 @ A2 ) )
          = ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_624_INF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr224028844097186518od_c_d,D2: $o > set_Pr224028844097186518od_c_d] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ C2 @ A2 ) )
          = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_625_INF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_Pr8357624050790843030od_c_a,D2: $o > set_Pr8357624050790843030od_c_a] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ A2 ) )
          = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_626_INF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > set_o,D2: $o > set_o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ C2 @ A2 ) )
          = ( comple3063163877087187839_set_o @ ( image_o_set_o @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_627_INF__cong,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,B2: set_Pr8357624050790843030od_c_a,C2: produc2669932201312201782od_c_a > $o,D2: produc2669932201312201782od_c_a > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Inf_Inf_o @ ( image_2427823168983691879_c_a_o @ C2 @ A2 ) )
          = ( complete_Inf_Inf_o @ ( image_2427823168983691879_c_a_o @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_628_INF__cong,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,B2: set_Pr224028844097186518od_c_d,C2: produc1399170498009216118od_c_d > $o,D2: produc1399170498009216118od_c_d > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Inf_Inf_o @ ( image_3816251319797307943_c_d_o @ C2 @ A2 ) )
          = ( complete_Inf_Inf_o @ ( image_3816251319797307943_c_d_o @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_629_INF__cong,axiom,
    ! [A2: set_o,B2: set_o,C2: $o > $o,D2: $o > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Inf_Inf_o @ ( image_o_o2 @ C2 @ A2 ) )
          = ( complete_Inf_Inf_o @ ( image_o_o2 @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_630_INF__cong,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,B2: set_Pr2898135061646675015od_c_d,C2: produc2736867335369716327od_c_d > $o,D2: produc2736867335369716327od_c_d > $o] :
      ( ( A2 = B2 )
     => ( ! [X2: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X2 @ B2 )
           => ( ( C2 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( complete_Inf_Inf_o @ ( image_1680233356757228022_c_d_o @ C2 @ A2 ) )
          = ( complete_Inf_Inf_o @ ( image_1680233356757228022_c_d_o @ D2 @ B2 ) ) ) ) ) ).

% INF_cong
thf(fact_631_refl__onD2,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R: set_Pr1454594181397332999od_c_a,X: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( refl_o2170641996227737909od_c_a @ A2 @ R )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Y3 ) @ R )
       => ( member5502040268208861279od_c_a @ Y3 @ A2 ) ) ) ).

% refl_onD2
thf(fact_632_refl__onD2,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R: set_Pr137242208728970247od_c_d,X: produc1399170498009216118od_c_d,Y3: produc1399170498009216118od_c_d] :
      ( ( refl_o899880292924752245od_c_d @ A2 @ R )
     => ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ X @ Y3 ) @ R )
       => ( member4231278564905875615od_c_d @ Y3 @ A2 ) ) ) ).

% refl_onD2
thf(fact_633_refl__onD2,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
      ( ( refl_on_o @ A2 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R )
       => ( member_o @ Y3 @ A2 ) ) ) ).

% refl_onD2
thf(fact_634_refl__onD2,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R: set_Pr8731163265463710919od_c_d,X: produc2736867335369716327od_c_d,Y3: produc2736867335369716327od_c_d] :
      ( ( refl_o2973842278997123942od_c_d @ A2 @ R )
     => ( ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ X @ Y3 ) @ R )
       => ( member4507666829407384208od_c_d @ Y3 @ A2 ) ) ) ).

% refl_onD2
thf(fact_635_refl__onD1,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R: set_Pr1454594181397332999od_c_a,X: produc2669932201312201782od_c_a,Y3: produc2669932201312201782od_c_a] :
      ( ( refl_o2170641996227737909od_c_a @ A2 @ R )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ X @ Y3 ) @ R )
       => ( member5502040268208861279od_c_a @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_636_refl__onD1,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R: set_Pr137242208728970247od_c_d,X: produc1399170498009216118od_c_d,Y3: produc1399170498009216118od_c_d] :
      ( ( refl_o899880292924752245od_c_d @ A2 @ R )
     => ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ X @ Y3 ) @ R )
       => ( member4231278564905875615od_c_d @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_637_refl__onD1,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,X: $o,Y3: $o] :
      ( ( refl_on_o @ A2 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y3 ) @ R )
       => ( member_o @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_638_refl__onD1,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R: set_Pr8731163265463710919od_c_d,X: produc2736867335369716327od_c_d,Y3: produc2736867335369716327od_c_d] :
      ( ( refl_o2973842278997123942od_c_d @ A2 @ R )
     => ( ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ X @ Y3 ) @ R )
       => ( member4507666829407384208od_c_d @ X @ A2 ) ) ) ).

% refl_onD1
thf(fact_639_refl__onD,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R: set_Pr1454594181397332999od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( refl_o2170641996227737909od_c_a @ A2 @ R )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_640_refl__onD,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R: set_Pr137242208728970247od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( refl_o899880292924752245od_c_d @ A2 @ R )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_641_refl__onD,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,A: $o] :
      ( ( refl_on_o @ A2 @ R )
     => ( ( member_o @ A @ A2 )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_642_refl__onD,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R: set_Pr8731163265463710919od_c_d,A: produc2736867335369716327od_c_d] :
      ( ( refl_o2973842278997123942od_c_d @ A2 @ R )
     => ( ( member4507666829407384208od_c_d @ A @ A2 )
       => ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_643_range__eqI,axiom,
    ! [B: $o,F: nat > $o,X: nat] :
      ( ( B
        = ( F @ X ) )
     => ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_644_range__eqI,axiom,
    ! [B: set_o,F: $o > set_o,X: $o] :
      ( ( B
        = ( F @ X ) )
     => ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_645_range__eqI,axiom,
    ! [B: produc2669932201312201782od_c_a,F: nat > produc2669932201312201782od_c_a,X: nat] :
      ( ( B
        = ( F @ X ) )
     => ( member5502040268208861279od_c_a @ B @ ( image_6077155709069503263od_c_a @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_646_range__eqI,axiom,
    ! [B: produc1399170498009216118od_c_d,F: nat > produc1399170498009216118od_c_d,X: nat] :
      ( ( B
        = ( F @ X ) )
     => ( member4231278564905875615od_c_d @ B @ ( image_4806394005766517599od_c_d @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_647_range__eqI,axiom,
    ! [B: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,X: $o] :
      ( ( B
        = ( F @ X ) )
     => ( member939977866535389951od_c_d @ B @ ( image_728402167444666691od_c_d @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_648_range__eqI,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a,X: $o] :
      ( ( B
        = ( F @ X ) )
     => ( member9073573073229046463od_c_a @ B @ ( image_8861997374138323203od_c_a @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_649_range__eqI,axiom,
    ! [B: produc2736867335369716327od_c_d,F: nat > produc2736867335369716327od_c_d,X: nat] :
      ( ( B
        = ( F @ X ) )
     => ( member4507666829407384208od_c_d @ B @ ( image_8700319210784927056od_c_d @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_650_range__eqI,axiom,
    ! [B: set_Pr2898135061646675015od_c_d,F: $o > set_Pr2898135061646675015od_c_d,X: $o] :
      ( ( B
        = ( F @ X ) )
     => ( member4652649041329350768od_c_d @ B @ ( image_8975626525747676852od_c_d @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_651_range__eqI,axiom,
    ! [B: produc2736867335369716327od_c_d,F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,X: produc1399170498009216118od_c_d] :
      ( ( B
        = ( F @ X ) )
     => ( member4507666829407384208od_c_d @ B @ ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d ) ) ) ).

% range_eqI
thf(fact_652_range__eqI,axiom,
    ! [B: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,X: produc2669932201312201782od_c_a] :
      ( ( B
        = ( F @ X ) )
     => ( member4652649041329350768od_c_d @ B @ ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a ) ) ) ).

% range_eqI
thf(fact_653_rangeI,axiom,
    ! [F: nat > $o,X: nat] : ( member_o @ ( F @ X ) @ ( image_nat_o @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_654_rangeI,axiom,
    ! [F: $o > set_o,X: $o] : ( member_set_o @ ( F @ X ) @ ( image_o_set_o @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_655_rangeI,axiom,
    ! [F: nat > produc2669932201312201782od_c_a,X: nat] : ( member5502040268208861279od_c_a @ ( F @ X ) @ ( image_6077155709069503263od_c_a @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_656_rangeI,axiom,
    ! [F: nat > produc1399170498009216118od_c_d,X: nat] : ( member4231278564905875615od_c_d @ ( F @ X ) @ ( image_4806394005766517599od_c_d @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_657_rangeI,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,X: $o] : ( member939977866535389951od_c_d @ ( F @ X ) @ ( image_728402167444666691od_c_d @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_658_rangeI,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,X: $o] : ( member9073573073229046463od_c_a @ ( F @ X ) @ ( image_8861997374138323203od_c_a @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_659_rangeI,axiom,
    ! [F: nat > produc2736867335369716327od_c_d,X: nat] : ( member4507666829407384208od_c_d @ ( F @ X ) @ ( image_8700319210784927056od_c_d @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_660_rangeI,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,X: $o] : ( member4652649041329350768od_c_d @ ( F @ X ) @ ( image_8975626525747676852od_c_d @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_661_rangeI,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,X: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( F @ X ) @ ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d ) ) ).

% rangeI
thf(fact_662_rangeI,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,X: produc2669932201312201782od_c_a] : ( member4652649041329350768od_c_d @ ( F @ X ) @ ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a ) ) ).

% rangeI
thf(fact_663_case__prodE,axiom,
    ! [C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o,P2: produc2736867335369716327od_c_d] :
      ( ( produc1656800403805021740_c_d_o @ C3 @ P2 )
     => ~ ! [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
            ( ( P2
              = ( produc2733640664908071511od_c_d @ X2 @ Y ) )
           => ~ ( C3 @ X2 @ Y ) ) ) ).

% case_prodE
thf(fact_664_case__prodE,axiom,
    ! [C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > $o,P2: produc3360334827542460560od_c_d] :
      ( ( produc7497122261040117353_c_d_o @ C3 @ P2 )
     => ~ ! [X2: mappin2657711750964880076od_c_a,Y: produc2736867335369716327od_c_d] :
            ( ( P2
              = ( produc5940215602367092546od_c_d @ X2 @ Y ) )
           => ~ ( C3 @ X2 @ Y ) ) ) ).

% case_prodE
thf(fact_665_case__prodE,axiom,
    ! [C3: d > produc2346461404597154448od_c_d > $o,P2: produc1399170498009216118od_c_d] :
      ( ( produc2693131752807987549_c_d_o @ C3 @ P2 )
     => ~ ! [X2: d,Y: produc2346461404597154448od_c_d] :
            ( ( P2
              = ( produc3045323491883059558od_c_d @ X2 @ Y ) )
           => ~ ( C3 @ X2 @ Y ) ) ) ).

% case_prodE
thf(fact_666_case__prodD,axiom,
    ! [F: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( produc1656800403805021740_c_d_o @ F @ ( produc2733640664908071511od_c_d @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_667_case__prodD,axiom,
    ! [F: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > $o,A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d] :
      ( ( produc7497122261040117353_c_d_o @ F @ ( produc5940215602367092546od_c_d @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_668_case__prodD,axiom,
    ! [F: d > produc2346461404597154448od_c_d > $o,A: d,B: produc2346461404597154448od_c_d] :
      ( ( produc2693131752807987549_c_d_o @ F @ ( produc3045323491883059558od_c_d @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_669_image__def,axiom,
    ( image_8721290246026763926od_c_d
    = ( ^ [F3: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A5: set_Pr8357624050790843030od_c_a] :
          ( collec6732257886551399090od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
            ? [X3: produc2669932201312201782od_c_a] :
              ( ( member5502040268208861279od_c_a @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_670_image__def,axiom,
    ( image_8975626525747676852od_c_d
    = ( ^ [F3: $o > set_Pr2898135061646675015od_c_d,A5: set_o] :
          ( collec6732257886551399090od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
            ? [X3: $o] :
              ( ( member_o @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_671_image__def,axiom,
    ( image_728402167444666691od_c_d
    = ( ^ [F3: $o > set_Pr224028844097186518od_c_d,A5: set_o] :
          ( collec4526355225760826689od_c_d
          @ ^ [Y2: set_Pr224028844097186518od_c_d] :
            ? [X3: $o] :
              ( ( member_o @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_672_image__def,axiom,
    ( image_8861997374138323203od_c_a
    = ( ^ [F3: $o > set_Pr8357624050790843030od_c_a,A5: set_o] :
          ( collec3436578395599707393od_c_a
          @ ^ [Y2: set_Pr8357624050790843030od_c_a] :
            ? [X3: $o] :
              ( ( member_o @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_673_image__def,axiom,
    ( image_o_set_o
    = ( ^ [F3: $o > set_o,A5: set_o] :
          ( collect_set_o
          @ ^ [Y2: set_o] :
            ? [X3: $o] :
              ( ( member_o @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_674_image__def,axiom,
    ( image_242321854666033093od_c_d
    = ( ^ [F3: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A5: set_se7324893142979633703od_c_d] :
          ( collec6732257886551399090od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
            ? [X3: set_Pr2898135061646675015od_c_d] :
              ( ( member4652649041329350768od_c_d @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_675_image__def,axiom,
    ( image_3436769108421412982od_c_d
    = ( ^ [F3: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A5: set_Pr224028844097186518od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [Y2: produc2736867335369716327od_c_d] :
            ? [X3: produc1399170498009216118od_c_d] :
              ( ( member4231278564905875615od_c_d @ X3 @ A5 )
              & ( Y2
                = ( F3 @ X3 ) ) ) ) ) ) ).

% image_def
thf(fact_676_Union__eq,axiom,
    ( comple3966289243721909940od_c_d
    = ( ^ [A5: set_se7324893142979633703od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [X3: produc2736867335369716327od_c_d] :
            ? [Y2: set_Pr2898135061646675015od_c_d] :
              ( ( member4652649041329350768od_c_d @ Y2 @ A5 )
              & ( member4507666829407384208od_c_d @ X3 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_677_Union__eq,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A5: set_set_o] :
          ( collect_o
          @ ^ [X3: $o] :
            ? [Y2: set_o] :
              ( ( member_set_o @ Y2 @ A5 )
              & ( member_o @ X3 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_678_Union__eq,axiom,
    ( comple7812139117791525187od_c_d
    = ( ^ [A5: set_se1756390050576932150od_c_d] :
          ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] :
            ? [Y2: set_Pr224028844097186518od_c_d] :
              ( ( member939977866535389951od_c_d @ Y2 @ A5 )
              & ( member4231278564905875615od_c_d @ X3 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_679_Union__eq,axiom,
    ( comple6722362287630405891od_c_a
    = ( ^ [A5: set_se312849170327590134od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] :
            ? [Y2: set_Pr8357624050790843030od_c_a] :
              ( ( member9073573073229046463od_c_a @ Y2 @ A5 )
              & ( member5502040268208861279od_c_a @ X3 @ Y2 ) ) ) ) ) ).

% Union_eq
thf(fact_680_INF__commute,axiom,
    ! [F: $o > $o > set_o,B2: set_o,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [I: $o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [J: $o] :
              ( comple3063163877087187839_set_o
              @ ( image_o_set_o
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_681_INF__commute,axiom,
    ! [F: $o > $o > set_Pr224028844097186518od_c_d,B2: set_o,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [I: $o] : ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [J: $o] :
              ( comple3677244080108858205od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_682_INF__commute,axiom,
    ! [F: $o > $o > set_Pr8357624050790843030od_c_a,B2: set_o,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [I: $o] : ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [J: $o] :
              ( comple2587467249947738909od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_683_INF__commute,axiom,
    ! [F: $o > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_o] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple5098958804508528846od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_684_INF__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple5098958804508528846od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_685_INF__commute,axiom,
    ! [F: $o > produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,B2: set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [J: produc2669932201312201782od_c_a] :
              ( comple5098958804508528846od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_686_INF__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,B2: set_Pr8357624050790843030od_c_a,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [J: produc2669932201312201782od_c_a] :
              ( comple5098958804508528846od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_687_INF__commute,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,B2: set_se7324893142979633703od_c_d,A2: set_o] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [I: $o] : ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [J: set_Pr2898135061646675015od_c_d] :
              ( comple5098958804508528846od_c_d
              @ ( image_8975626525747676852od_c_d
                @ ^ [I: $o] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_688_INF__commute,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > $o > set_Pr2898135061646675015od_c_d,B2: set_o,A2: set_se7324893142979633703od_c_d] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [I: set_Pr2898135061646675015od_c_d] : ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [J: $o] :
              ( comple5098958804508528846od_c_d
              @ ( image_242321854666033093od_c_d
                @ ^ [I: set_Pr2898135061646675015od_c_d] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_689_INF__commute,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,B2: set_se7324893142979633703od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [I: produc2669932201312201782od_c_a] : ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ ( F @ I ) @ B2 ) )
          @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [J: set_Pr2898135061646675015od_c_d] :
              ( comple5098958804508528846od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [I: produc2669932201312201782od_c_a] : ( F @ I @ J )
                @ A2 ) )
          @ B2 ) ) ) ).

% INF_commute
thf(fact_690_INT__E,axiom,
    ! [B: $o,B2: $o > set_o,A2: set_o,A: $o] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
     => ( ~ ( member_o @ B @ ( B2 @ A ) )
       => ~ ( member_o @ A @ A2 ) ) ) ).

% INT_E
thf(fact_691_INT__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,A: $o] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ~ ( member_o @ A @ A2 ) ) ) ).

% INT_E
thf(fact_692_INT__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,A: $o] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
     => ( ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ~ ( member_o @ A @ A2 ) ) ) ).

% INT_E
thf(fact_693_INT__E,axiom,
    ! [B: $o,B2: produc2669932201312201782od_c_a > set_o,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ B2 @ A2 ) ) )
     => ( ~ ( member_o @ B @ ( B2 @ A ) )
       => ~ ( member5502040268208861279od_c_a @ A @ A2 ) ) ) ).

% INT_E
thf(fact_694_INT__E,axiom,
    ! [B: $o,B2: produc1399170498009216118od_c_d > set_o,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ B2 @ A2 ) ) )
     => ( ~ ( member_o @ B @ ( B2 @ A ) )
       => ~ ( member4231278564905875615od_c_d @ A @ A2 ) ) ) ).

% INT_E
thf(fact_695_INT__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ A2 ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ~ ( member5502040268208861279od_c_a @ A @ A2 ) ) ) ).

% INT_E
thf(fact_696_INT__E,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ A2 ) ) )
     => ( ~ ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) )
       => ~ ( member4231278564905875615od_c_d @ A @ A2 ) ) ) ).

% INT_E
thf(fact_697_INT__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ A2 ) ) )
     => ( ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ~ ( member5502040268208861279od_c_a @ A @ A2 ) ) ) ).

% INT_E
thf(fact_698_INT__E,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ A2 ) ) )
     => ( ~ ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) )
       => ~ ( member4231278564905875615od_c_d @ A @ A2 ) ) ) ).

% INT_E
thf(fact_699_INT__E,axiom,
    ! [B: $o,B2: produc2736867335369716327od_c_d > set_o,A2: set_Pr2898135061646675015od_c_d,A: produc2736867335369716327od_c_d] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_341555330545068246_set_o @ B2 @ A2 ) ) )
     => ( ~ ( member_o @ B @ ( B2 @ A ) )
       => ~ ( member4507666829407384208od_c_d @ A @ A2 ) ) ) ).

% INT_E
thf(fact_700_INT__D,axiom,
    ! [B: $o,B2: $o > set_o,A2: set_o,A: $o] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( member_o @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_701_INT__D,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o,A: $o] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_702_INT__D,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o,A: $o] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
     => ( ( member_o @ A @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_703_INT__D,axiom,
    ! [B: $o,B2: produc2669932201312201782od_c_a > set_o,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ B2 @ A2 ) ) )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member_o @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_704_INT__D,axiom,
    ! [B: $o,B2: produc1399170498009216118od_c_d > set_o,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ B2 @ A2 ) ) )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member_o @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_705_INT__D,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_4003539448451559653od_c_a @ B2 @ A2 ) ) )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_706_INT__D,axiom,
    ! [B: produc2669932201312201782od_c_a,B2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ B @ ( comple2587467249947738909od_c_a @ ( image_2410488862838995109od_c_a @ B2 @ A2 ) ) )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member5502040268208861279od_c_a @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_707_INT__D,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ B2 @ A2 ) ) )
     => ( ( member5502040268208861279od_c_a @ A @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_708_INT__D,axiom,
    ! [B: produc1399170498009216118od_c_d,B2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ B @ ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ B2 @ A2 ) ) )
     => ( ( member4231278564905875615od_c_d @ A @ A2 )
       => ( member4231278564905875615od_c_d @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_709_INT__D,axiom,
    ! [B: $o,B2: produc2736867335369716327od_c_d > set_o,A2: set_Pr2898135061646675015od_c_d,A: produc2736867335369716327od_c_d] :
      ( ( member_o @ B @ ( comple3063163877087187839_set_o @ ( image_341555330545068246_set_o @ B2 @ A2 ) ) )
     => ( ( member4507666829407384208od_c_d @ A @ A2 )
       => ( member_o @ B @ ( B2 @ A ) ) ) ) ).

% INT_D
thf(fact_710_range__composition,axiom,
    ! [F: $o > set_o,G: $o > $o] :
      ( ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_o_set_o @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_711_range__composition,axiom,
    ! [F: $o > set_o,G: nat > $o] :
      ( ( image_nat_set_o
        @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
        @ top_top_set_nat )
      = ( image_o_set_o @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_712_range__composition,axiom,
    ! [F: set_o > set_o,G: $o > set_o] :
      ( ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_set_o_set_o @ F @ ( image_o_set_o @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_713_range__composition,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,G: $o > $o] :
      ( ( image_728402167444666691od_c_d
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_728402167444666691od_c_d @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_714_range__composition,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,G: $o > $o] :
      ( ( image_8861997374138323203od_c_a
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_8861997374138323203od_c_a @ F @ ( image_o_o2 @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_715_range__composition,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,G: nat > $o] :
      ( ( image_8269028152250893759od_c_d
        @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
        @ top_top_set_nat )
      = ( image_728402167444666691od_c_d @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_716_range__composition,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,G: nat > $o] :
      ( ( image_7179251322089774463od_c_a
        @ ^ [X3: nat] : ( F @ ( G @ X3 ) )
        @ top_top_set_nat )
      = ( image_8861997374138323203od_c_a @ F @ ( image_nat_o @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_717_range__composition,axiom,
    ! [F: set_o > set_Pr224028844097186518od_c_d,G: $o > set_o] :
      ( ( image_728402167444666691od_c_d
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_564716939699073635od_c_d @ F @ ( image_o_set_o @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_718_range__composition,axiom,
    ! [F: set_o > set_Pr8357624050790843030od_c_a,G: $o > set_o] :
      ( ( image_8861997374138323203od_c_a
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_8698312146392730147od_c_a @ F @ ( image_o_set_o @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_719_range__composition,axiom,
    ! [F: set_Pr224028844097186518od_c_d > set_o,G: $o > set_Pr224028844097186518od_c_d] :
      ( ( image_o_set_o
        @ ^ [X3: $o] : ( F @ ( G @ X3 ) )
        @ top_top_set_o )
      = ( image_2583072847733522087_set_o @ F @ ( image_728402167444666691od_c_d @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_720_rangeE,axiom,
    ! [B: $o,F: nat > $o] :
      ( ( member_o @ B @ ( image_nat_o @ F @ top_top_set_nat ) )
     => ~ ! [X2: nat] :
            ( B
            = ( ~ ( F @ X2 ) ) ) ) ).

% rangeE
thf(fact_721_rangeE,axiom,
    ! [B: set_o,F: $o > set_o] :
      ( ( member_set_o @ B @ ( image_o_set_o @ F @ top_top_set_o ) )
     => ~ ! [X2: $o] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_722_rangeE,axiom,
    ! [B: produc2669932201312201782od_c_a,F: nat > produc2669932201312201782od_c_a] :
      ( ( member5502040268208861279od_c_a @ B @ ( image_6077155709069503263od_c_a @ F @ top_top_set_nat ) )
     => ~ ! [X2: nat] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_723_rangeE,axiom,
    ! [B: produc1399170498009216118od_c_d,F: nat > produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ B @ ( image_4806394005766517599od_c_d @ F @ top_top_set_nat ) )
     => ~ ! [X2: nat] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_724_rangeE,axiom,
    ! [B: set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d] :
      ( ( member939977866535389951od_c_d @ B @ ( image_728402167444666691od_c_d @ F @ top_top_set_o ) )
     => ~ ! [X2: $o] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_725_rangeE,axiom,
    ! [B: set_Pr8357624050790843030od_c_a,F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( member9073573073229046463od_c_a @ B @ ( image_8861997374138323203od_c_a @ F @ top_top_set_o ) )
     => ~ ! [X2: $o] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_726_rangeE,axiom,
    ! [B: produc2736867335369716327od_c_d,F: nat > produc2736867335369716327od_c_d] :
      ( ( member4507666829407384208od_c_d @ B @ ( image_8700319210784927056od_c_d @ F @ top_top_set_nat ) )
     => ~ ! [X2: nat] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_727_rangeE,axiom,
    ! [B: set_Pr2898135061646675015od_c_d,F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( member4652649041329350768od_c_d @ B @ ( image_8975626525747676852od_c_d @ F @ top_top_set_o ) )
     => ~ ! [X2: $o] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_728_rangeE,axiom,
    ! [B: produc2736867335369716327od_c_d,F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d] :
      ( ( member4507666829407384208od_c_d @ B @ ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d ) )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_729_rangeE,axiom,
    ! [B: set_Pr2898135061646675015od_c_d,F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( member4652649041329350768od_c_d @ B @ ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a ) )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( B
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_730_Collect__bex__eq,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > produc2669932201312201782od_c_a > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [Y2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [Y2: produc2669932201312201782od_c_a] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_731_Collect__bex__eq,axiom,
    ! [A2: set_o,P: produc2736867335369716327od_c_d > $o > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [Y2: $o] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_732_Collect__bex__eq,axiom,
    ! [A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > set_Pr2898135061646675015od_c_d > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ? [Y2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_733_Collect__bex__eq,axiom,
    ! [A2: set_o,P: $o > $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
          ? [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( collect_o
              @ ^ [X3: $o] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_734_Collect__bex__eq,axiom,
    ! [A2: set_o,P: produc1399170498009216118od_c_d > $o > $o] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] :
          ? [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( collec7134311822446639841od_c_d
              @ ^ [X3: produc1399170498009216118od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_735_Collect__bex__eq,axiom,
    ! [A2: set_o,P: produc2669932201312201782od_c_a > $o > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
          ? [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
            & ( P @ X3 @ Y2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] :
              ( collec8405073525749625505od_c_a
              @ ^ [X3: produc2669932201312201782od_c_a] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_bex_eq
thf(fact_736_UNION__eq,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_737_UNION__eq,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_738_UNION__eq,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ? [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
            & ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_739_UNION__eq,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
      = ( collect_o
        @ ^ [Y2: $o] :
          ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member_o @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_740_UNION__eq,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
      = ( collec7134311822446639841od_c_d
        @ ^ [Y2: produc1399170498009216118od_c_d] :
          ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_741_UNION__eq,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [Y2: produc2669932201312201782od_c_a] :
          ? [X3: $o] :
            ( ( member_o @ X3 @ A2 )
            & ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% UNION_eq
thf(fact_742_Image__UN,axiom,
    ! [R: set_Product_prod_o_o,B2: $o > set_o,A2: set_o] :
      ( ( image_o_o @ R @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_o_o @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_743_Image__UN,axiom,
    ! [R: set_Pr1302033441633612133od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( image_9190242693070855089od_c_d @ R @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( image_9190242693070855089od_c_d @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_744_Image__UN,axiom,
    ! [R: set_Pr9081864598239045925od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( image_1237632359519064945od_c_a @ R @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( image_1237632359519064945od_c_a @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_745_Image__UN,axiom,
    ! [R: set_Pr1828509812422786985_c_d_o,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_125310442963549941_c_d_o @ R @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_125310442963549941_c_d_o @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_746_Image__UN,axiom,
    ! [R: set_Pr3612504503696799721_c_a_o,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( image_7960254329004709685_c_a_o @ R @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_7960254329004709685_c_a_o @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_747_Image__UN,axiom,
    ! [R: set_Pr137242208728970247od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_4797793951595579987od_c_d @ R @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( image_4797793951595579987od_c_d @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_748_Image__UN,axiom,
    ! [R: set_Pr7917073365334404039od_c_a,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( image_6068555654898565651od_c_a @ R @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( image_6068555654898565651od_c_a @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_749_Image__UN,axiom,
    ! [R: set_Pr2898135061646675015od_c_d,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( image_5265990627690709651od_c_d @ R @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( image_5265990627690709651od_c_d @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_750_Image__UN,axiom,
    ! [R: set_Pr1454594181397332999od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( image_6536752330993695315od_c_a @ R @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( image_6536752330993695315od_c_a @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_751_Image__UN,axiom,
    ! [R: set_Pr5824630068737222648_c_d_o,B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( image_7152550133608335044_c_d_o @ R @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( image_7152550133608335044_c_d_o @ R @ ( B2 @ X3 ) )
          @ A2 ) ) ) ).

% Image_UN
thf(fact_752_INT__extend__simps_I10_J,axiom,
    ! [B2: $o > set_o,F: $o > $o,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_753_INT__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_o,F: $o > set_o,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_set_o_set_o @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_754_INT__extend__simps_I10_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,F: $o > $o,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_755_INT__extend__simps_I10_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,F: $o > $o,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ ( image_o_o2 @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_756_INT__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_Pr224028844097186518od_c_d,F: $o > set_o,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_564716939699073635od_c_d @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_757_INT__extend__simps_I10_J,axiom,
    ! [B2: set_o > set_Pr8357624050790843030od_c_a,F: $o > set_o,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_8698312146392730147od_c_a @ B2 @ ( image_o_set_o @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_758_INT__extend__simps_I10_J,axiom,
    ! [B2: set_Pr224028844097186518od_c_d > set_o,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_2583072847733522087_set_o @ B2 @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_759_INT__extend__simps_I10_J,axiom,
    ! [B2: set_Pr8357624050790843030od_c_a > set_o,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_1108906476932298471_set_o @ B2 @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_760_INT__extend__simps_I10_J,axiom,
    ! [B2: set_Pr224028844097186518od_c_d > set_Pr224028844097186518od_c_d,F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_563476247564946053od_c_d @ B2 @ ( image_728402167444666691od_c_d @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_761_INT__extend__simps_I10_J,axiom,
    ! [B2: set_Pr8357624050790843030od_c_a > set_Pr224028844097186518od_c_d,F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [A6: $o] : ( B2 @ ( F @ A6 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_4444820494315463365od_c_d @ B2 @ ( image_8861997374138323203od_c_a @ F @ A2 ) ) ) ) ).

% INT_extend_simps(10)
thf(fact_762_INT__extend__simps_I9_J,axiom,
    ! [C2: $o > set_o,B2: $o > set_o,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_o_set_o @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_763_INT__extend__simps_I9_J,axiom,
    ! [C2: $o > set_Pr224028844097186518od_c_d,B2: $o > set_o,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_764_INT__extend__simps_I9_J,axiom,
    ! [C2: $o > set_Pr8357624050790843030od_c_a,B2: $o > set_o,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ C2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_765_INT__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_o,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_766_INT__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_o,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_767_INT__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_768_INT__extend__simps_I9_J,axiom,
    ! [C2: produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple2587467249947738909od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_2410488862838995109od_c_a @ C2 @ ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_769_INT__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_770_INT__extend__simps_I9_J,axiom,
    ! [C2: produc2669932201312201782od_c_a > set_Pr8357624050790843030od_c_a,B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : ( comple2587467249947738909od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a @ ( image_4003539448451559653od_c_a @ C2 @ ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_771_INT__extend__simps_I9_J,axiom,
    ! [C2: produc2736867335369716327od_c_d > set_o,B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : ( comple3063163877087187839_set_o @ ( image_341555330545068246_set_o @ C2 @ ( B2 @ X3 ) ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o @ ( image_341555330545068246_set_o @ C2 @ ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) ) ) ) ).

% INT_extend_simps(9)
thf(fact_772_full__SetCompr__eq,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [U: set_Pr2898135061646675015od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a ) ) ).

% full_SetCompr_eq
thf(fact_773_full__SetCompr__eq,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [U: set_Pr2898135061646675015od_c_d] :
          ? [X3: $o] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_8975626525747676852od_c_d @ F @ top_top_set_o ) ) ).

% full_SetCompr_eq
thf(fact_774_full__SetCompr__eq,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d] :
      ( ( collec4526355225760826689od_c_d
        @ ^ [U: set_Pr224028844097186518od_c_d] :
          ? [X3: $o] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_728402167444666691od_c_d @ F @ top_top_set_o ) ) ).

% full_SetCompr_eq
thf(fact_775_full__SetCompr__eq,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( collec3436578395599707393od_c_a
        @ ^ [U: set_Pr8357624050790843030od_c_a] :
          ? [X3: $o] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_8861997374138323203od_c_a @ F @ top_top_set_o ) ) ).

% full_SetCompr_eq
thf(fact_776_full__SetCompr__eq,axiom,
    ! [F: $o > set_o] :
      ( ( collect_set_o
        @ ^ [U: set_o] :
          ? [X3: $o] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_o_set_o @ F @ top_top_set_o ) ) ).

% full_SetCompr_eq
thf(fact_777_full__SetCompr__eq,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( collec6732257886551399090od_c_d
        @ ^ [U: set_Pr2898135061646675015od_c_d] :
          ? [X3: set_Pr2898135061646675015od_c_d] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_242321854666033093od_c_d @ F @ top_to8663423184816037751od_c_d ) ) ).

% full_SetCompr_eq
thf(fact_778_full__SetCompr__eq,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [U: produc2736867335369716327od_c_d] :
          ? [X3: produc1399170498009216118od_c_d] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d ) ) ).

% full_SetCompr_eq
thf(fact_779_full__SetCompr__eq,axiom,
    ! [F: nat > produc2736867335369716327od_c_d] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [U: produc2736867335369716327od_c_d] :
          ? [X3: nat] :
            ( U
            = ( F @ X3 ) ) )
      = ( image_8700319210784927056od_c_d @ F @ top_top_set_nat ) ) ).

% full_SetCompr_eq
thf(fact_780_SUP__INF,axiom,
    ! [P: nat > nat > $o] :
      ( ( complete_Sup_Sup_o
        @ ( image_nat_o
          @ ^ [Y2: nat] :
              ( complete_Inf_Inf_o
              @ ( image_nat_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( complete_Inf_Inf_o
        @ ( image_nat_nat_o
          @ ^ [X3: nat > nat] :
              ( complete_Sup_Sup_o
              @ ( image_nat_o
                @ ^ [Y2: nat] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% SUP_INF
thf(fact_781_SUP__INF,axiom,
    ! [P: $o > $o > set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( comple3063163877087187839_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple3063163877087187839_set_o
        @ ( image_o_o_set_o
          @ ^ [X3: $o > $o] :
              ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [Y2: $o] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% SUP_INF
thf(fact_782_SUP__INF,axiom,
    ! [P: $o > nat > set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_nat_set_o
          @ ^ [Y2: nat] :
              ( comple3063163877087187839_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_nat ) )
      = ( comple3063163877087187839_set_o
        @ ( image_nat_o_set_o
          @ ^ [X3: nat > $o] :
              ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [Y2: nat] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_o ) ) ) ).

% SUP_INF
thf(fact_783_SUP__INF,axiom,
    ! [P: nat > $o > set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( comple3063163877087187839_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_o ) )
      = ( comple3063163877087187839_set_o
        @ ( image_o_nat_set_o
          @ ^ [X3: $o > nat] :
              ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [Y2: $o] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o_nat ) ) ) ).

% SUP_INF
thf(fact_784_SUP__INF,axiom,
    ! [P: nat > nat > set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_nat_set_o
          @ ^ [Y2: nat] :
              ( comple3063163877087187839_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple3063163877087187839_set_o
        @ ( image_nat_nat_set_o
          @ ^ [X3: nat > nat] :
              ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [Y2: nat] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% SUP_INF
thf(fact_785_SUP__INF,axiom,
    ! [P: $o > $o > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( comple3677244080108858205od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_4634548954541275498od_c_d
          @ ^ [X3: $o > $o] :
              ( comple7812139117791525187od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [Y2: $o] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% SUP_INF
thf(fact_786_SUP__INF,axiom,
    ! [P: $o > nat > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_8269028152250893759od_c_d
          @ ^ [Y2: nat] :
              ( comple3677244080108858205od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_nat ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_7019777547377972252od_c_d
          @ ^ [X3: nat > $o] :
              ( comple7812139117791525187od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [Y2: nat] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_o ) ) ) ).

% SUP_INF
thf(fact_787_SUP__INF,axiom,
    ! [P: nat > $o > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( comple3677244080108858205od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_o ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_4559814919464984130od_c_d
          @ ^ [X3: $o > nat] :
              ( comple7812139117791525187od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [Y2: $o] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o_nat ) ) ) ).

% SUP_INF
thf(fact_788_SUP__INF,axiom,
    ! [P: nat > nat > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_8269028152250893759od_c_d
          @ ^ [Y2: nat] :
              ( comple3677244080108858205od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_4465398921632896016od_c_d
          @ ^ [X3: nat > nat] :
              ( comple7812139117791525187od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [Y2: nat] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% SUP_INF
thf(fact_789_SUP__INF,axiom,
    ! [P: $o > $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] :
              ( comple2587467249947738909od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple2587467249947738909od_c_a
        @ ( image_3544772124380156202od_c_a
          @ ^ [X3: $o > $o] :
              ( comple6722362287630405891od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [Y2: $o] : ( P @ ( X3 @ Y2 ) @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% SUP_INF
thf(fact_790_INF__SUP,axiom,
    ! [P: nat > nat > $o] :
      ( ( complete_Inf_Inf_o
        @ ( image_nat_o
          @ ^ [Y2: nat] :
              ( complete_Sup_Sup_o
              @ ( image_nat_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( complete_Sup_Sup_o
        @ ( image_nat_nat_o
          @ ^ [F3: nat > nat] :
              ( complete_Inf_Inf_o
              @ ( image_nat_o
                @ ^ [X3: nat] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% INF_SUP
thf(fact_791_INF__SUP,axiom,
    ! [P: $o > $o > set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_o_set_o
          @ ^ [F3: $o > $o] :
              ( comple3063163877087187839_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% INF_SUP
thf(fact_792_INF__SUP,axiom,
    ! [P: $o > nat > set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_nat_set_o
          @ ^ [Y2: nat] :
              ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_nat ) )
      = ( comple90263536869209701_set_o
        @ ( image_nat_o_set_o
          @ ^ [F3: nat > $o] :
              ( comple3063163877087187839_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_o ) ) ) ).

% INF_SUP
thf(fact_793_INF__SUP,axiom,
    ! [P: nat > $o > set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_o ) )
      = ( comple90263536869209701_set_o
        @ ( image_o_nat_set_o
          @ ^ [F3: $o > nat] :
              ( comple3063163877087187839_set_o
              @ ( image_o_set_o
                @ ^ [X3: $o] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_o ) )
          @ top_top_set_o_nat ) ) ) ).

% INF_SUP
thf(fact_794_INF__SUP,axiom,
    ! [P: nat > nat > set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_nat_set_o
          @ ^ [Y2: nat] :
              ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple90263536869209701_set_o
        @ ( image_nat_nat_set_o
          @ ^ [F3: nat > nat] :
              ( comple3063163877087187839_set_o
              @ ( image_nat_set_o
                @ ^ [X3: nat] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% INF_SUP
thf(fact_795_INF__SUP,axiom,
    ! [P: $o > $o > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( comple7812139117791525187od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_4634548954541275498od_c_d
          @ ^ [F3: $o > $o] :
              ( comple3677244080108858205od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% INF_SUP
thf(fact_796_INF__SUP,axiom,
    ! [P: $o > nat > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_8269028152250893759od_c_d
          @ ^ [Y2: nat] :
              ( comple7812139117791525187od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_nat ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_7019777547377972252od_c_d
          @ ^ [F3: nat > $o] :
              ( comple3677244080108858205od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_o ) ) ) ).

% INF_SUP
thf(fact_797_INF__SUP,axiom,
    ! [P: nat > $o > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( comple7812139117791525187od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_o ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_4559814919464984130od_c_d
          @ ^ [F3: $o > nat] :
              ( comple3677244080108858205od_c_d
              @ ( image_728402167444666691od_c_d
                @ ^ [X3: $o] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_o ) )
          @ top_top_set_o_nat ) ) ) ).

% INF_SUP
thf(fact_798_INF__SUP,axiom,
    ! [P: nat > nat > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_8269028152250893759od_c_d
          @ ^ [Y2: nat] :
              ( comple7812139117791525187od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ X3 @ Y2 )
                @ top_top_set_nat ) )
          @ top_top_set_nat ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_4465398921632896016od_c_d
          @ ^ [F3: nat > nat] :
              ( comple3677244080108858205od_c_d
              @ ( image_8269028152250893759od_c_d
                @ ^ [X3: nat] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_nat ) )
          @ top_top_set_nat_nat ) ) ) ).

% INF_SUP
thf(fact_799_INF__SUP,axiom,
    ! [P: $o > $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] :
              ( comple6722362287630405891od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [X3: $o] : ( P @ X3 @ Y2 )
                @ top_top_set_o ) )
          @ top_top_set_o ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_3544772124380156202od_c_a
          @ ^ [F3: $o > $o] :
              ( comple2587467249947738909od_c_a
              @ ( image_8861997374138323203od_c_a
                @ ^ [X3: $o] : ( P @ ( F3 @ X3 ) @ X3 )
                @ top_top_set_o ) )
          @ top_top_set_o_o ) ) ) ).

% INF_SUP
thf(fact_800_Bex__def__raw,axiom,
    ( bex_Pr8276177465575388054od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,P3: produc2669932201312201782od_c_a > $o] :
        ? [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def_raw
thf(fact_801_Bex__def__raw,axiom,
    ( bex_Pr7005415762272402390od_c_d
    = ( ^ [A5: set_Pr224028844097186518od_c_d,P3: produc1399170498009216118od_c_d > $o] :
        ? [X3: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def_raw
thf(fact_802_Bex__def__raw,axiom,
    ( bex_o
    = ( ^ [A5: set_o,P3: $o > $o] :
        ? [X3: $o] :
          ( ( member_o @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def_raw
thf(fact_803_Bex__def__raw,axiom,
    ( bex_Pr679877479140882887od_c_d
    = ( ^ [A5: set_Pr2898135061646675015od_c_d,P3: produc2736867335369716327od_c_d > $o] :
        ? [X3: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X3 @ A5 )
          & ( P3 @ X3 ) ) ) ) ).

% Bex_def_raw
thf(fact_804_iso__tuple__UNIV__I,axiom,
    ! [X: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X @ top_to7710394033828511718od_c_a ) ).

% iso_tuple_UNIV_I
thf(fact_805_iso__tuple__UNIV__I,axiom,
    ! [X: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X @ top_to8800170863989631014od_c_d ) ).

% iso_tuple_UNIV_I
thf(fact_806_iso__tuple__UNIV__I,axiom,
    ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_807_iso__tuple__UNIV__I,axiom,
    ! [X: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X @ top_to3206388957835446679od_c_d ) ).

% iso_tuple_UNIV_I
thf(fact_808_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_809_refl__on__domain,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,R: set_Pr1454594181397332999od_c_a,A: produc2669932201312201782od_c_a,B: produc2669932201312201782od_c_a] :
      ( ( refl_o2170641996227737909od_c_a @ A2 @ R )
     => ( ( member3417889999246264912od_c_a @ ( produc4004402368211057175od_c_a @ A @ B ) @ R )
       => ( ( member5502040268208861279od_c_a @ A @ A2 )
          & ( member5502040268208861279od_c_a @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_810_refl__on__domain,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,R: set_Pr137242208728970247od_c_d,A: produc1399170498009216118od_c_d,B: produc1399170498009216118od_c_d] :
      ( ( refl_o899880292924752245od_c_d @ A2 @ R )
     => ( ( member4039470153312254544od_c_d @ ( produc2265443988812941847od_c_d @ A @ B ) @ R )
       => ( ( member4231278564905875615od_c_d @ A @ A2 )
          & ( member4231278564905875615od_c_d @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_811_refl__on__domain,axiom,
    ! [A2: set_o,R: set_Product_prod_o_o,A: $o,B: $o] :
      ( ( refl_on_o @ A2 @ R )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
       => ( ( member_o @ A @ A2 )
          & ( member_o @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_812_refl__on__domain,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,R: set_Pr8731163265463710919od_c_d,A: produc2736867335369716327od_c_d,B: produc2736867335369716327od_c_d] :
      ( ( refl_o2973842278997123942od_c_d @ A2 @ R )
     => ( ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ A @ B ) @ R )
       => ( ( member4507666829407384208od_c_d @ A @ A2 )
          & ( member4507666829407384208od_c_d @ B @ A2 ) ) ) ) ).

% refl_on_domain
thf(fact_813_DEADID_Orel__reflp,axiom,
    ( reflp_on_nat @ top_top_set_nat
    @ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) ) ).

% DEADID.rel_reflp
thf(fact_814_bex__reg__left,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,Q2: produc2669932201312201782od_c_a > $o,P: produc2669932201312201782od_c_a > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ R2 )
         => ( ( Q2 @ X2 )
           => ( P @ X2 ) ) )
     => ( ? [X6: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X6 @ R2 )
            & ( Q2 @ X6 ) )
       => ? [X_1: produc2669932201312201782od_c_a] : ( P @ X_1 ) ) ) ).

% bex_reg_left
thf(fact_815_bex__reg__left,axiom,
    ! [R2: set_Pr224028844097186518od_c_d,Q2: produc1399170498009216118od_c_d > $o,P: produc1399170498009216118od_c_d > $o] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ R2 )
         => ( ( Q2 @ X2 )
           => ( P @ X2 ) ) )
     => ( ? [X6: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X6 @ R2 )
            & ( Q2 @ X6 ) )
       => ? [X_1: produc1399170498009216118od_c_d] : ( P @ X_1 ) ) ) ).

% bex_reg_left
thf(fact_816_bex__reg__left,axiom,
    ! [R2: set_o,Q2: $o > $o,P: $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ R2 )
         => ( ( Q2 @ X2 )
           => ( P @ X2 ) ) )
     => ( ? [X6: $o] :
            ( ( member_o @ X6 @ R2 )
            & ( Q2 @ X6 ) )
       => ? [X_1: $o] : ( P @ X_1 ) ) ) ).

% bex_reg_left
thf(fact_817_bex__reg__left,axiom,
    ! [R2: set_Pr2898135061646675015od_c_d,Q2: produc2736867335369716327od_c_d > $o,P: produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X2 @ R2 )
         => ( ( Q2 @ X2 )
           => ( P @ X2 ) ) )
     => ( ? [X6: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X6 @ R2 )
            & ( Q2 @ X6 ) )
       => ? [X_1: produc2736867335369716327od_c_d] : ( P @ X_1 ) ) ) ).

% bex_reg_left
thf(fact_818_surj__def,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a )
        = top_to8663423184816037751od_c_d )
      = ( ! [Y2: set_Pr2898135061646675015od_c_d] :
          ? [X3: produc2669932201312201782od_c_a] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_819_surj__def,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d] :
      ( ( ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d )
        = top_to3206388957835446679od_c_d )
      = ( ! [Y2: produc2736867335369716327od_c_d] :
          ? [X3: produc1399170498009216118od_c_d] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_820_surj__def,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8975626525747676852od_c_d @ F @ top_top_set_o )
        = top_to8663423184816037751od_c_d )
      = ( ! [Y2: set_Pr2898135061646675015od_c_d] :
          ? [X3: $o] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_821_surj__def,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d] :
      ( ( ( image_728402167444666691od_c_d @ F @ top_top_set_o )
        = top_to2093899158997654150od_c_d )
      = ( ! [Y2: set_Pr224028844097186518od_c_d] :
          ? [X3: $o] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_822_surj__def,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( ( image_8861997374138323203od_c_a @ F @ top_top_set_o )
        = top_to650358278748312134od_c_a )
      = ( ! [Y2: set_Pr8357624050790843030od_c_a] :
          ? [X3: $o] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_823_surj__def,axiom,
    ! [F: $o > set_o] :
      ( ( ( image_o_set_o @ F @ top_top_set_o )
        = top_top_set_set_o )
      = ( ! [Y2: set_o] :
          ? [X3: $o] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_824_surj__def,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( ( image_242321854666033093od_c_d @ F @ top_to8663423184816037751od_c_d )
        = top_to8663423184816037751od_c_d )
      = ( ! [Y2: set_Pr2898135061646675015od_c_d] :
          ? [X3: set_Pr2898135061646675015od_c_d] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_825_surj__def,axiom,
    ! [F: nat > nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y2: nat] :
          ? [X3: nat] :
            ( Y2
            = ( F @ X3 ) ) ) ) ).

% surj_def
thf(fact_826_mem__case__prodI,axiom,
    ! [Z: $o,C3: d > produc2346461404597154448od_c_d > set_o,A: d,B: produc2346461404597154448od_c_d] :
      ( ( member_o @ Z @ ( C3 @ A @ B ) )
     => ( member_o @ Z @ ( produc4258105878261640253_set_o @ C3 @ ( produc3045323491883059558od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_827_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: d > produc2346461404597154448od_c_d > set_Pr8357624050790843030od_c_a,A: d,B: produc2346461404597154448od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C3 @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc794508062243443419od_c_a @ C3 @ ( produc3045323491883059558od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_828_mem__case__prodI,axiom,
    ! [Z: produc1399170498009216118od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr224028844097186518od_c_d,A: d,B: produc2346461404597154448od_c_d] :
      ( ( member4231278564905875615od_c_d @ Z @ ( C3 @ A @ B ) )
     => ( member4231278564905875615od_c_d @ Z @ ( produc1884284892404562715od_c_d @ C3 @ ( produc3045323491883059558od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_829_mem__case__prodI,axiom,
    ! [Z: $o,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_o,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( member_o @ Z @ ( C3 @ A @ B ) )
     => ( member_o @ Z @ ( produc7078235429099183372_set_o @ C3 @ ( produc2733640664908071511od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_830_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C3 @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc7961053910402946858od_c_a @ C3 @ ( produc2733640664908071511od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_831_mem__case__prodI,axiom,
    ! [Z: produc1399170498009216118od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ Z @ ( C3 @ A @ B ) )
     => ( member4231278564905875615od_c_d @ Z @ ( produc9050830740564066154od_c_d @ C3 @ ( produc2733640664908071511od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_832_mem__case__prodI,axiom,
    ! [Z: produc2736867335369716327od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr2898135061646675015od_c_d,A: d,B: produc2346461404597154448od_c_d] :
      ( ( member4507666829407384208od_c_d @ Z @ ( C3 @ A @ B ) )
     => ( member4507666829407384208od_c_d @ Z @ ( produc3280351911558244492od_c_d @ C3 @ ( produc3045323491883059558od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_833_mem__case__prodI,axiom,
    ! [Z: produc2736867335369716327od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr2898135061646675015od_c_d,A: produc2669932201312201782od_c_a,B: produc1399170498009216118od_c_d] :
      ( ( member4507666829407384208od_c_d @ Z @ ( C3 @ A @ B ) )
     => ( member4507666829407384208od_c_d @ Z @ ( produc2582946828107796699od_c_d @ C3 @ ( produc2733640664908071511od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_834_mem__case__prodI,axiom,
    ! [Z: $o,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_o,A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d] :
      ( ( member_o @ Z @ ( C3 @ A @ B ) )
     => ( member_o @ Z @ ( produc1175707102232295753_set_o @ C3 @ ( produc5940215602367092546od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_835_mem__case__prodI,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_Pr8357624050790843030od_c_a,A: mappin2657711750964880076od_c_a,B: produc2736867335369716327od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( C3 @ A @ B ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc1506925131489990119od_c_a @ C3 @ ( produc5940215602367092546od_c_d @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_836_mem__case__prodI2,axiom,
    ! [P2: produc1399170498009216118od_c_d,Z: $o,C3: d > produc2346461404597154448od_c_d > set_o] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( ( P2
            = ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
         => ( member_o @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member_o @ Z @ ( produc4258105878261640253_set_o @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_837_mem__case__prodI2,axiom,
    ! [P2: produc1399170498009216118od_c_d,Z: produc2669932201312201782od_c_a,C3: d > produc2346461404597154448od_c_d > set_Pr8357624050790843030od_c_a] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( ( P2
            = ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc794508062243443419od_c_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_838_mem__case__prodI2,axiom,
    ! [P2: produc1399170498009216118od_c_d,Z: produc1399170498009216118od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr224028844097186518od_c_d] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( ( P2
            = ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
         => ( member4231278564905875615od_c_d @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member4231278564905875615od_c_d @ Z @ ( produc1884284892404562715od_c_d @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_839_mem__case__prodI2,axiom,
    ! [P2: produc2736867335369716327od_c_d,Z: $o,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_o] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( ( P2
            = ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
         => ( member_o @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member_o @ Z @ ( produc7078235429099183372_set_o @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_840_mem__case__prodI2,axiom,
    ! [P2: produc2736867335369716327od_c_d,Z: produc2669932201312201782od_c_a,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( ( P2
            = ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc7961053910402946858od_c_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_841_mem__case__prodI2,axiom,
    ! [P2: produc2736867335369716327od_c_d,Z: produc1399170498009216118od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( ( P2
            = ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
         => ( member4231278564905875615od_c_d @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member4231278564905875615od_c_d @ Z @ ( produc9050830740564066154od_c_d @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_842_mem__case__prodI2,axiom,
    ! [P2: produc1399170498009216118od_c_d,Z: produc2736867335369716327od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ! [A3: d,B3: produc2346461404597154448od_c_d] :
          ( ( P2
            = ( produc3045323491883059558od_c_d @ A3 @ B3 ) )
         => ( member4507666829407384208od_c_d @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member4507666829407384208od_c_d @ Z @ ( produc3280351911558244492od_c_d @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_843_mem__case__prodI2,axiom,
    ! [P2: produc2736867335369716327od_c_d,Z: produc2736867335369716327od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ! [A3: produc2669932201312201782od_c_a,B3: produc1399170498009216118od_c_d] :
          ( ( P2
            = ( produc2733640664908071511od_c_d @ A3 @ B3 ) )
         => ( member4507666829407384208od_c_d @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member4507666829407384208od_c_d @ Z @ ( produc2582946828107796699od_c_d @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_844_mem__case__prodI2,axiom,
    ! [P2: produc3360334827542460560od_c_d,Z: $o,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_o] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2736867335369716327od_c_d] :
          ( ( P2
            = ( produc5940215602367092546od_c_d @ A3 @ B3 ) )
         => ( member_o @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member_o @ Z @ ( produc1175707102232295753_set_o @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_845_mem__case__prodI2,axiom,
    ! [P2: produc3360334827542460560od_c_d,Z: produc2669932201312201782od_c_a,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_Pr8357624050790843030od_c_a] :
      ( ! [A3: mappin2657711750964880076od_c_a,B3: produc2736867335369716327od_c_d] :
          ( ( P2
            = ( produc5940215602367092546od_c_d @ A3 @ B3 ) )
         => ( member5502040268208861279od_c_a @ Z @ ( C3 @ A3 @ B3 ) ) )
     => ( member5502040268208861279od_c_a @ Z @ ( produc1506925131489990119od_c_a @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_846_top__set__def,axiom,
    ( top_to3206388957835446679od_c_d
    = ( collec2789978228259155154od_c_d @ top_to2551724712082855430_c_d_o ) ) ).

% top_set_def
thf(fact_847_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_848_top__empty__eq,axiom,
    ( top_to6955461090368098935_c_a_o
    = ( ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ top_to7710394033828511718od_c_a ) ) ) ).

% top_empty_eq
thf(fact_849_top__empty__eq,axiom,
    ( top_to8343889241181714999_c_d_o
    = ( ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ top_to8800170863989631014od_c_d ) ) ) ).

% top_empty_eq
thf(fact_850_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_851_top__empty__eq,axiom,
    ( top_to2551724712082855430_c_d_o
    = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ top_to3206388957835446679od_c_d ) ) ) ).

% top_empty_eq
thf(fact_852_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_853_mem__case__prodE,axiom,
    ! [Z: $o,C3: d > produc2346461404597154448od_c_d > set_o,P2: produc1399170498009216118od_c_d] :
      ( ( member_o @ Z @ ( produc4258105878261640253_set_o @ C3 @ P2 ) )
     => ~ ! [X2: d,Y: produc2346461404597154448od_c_d] :
            ( ( P2
              = ( produc3045323491883059558od_c_d @ X2 @ Y ) )
           => ~ ( member_o @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_854_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: d > produc2346461404597154448od_c_d > set_Pr8357624050790843030od_c_a,P2: produc1399170498009216118od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc794508062243443419od_c_a @ C3 @ P2 ) )
     => ~ ! [X2: d,Y: produc2346461404597154448od_c_d] :
            ( ( P2
              = ( produc3045323491883059558od_c_d @ X2 @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_855_mem__case__prodE,axiom,
    ! [Z: produc1399170498009216118od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr224028844097186518od_c_d,P2: produc1399170498009216118od_c_d] :
      ( ( member4231278564905875615od_c_d @ Z @ ( produc1884284892404562715od_c_d @ C3 @ P2 ) )
     => ~ ! [X2: d,Y: produc2346461404597154448od_c_d] :
            ( ( P2
              = ( produc3045323491883059558od_c_d @ X2 @ Y ) )
           => ~ ( member4231278564905875615od_c_d @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_856_mem__case__prodE,axiom,
    ! [Z: $o,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_o,P2: produc2736867335369716327od_c_d] :
      ( ( member_o @ Z @ ( produc7078235429099183372_set_o @ C3 @ P2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
            ( ( P2
              = ( produc2733640664908071511od_c_d @ X2 @ Y ) )
           => ~ ( member_o @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_857_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr8357624050790843030od_c_a,P2: produc2736867335369716327od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc7961053910402946858od_c_a @ C3 @ P2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
            ( ( P2
              = ( produc2733640664908071511od_c_d @ X2 @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_858_mem__case__prodE,axiom,
    ! [Z: produc1399170498009216118od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,P2: produc2736867335369716327od_c_d] :
      ( ( member4231278564905875615od_c_d @ Z @ ( produc9050830740564066154od_c_d @ C3 @ P2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
            ( ( P2
              = ( produc2733640664908071511od_c_d @ X2 @ Y ) )
           => ~ ( member4231278564905875615od_c_d @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_859_mem__case__prodE,axiom,
    ! [Z: produc2736867335369716327od_c_d,C3: d > produc2346461404597154448od_c_d > set_Pr2898135061646675015od_c_d,P2: produc1399170498009216118od_c_d] :
      ( ( member4507666829407384208od_c_d @ Z @ ( produc3280351911558244492od_c_d @ C3 @ P2 ) )
     => ~ ! [X2: d,Y: produc2346461404597154448od_c_d] :
            ( ( P2
              = ( produc3045323491883059558od_c_d @ X2 @ Y ) )
           => ~ ( member4507666829407384208od_c_d @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_860_mem__case__prodE,axiom,
    ! [Z: produc2736867335369716327od_c_d,C3: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > set_Pr2898135061646675015od_c_d,P2: produc2736867335369716327od_c_d] :
      ( ( member4507666829407384208od_c_d @ Z @ ( produc2582946828107796699od_c_d @ C3 @ P2 ) )
     => ~ ! [X2: produc2669932201312201782od_c_a,Y: produc1399170498009216118od_c_d] :
            ( ( P2
              = ( produc2733640664908071511od_c_d @ X2 @ Y ) )
           => ~ ( member4507666829407384208od_c_d @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_861_mem__case__prodE,axiom,
    ! [Z: $o,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_o,P2: produc3360334827542460560od_c_d] :
      ( ( member_o @ Z @ ( produc1175707102232295753_set_o @ C3 @ P2 ) )
     => ~ ! [X2: mappin2657711750964880076od_c_a,Y: produc2736867335369716327od_c_d] :
            ( ( P2
              = ( produc5940215602367092546od_c_d @ X2 @ Y ) )
           => ~ ( member_o @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_862_mem__case__prodE,axiom,
    ! [Z: produc2669932201312201782od_c_a,C3: mappin2657711750964880076od_c_a > produc2736867335369716327od_c_d > set_Pr8357624050790843030od_c_a,P2: produc3360334827542460560od_c_d] :
      ( ( member5502040268208861279od_c_a @ Z @ ( produc1506925131489990119od_c_a @ C3 @ P2 ) )
     => ~ ! [X2: mappin2657711750964880076od_c_a,Y: produc2736867335369716327od_c_d] :
            ( ( P2
              = ( produc5940215602367092546od_c_d @ X2 @ Y ) )
           => ~ ( member5502040268208861279od_c_a @ Z @ ( C3 @ X2 @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_863_top__empty__eq2,axiom,
    ( top_to1994068701486593775_c_a_o
    = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ top_to7710394033828511718od_c_a ) ) ) ).

% top_empty_eq2
thf(fact_864_top__empty__eq2,axiom,
    ( top_to4146800493285002110_c_d_o
    = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ top_to3206388957835446679od_c_d ) ) ) ).

% top_empty_eq2
thf(fact_865_top__empty__eq2,axiom,
    ( top_to5361376997215501169_c_d_o
    = ( ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ top_to7512568406804807414od_c_d ) ) ) ).

% top_empty_eq2
thf(fact_866_top__empty__eq2,axiom,
    ( top_to5396431732723345967_c_d_o
    = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ top_to8800170863989631014od_c_d ) ) ) ).

% top_empty_eq2
thf(fact_867_Inf__INT__eq,axiom,
    ( comple4397882455294035456_c_a_o
    = ( ^ [S2: set_Pr7260878174465401469_c_a_o,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple2587467249947738909od_c_a @ ( image_1542064655996835144od_c_a @ collec8405073525749625505od_c_a @ S2 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_868_Inf__INT__eq,axiom,
    ( comple5786310606107651520_c_d_o
    = ( ^ [S2: set_Pr5476883483191388733_c_d_o,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple3677244080108858205od_c_d @ ( image_1266220945562961352od_c_d @ collec7134311822446639841od_c_d @ S2 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_869_Inf__INT__eq,axiom,
    ( complete_Inf_Inf_o_o
    = ( ^ [S2: set_o_o,X3: $o] : ( member_o @ X3 @ ( comple3063163877087187839_set_o @ ( image_o_o_set_o @ collect_o @ S2 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_870_Inf__INT__eq,axiom,
    ( comple2892348415514068367_c_d_o
    = ( ^ [S2: set_Pr4683839544804131468_c_d_o,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple5098958804508528846od_c_d @ ( image_7682309925354437162od_c_d @ collec2789978228259155154od_c_d @ S2 ) ) ) ) ) ).

% Inf_INT_eq
thf(fact_871_UN__UN__split__split__eq,axiom,
    ! [A2: a > produc2133359284458116493od_c_a > a > produc2133359284458116493od_c_a > set_Pr2898135061646675015od_c_d,Y4: set_Pr8357624050790843030od_c_a,X4: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ( produc8587555734718091852od_c_d
            @ ^ [X12: a,X23: produc2133359284458116493od_c_a] : ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ ( produc8587555734718091852od_c_d @ ( A2 @ X12 @ X23 ) ) @ Y4 ) ) )
          @ X4 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [X3: produc2669932201312201782od_c_a] :
              ( comple3966289243721909940od_c_d
              @ ( image_8721290246026763926od_c_d
                @ ^ [Y2: produc2669932201312201782od_c_a] :
                    ( produc8587555734718091852od_c_d
                    @ ^ [X12: a,X23: produc2133359284458116493od_c_a] : ( produc8587555734718091852od_c_d @ ( A2 @ X12 @ X23 ) @ Y2 )
                    @ X3 )
                @ Y4 ) )
          @ X4 ) ) ) ).

% UN_UN_split_split_eq
thf(fact_872_INF__Int__eq2,axiom,
    ! [S: set_se312849170327590134od_c_a] :
      ( ( comple6780498534898304038_c_a_o
        @ ( image_34615054399089550_c_a_o
          @ ^ [I: set_Pr8357624050790843030od_c_a,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple2587467249947738909od_c_a @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_873_INF__Int__eq2,axiom,
    ! [S: set_se7324893142979633703od_c_d] :
      ( ( comple5461560290627148981_c_d_o
        @ ( image_5533172864875290028_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple5098958804508528846od_c_d @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_874_INF__Int__eq2,axiom,
    ! [S: set_se7355440763634802940od_c_d] :
      ( ( comple1298336921213068666_c_d_o
        @ ( image_4928181473391346958_c_d_o
          @ ^ [I: set_Pr5052118491147477062od_c_d,X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ ( comple842285552199152383od_c_d @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_875_INF__Int__eq2,axiom,
    ! [S: set_se1756390050576932150od_c_d] :
      ( ( comple959489529280280422_c_d_o
        @ ( image_1218452421501306510_c_d_o
          @ ^ [I: set_Pr224028844097186518od_c_d,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ I )
          @ S ) )
      = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple3677244080108858205od_c_d @ S ) ) ) ) ).

% INF_Int_eq2
thf(fact_876_INF__Int__eq,axiom,
    ! [S: set_se312849170327590134od_c_a] :
      ( ( comple4397882455294035456_c_a_o
        @ ( image_7664211964329984152_c_a_o
          @ ^ [I: set_Pr8357624050790843030od_c_a,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple2587467249947738909od_c_a @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_877_INF__Int__eq,axiom,
    ! [S: set_se1756390050576932150od_c_d] :
      ( ( comple5786310606107651520_c_d_o
        @ ( image_1484836191878656152_c_d_o
          @ ^ [I: set_Pr224028844097186518od_c_d,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple3677244080108858205od_c_d @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_878_INF__Int__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Inf_Inf_o_o
        @ ( image_set_o_o_o
          @ ^ [I: set_o,X3: $o] : ( member_o @ X3 @ I )
          @ S ) )
      = ( ^ [X3: $o] : ( member_o @ X3 @ ( comple3063163877087187839_set_o @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_879_INF__Int__eq,axiom,
    ! [S: set_se7324893142979633703od_c_d] :
      ( ( comple2892348415514068367_c_d_o
        @ ( image_6411191663572841816_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ I )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple5098958804508528846od_c_d @ S ) ) ) ) ).

% INF_Int_eq
thf(fact_880_bex__reg,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ? [X6: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X6 @ R2 )
            & ( P @ X6 ) )
       => ? [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ R2 )
            & ( Q2 @ X2 ) ) ) ) ).

% bex_reg
thf(fact_881_bex__reg,axiom,
    ! [R2: set_Pr224028844097186518od_c_d,P: produc1399170498009216118od_c_d > $o,Q2: produc1399170498009216118od_c_d > $o] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ? [X6: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X6 @ R2 )
            & ( P @ X6 ) )
       => ? [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ R2 )
            & ( Q2 @ X2 ) ) ) ) ).

% bex_reg
thf(fact_882_bex__reg,axiom,
    ! [R2: set_o,P: $o > $o,Q2: $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ? [X6: $o] :
            ( ( member_o @ X6 @ R2 )
            & ( P @ X6 ) )
       => ? [X2: $o] :
            ( ( member_o @ X2 @ R2 )
            & ( Q2 @ X2 ) ) ) ) ).

% bex_reg
thf(fact_883_bex__reg,axiom,
    ! [R2: set_Pr2898135061646675015od_c_d,P: produc2736867335369716327od_c_d > $o,Q2: produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ? [X6: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X6 @ R2 )
            & ( P @ X6 ) )
       => ? [X2: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X2 @ R2 )
            & ( Q2 @ X2 ) ) ) ) ).

% bex_reg
thf(fact_884_INF__INT__eq2,axiom,
    ! [R: $o > set_Pr8357624050790843030od_c_a,S: set_o] :
      ( ( comple6780498534898304038_c_a_o
        @ ( image_4646072590069495308_c_a_o
          @ ^ [I: $o,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ R @ S ) ) ) ) ) ).

% INF_INT_eq2
thf(fact_885_INF__INT__eq2,axiom,
    ! [R: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,S: set_Pr8357624050790843030od_c_a] :
      ( ( comple5461560290627148981_c_d_o
        @ ( image_393729682460826237_c_d_o
          @ ^ [I: produc2669932201312201782od_c_a,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq2
thf(fact_886_INF__INT__eq2,axiom,
    ! [R: $o > set_Pr2898135061646675015od_c_d,S: set_o] :
      ( ( comple5461560290627148981_c_d_o
        @ ( image_4475099463610824347_c_d_o
          @ ^ [I: $o,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq2
thf(fact_887_INF__INT__eq2,axiom,
    ! [R: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,S: set_se7324893142979633703od_c_d] :
      ( ( comple5461560290627148981_c_d_o
        @ ( image_5533172864875290028_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq2
thf(fact_888_INF__INT__eq2,axiom,
    ! [R: $o > set_Pr224028844097186518od_c_d,S: set_o] :
      ( ( comple959489529280280422_c_d_o
        @ ( image_8048435621306247500_c_d_o
          @ ^ [I: $o,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq2
thf(fact_889_INF__INT__eq,axiom,
    ! [R: $o > set_Pr8357624050790843030od_c_a,S: set_o] :
      ( ( comple4397882455294035456_c_a_o
        @ ( image_5467652703200512922_c_a_o
          @ ^ [I: $o,X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_890_INF__INT__eq,axiom,
    ! [R: $o > set_Pr224028844097186518od_c_d,S: set_o] :
      ( ( comple5786310606107651520_c_d_o
        @ ( image_6856080854014128986_c_d_o
          @ ^ [I: $o,X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_891_INF__INT__eq,axiom,
    ! [R: $o > set_o,S: set_o] :
      ( ( complete_Inf_Inf_o_o
        @ ( image_o_o_o
          @ ^ [I: $o,X3: $o] : ( member_o @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: $o] : ( member_o @ X3 @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_892_INF__INT__eq,axiom,
    ! [R: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,S: set_Pr8357624050790843030od_c_a] :
      ( ( comple2892348415514068367_c_d_o
        @ ( image_4757571491694354887_c_d_o
          @ ^ [I: produc2669932201312201782od_c_a,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_893_INF__INT__eq,axiom,
    ! [R: $o > set_Pr2898135061646675015od_c_d,S: set_o] :
      ( ( comple2892348415514068367_c_d_o
        @ ( image_5121210478446633769_c_d_o
          @ ^ [I: $o,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_894_INF__INT__eq,axiom,
    ! [R: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,S: set_se7324893142979633703od_c_d] :
      ( ( comple2892348415514068367_c_d_o
        @ ( image_6411191663572841816_c_d_o
          @ ^ [I: set_Pr2898135061646675015od_c_d,X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( R @ I ) )
          @ S ) )
      = ( ^ [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ R @ S ) ) ) ) ) ).

% INF_INT_eq
thf(fact_895_Inf__set__def,axiom,
    ( comple2587467249947738909od_c_a
    = ( ^ [A5: set_se312849170327590134od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] : ( complete_Inf_Inf_o @ ( image_3105936341704502279_c_a_o @ ( member5502040268208861279od_c_a @ X3 ) @ A5 ) ) ) ) ) ).

% Inf_set_def
thf(fact_896_Inf__set__def,axiom,
    ( comple3677244080108858205od_c_d
    = ( ^ [A5: set_se1756390050576932150od_c_d] :
          ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] : ( complete_Inf_Inf_o @ ( image_1321941650430489543_c_d_o @ ( member4231278564905875615od_c_d @ X3 ) @ A5 ) ) ) ) ) ).

% Inf_set_def
thf(fact_897_Inf__set__def,axiom,
    ( comple3063163877087187839_set_o
    = ( ^ [A5: set_set_o] :
          ( collect_o
          @ ^ [X3: $o] : ( complete_Inf_Inf_o @ ( image_set_o_o @ ( member_o @ X3 ) @ A5 ) ) ) ) ) ).

% Inf_set_def
thf(fact_898_Inf__set__def,axiom,
    ( comple5098958804508528846od_c_d
    = ( ^ [A5: set_se7324893142979633703od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [X3: produc2736867335369716327od_c_d] : ( complete_Inf_Inf_o @ ( image_2471105756073848854_c_d_o @ ( member4507666829407384208od_c_d @ X3 ) @ A5 ) ) ) ) ) ).

% Inf_set_def
thf(fact_899_reflp__on__Inf,axiom,
    ! [S: set_Pr8357624050790843030od_c_a,A2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,R2: produc2669932201312201782od_c_a > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ A2 @ S ) ) @ ( comple4625517889965756085_c_d_o @ ( image_8192407713068371069_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_900_reflp__on__Inf,axiom,
    ! [S: set_o,A2: $o > set_Pr2898135061646675015od_c_d,R2: $o > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ A2 @ S ) ) @ ( comple4625517889965756085_c_d_o @ ( image_8434443802585561243_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_901_reflp__on__Inf,axiom,
    ! [S: set_o,A2: $o > set_Pr224028844097186518od_c_d,R2: $o > produc1399170498009216118od_c_d > produc1399170498009216118od_c_d > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_8343749969777128071od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_8343749969777128071od_c_d @ ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ A2 @ S ) ) @ ( comple8674126625377502453_c_d_o @ ( image_7687665798361177819_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_902_reflp__on__Inf,axiom,
    ! [S: set_o,A2: $o > set_Pr8357624050790843030od_c_a,R2: $o > produc2669932201312201782od_c_a > produc2669932201312201782od_c_a > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_391139636225337927od_c_a @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_391139636225337927od_c_a @ ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ A2 @ S ) ) @ ( comple7245554981901161717_c_a_o @ ( image_6259094154884837083_c_a_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_903_reflp__on__Inf,axiom,
    ! [S: set_o,A2: $o > set_o,R2: $o > $o > $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ S )
         => ( reflp_on_o @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_on_o @ ( comple3063163877087187839_set_o @ ( image_o_set_o @ A2 @ S ) ) @ ( comple6893622634835925621_o_o_o @ ( image_o_o_o_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_904_reflp__on__Inf,axiom,
    ! [S: set_se7324893142979633703od_c_d,A2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,R2: set_Pr2898135061646675015od_c_d > produc2736867335369716327od_c_d > produc2736867335369716327od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( member4652649041329350768od_c_d @ X2 @ S )
         => ( reflp_4191898174370630776od_c_d @ ( A2 @ X2 ) @ ( R2 @ X2 ) ) )
     => ( reflp_4191898174370630776od_c_d @ ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ A2 @ S ) ) @ ( comple4625517889965756085_c_d_o @ ( image_1525462150832592812_c_d_o @ R2 @ S ) ) ) ) ).

% reflp_on_Inf
thf(fact_905_Inf__INT__eq2,axiom,
    ( comple6780498534898304038_c_a_o
    = ( ^ [S2: set_a_6354160464449729919_c_a_o,X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ ( comple2587467249947738909od_c_a @ ( image_1542064655996835144od_c_a @ collec8405073525749625505od_c_a @ ( image_3449380194339102497_c_a_o @ produc1940289858450342685_c_a_o @ S2 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_906_Inf__INT__eq2,axiom,
    ( comple5461560290627148981_c_d_o
    = ( ^ [S2: set_Pr6674123571878231694_c_d_o,X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ ( comple5098958804508528846od_c_d @ ( image_7682309925354437162od_c_d @ collec2789978228259155154od_c_d @ ( image_6408603934379594687_c_d_o @ produc1656800403805021740_c_d_o @ S2 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_907_Inf__INT__eq2,axiom,
    ( comple1298336921213068666_c_d_o
    = ( ^ [S2: set_ma5789154285582676471_c_d_o,X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ ( comple842285552199152383od_c_d @ ( image_7829844492532784832od_c_d @ collec1512216693319809125od_c_d @ ( image_2436371738898717431_c_d_o @ produc7497122261040117353_c_d_o @ S2 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_908_Inf__INT__eq2,axiom,
    ( comple959489529280280422_c_d_o
    = ( ^ [S2: set_d_4343312612943956159_c_d_o,X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ ( comple3677244080108858205od_c_d @ ( image_1266220945562961352od_c_d @ collec7134311822446639841od_c_d @ ( image_8921174598359947809_c_d_o @ produc2693131752807987549_c_d_o @ S2 ) ) ) ) ) ) ).

% Inf_INT_eq2
thf(fact_909_surjD,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a )
        = top_to8663423184816037751od_c_d )
     => ? [X2: produc2669932201312201782od_c_a] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_910_surjD,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,Y3: produc2736867335369716327od_c_d] :
      ( ( ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d )
        = top_to3206388957835446679od_c_d )
     => ? [X2: produc1399170498009216118od_c_d] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_911_surjD,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8975626525747676852od_c_d @ F @ top_top_set_o )
        = top_to8663423184816037751od_c_d )
     => ? [X2: $o] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_912_surjD,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,Y3: set_Pr224028844097186518od_c_d] :
      ( ( ( image_728402167444666691od_c_d @ F @ top_top_set_o )
        = top_to2093899158997654150od_c_d )
     => ? [X2: $o] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_913_surjD,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( image_8861997374138323203od_c_a @ F @ top_top_set_o )
        = top_to650358278748312134od_c_a )
     => ? [X2: $o] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_914_surjD,axiom,
    ! [F: $o > set_o,Y3: set_o] :
      ( ( ( image_o_set_o @ F @ top_top_set_o )
        = top_top_set_set_o )
     => ? [X2: $o] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_915_surjD,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_242321854666033093od_c_d @ F @ top_to8663423184816037751od_c_d )
        = top_to8663423184816037751od_c_d )
     => ? [X2: set_Pr2898135061646675015od_c_d] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_916_surjD,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X2: nat] :
          ( Y3
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_917_surjE,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8721290246026763926od_c_d @ F @ top_to7710394033828511718od_c_a )
        = top_to8663423184816037751od_c_d )
     => ~ ! [X2: produc2669932201312201782od_c_a] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_918_surjE,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,Y3: produc2736867335369716327od_c_d] :
      ( ( ( image_3436769108421412982od_c_d @ F @ top_to8800170863989631014od_c_d )
        = top_to3206388957835446679od_c_d )
     => ~ ! [X2: produc1399170498009216118od_c_d] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_919_surjE,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_8975626525747676852od_c_d @ F @ top_top_set_o )
        = top_to8663423184816037751od_c_d )
     => ~ ! [X2: $o] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_920_surjE,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,Y3: set_Pr224028844097186518od_c_d] :
      ( ( ( image_728402167444666691od_c_d @ F @ top_top_set_o )
        = top_to2093899158997654150od_c_d )
     => ~ ! [X2: $o] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_921_surjE,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( image_8861997374138323203od_c_a @ F @ top_top_set_o )
        = top_to650358278748312134od_c_a )
     => ~ ! [X2: $o] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_922_surjE,axiom,
    ! [F: $o > set_o,Y3: set_o] :
      ( ( ( image_o_set_o @ F @ top_top_set_o )
        = top_top_set_set_o )
     => ~ ! [X2: $o] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_923_surjE,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,Y3: set_Pr2898135061646675015od_c_d] :
      ( ( ( image_242321854666033093od_c_d @ F @ top_to8663423184816037751od_c_d )
        = top_to8663423184816037751od_c_d )
     => ~ ! [X2: set_Pr2898135061646675015od_c_d] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_924_surjE,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X2: nat] :
            ( Y3
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_925_surjI,axiom,
    ! [G: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,F: set_Pr2898135061646675015od_c_d > produc2669932201312201782od_c_a] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_8721290246026763926od_c_d @ G @ top_to7710394033828511718od_c_a )
        = top_to8663423184816037751od_c_d ) ) ).

% surjI
thf(fact_926_surjI,axiom,
    ! [G: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,F: produc2736867335369716327od_c_d > produc1399170498009216118od_c_d] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_3436769108421412982od_c_d @ G @ top_to8800170863989631014od_c_d )
        = top_to3206388957835446679od_c_d ) ) ).

% surjI
thf(fact_927_surjI,axiom,
    ! [G: $o > set_Pr2898135061646675015od_c_d,F: set_Pr2898135061646675015od_c_d > $o] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_8975626525747676852od_c_d @ G @ top_top_set_o )
        = top_to8663423184816037751od_c_d ) ) ).

% surjI
thf(fact_928_surjI,axiom,
    ! [G: $o > set_Pr224028844097186518od_c_d,F: set_Pr224028844097186518od_c_d > $o] :
      ( ! [X2: set_Pr224028844097186518od_c_d] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_728402167444666691od_c_d @ G @ top_top_set_o )
        = top_to2093899158997654150od_c_d ) ) ).

% surjI
thf(fact_929_surjI,axiom,
    ! [G: $o > set_Pr8357624050790843030od_c_a,F: set_Pr8357624050790843030od_c_a > $o] :
      ( ! [X2: set_Pr8357624050790843030od_c_a] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_8861997374138323203od_c_a @ G @ top_top_set_o )
        = top_to650358278748312134od_c_a ) ) ).

% surjI
thf(fact_930_surjI,axiom,
    ! [G: $o > set_o,F: set_o > $o] :
      ( ! [X2: set_o] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_o_set_o @ G @ top_top_set_o )
        = top_top_set_set_o ) ) ).

% surjI
thf(fact_931_surjI,axiom,
    ! [G: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ! [X2: set_Pr2898135061646675015od_c_d] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_242321854666033093od_c_d @ G @ top_to8663423184816037751od_c_d )
        = top_to8663423184816037751od_c_d ) ) ).

% surjI
thf(fact_932_surjI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ! [X2: nat] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_nat_nat @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_933_transitions__from_H_Oelims,axiom,
    ! [X: produc8383368745218977101od_c_a > set_Pr224028844097186518od_c_d,Xa: set_Pr8357624050790843030od_c_a,Xb: produc2346461404597154448od_c_d,Y3: set_Pr8765919954514477662od_c_d] :
      ( ( ( transi3152886439771060669_c_a_d @ X @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple3601077897935609291od_c_d
          @ ( image_878321926469689261od_c_d
            @ ^ [X3: produc2669932201312201782od_c_a] :
                ( image_2084376701796949005od_c_d
                @ ( produc8862148518236521539od_c_d
                  @ ^ [Y2: d,Q3: produc2346461404597154448od_c_d] : ( produc4256147637767519854od_c_d @ Xb @ ( produc2733640664908071511od_c_d @ X3 @ ( produc3045323491883059558od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( X @ ( produc8589540272964826173od_c_a @ Xb @ X3 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_934_transitions__from_H_Oelims,axiom,
    ! [X: produc3653416647907290097od_c_a > set_Pr2898135061646675015od_c_d,Xa: set_ma4406981705840051330od_c_a,Xb: produc1399170498009216118od_c_d,Y3: set_Pr6226829810622818539od_c_d] :
      ( ( ( transi7632266320214356496od_c_a @ X @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple221672675422206526od_c_d
          @ ( image_3565131711254137278od_c_d
            @ ^ [X3: mappin2657711750964880076od_c_a] :
                ( image_2731808243408159959od_c_d
                @ ( produc4688940134641226273od_c_d
                  @ ^ [Y2: produc2669932201312201782od_c_a,Q3: produc1399170498009216118od_c_d] : ( produc7045801625143393711od_c_d @ Xb @ ( produc5940215602367092546od_c_d @ X3 @ ( produc2733640664908071511od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( X @ ( produc7298481920129392619od_c_a @ Xb @ X3 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_935_transitions__from_H_Oelims,axiom,
    ! [X: produc3865010254569102336_c_d_d > set_Pr1931820589448984816od_c_d,Xa: set_d,Xb: product_prod_c_d,Y3: set_Pr8911588288622154709od_c_d] :
      ( ( ( transi1060146855332110408_d_d_b @ X @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple3178548425920542786od_c_d
          @ ( image_738897005216940825od_c_d
            @ ^ [X3: d] :
                ( image_4957440889986854762od_c_d
                @ ( produc2490630522695257248od_c_d
                  @ ^ [Y2: b,Q3: product_prod_c_d] : ( produc860619188946429157od_c_d @ Xb @ ( produc3045323491883059558od_c_d @ X3 @ ( produc3440865926910568064od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( X @ ( produc5754655040294037616_c_d_d @ Xb @ X3 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_936_transitions__from_H_Oelims,axiom,
    ! [X: product_prod_d_b > set_Product_prod_c_d,Xa: set_b,Xb: d,Y3: set_Pr224028844097186518od_c_d] :
      ( ( ( transi6114789321072267300_d_b_c @ X @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple7812139117791525187od_c_d
          @ ( image_1627772815154405276od_c_d
            @ ^ [X3: b] :
                ( image_5617561763891002707od_c_d
                @ ( produc8093752837941898505od_c_d
                  @ ^ [Y2: c,Q3: d] : ( produc3045323491883059558od_c_d @ Xb @ ( produc3440865926910568064od_c_d @ X3 @ ( product_Pair_c_d @ Y2 @ Q3 ) ) ) )
                @ ( X @ ( product_Pair_d_b @ Xb @ X3 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_937_transitions__from_H_Oelims,axiom,
    ! [X: product_prod_a_b > set_Product_prod_c_a,Xa: set_b,Xb: a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( transi7772877549217426849_a_b_c @ X @ Xa @ Xb )
        = Y3 )
     => ( Y3
        = ( comple6722362287630405891od_c_a
          @ ( image_537995984993285980od_c_a
            @ ^ [X3: b] :
                ( image_3444923613702048726od_c_a
                @ ( produc3316798848510373132od_c_a
                  @ ^ [Y2: c,Q3: a] : ( produc5762801557034676134od_c_a @ Xb @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y2 @ Q3 ) ) ) )
                @ ( X @ ( product_Pair_a_b @ Xb @ X3 ) ) )
            @ Xa ) ) ) ) ).

% transitions_from'.elims
thf(fact_938_transitions__from_H_Osimps,axiom,
    ( transi3152886439771060669_c_a_d
    = ( ^ [HM: produc8383368745218977101od_c_a > set_Pr224028844097186518od_c_d,IM: set_Pr8357624050790843030od_c_a,Q4: produc2346461404597154448od_c_d] :
          ( comple3601077897935609291od_c_d
          @ ( image_878321926469689261od_c_d
            @ ^ [X3: produc2669932201312201782od_c_a] :
                ( image_2084376701796949005od_c_d
                @ ( produc8862148518236521539od_c_d
                  @ ^ [Y2: d,Q3: produc2346461404597154448od_c_d] : ( produc4256147637767519854od_c_d @ Q4 @ ( produc2733640664908071511od_c_d @ X3 @ ( produc3045323491883059558od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( HM @ ( produc8589540272964826173od_c_a @ Q4 @ X3 ) ) )
            @ IM ) ) ) ) ).

% transitions_from'.simps
thf(fact_939_transitions__from_H_Osimps,axiom,
    ( transi7632266320214356496od_c_a
    = ( ^ [HM: produc3653416647907290097od_c_a > set_Pr2898135061646675015od_c_d,IM: set_ma4406981705840051330od_c_a,Q4: produc1399170498009216118od_c_d] :
          ( comple221672675422206526od_c_d
          @ ( image_3565131711254137278od_c_d
            @ ^ [X3: mappin2657711750964880076od_c_a] :
                ( image_2731808243408159959od_c_d
                @ ( produc4688940134641226273od_c_d
                  @ ^ [Y2: produc2669932201312201782od_c_a,Q3: produc1399170498009216118od_c_d] : ( produc7045801625143393711od_c_d @ Q4 @ ( produc5940215602367092546od_c_d @ X3 @ ( produc2733640664908071511od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( HM @ ( produc7298481920129392619od_c_a @ Q4 @ X3 ) ) )
            @ IM ) ) ) ) ).

% transitions_from'.simps
thf(fact_940_transitions__from_H_Osimps,axiom,
    ( transi1060146855332110408_d_d_b
    = ( ^ [HM: produc3865010254569102336_c_d_d > set_Pr1931820589448984816od_c_d,IM: set_d,Q4: product_prod_c_d] :
          ( comple3178548425920542786od_c_d
          @ ( image_738897005216940825od_c_d
            @ ^ [X3: d] :
                ( image_4957440889986854762od_c_d
                @ ( produc2490630522695257248od_c_d
                  @ ^ [Y2: b,Q3: product_prod_c_d] : ( produc860619188946429157od_c_d @ Q4 @ ( produc3045323491883059558od_c_d @ X3 @ ( produc3440865926910568064od_c_d @ Y2 @ Q3 ) ) ) )
                @ ( HM @ ( produc5754655040294037616_c_d_d @ Q4 @ X3 ) ) )
            @ IM ) ) ) ) ).

% transitions_from'.simps
thf(fact_941_transitions__from_H_Osimps,axiom,
    ( transi6114789321072267300_d_b_c
    = ( ^ [HM: product_prod_d_b > set_Product_prod_c_d,IM: set_b,Q4: d] :
          ( comple7812139117791525187od_c_d
          @ ( image_1627772815154405276od_c_d
            @ ^ [X3: b] :
                ( image_5617561763891002707od_c_d
                @ ( produc8093752837941898505od_c_d
                  @ ^ [Y2: c,Q3: d] : ( produc3045323491883059558od_c_d @ Q4 @ ( produc3440865926910568064od_c_d @ X3 @ ( product_Pair_c_d @ Y2 @ Q3 ) ) ) )
                @ ( HM @ ( product_Pair_d_b @ Q4 @ X3 ) ) )
            @ IM ) ) ) ) ).

% transitions_from'.simps
thf(fact_942_transitions__from_H_Osimps,axiom,
    ( transi7772877549217426849_a_b_c
    = ( ^ [HM: product_prod_a_b > set_Product_prod_c_a,IM: set_b,Q4: a] :
          ( comple6722362287630405891od_c_a
          @ ( image_537995984993285980od_c_a
            @ ^ [X3: b] :
                ( image_3444923613702048726od_c_a
                @ ( produc3316798848510373132od_c_a
                  @ ^ [Y2: c,Q3: a] : ( produc5762801557034676134od_c_a @ Q4 @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y2 @ Q3 ) ) ) )
                @ ( HM @ ( product_Pair_a_b @ Q4 @ X3 ) ) )
            @ IM ) ) ) ) ).

% transitions_from'.simps
thf(fact_943_top__prod__def,axiom,
    ( top_to7593079806418470589et_nat
    = ( produc4532415448927165861et_nat @ top_top_set_nat @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_944_aboveS__def,axiom,
    ( order_7031714650664417362od_c_d
    = ( ^ [R3: set_Pr8731163265463710919od_c_d,A6: produc2736867335369716327od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [B5: produc2736867335369716327od_c_d] :
              ( ( B5 != A6 )
              & ( member1676691871494601488od_c_d @ ( produc1518160538322555735od_c_d @ A6 @ B5 ) @ R3 ) ) ) ) ) ).

% aboveS_def
thf(fact_945_INF__SUP__set,axiom,
    ! [G: $o > $o,A2: set_set_o] :
      ( ( complete_Inf_Inf_o
        @ ( image_set_o_o
          @ ^ [B6: set_o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B6 ) )
          @ A2 ) )
      = ( complete_Sup_Sup_o
        @ ( image_set_o_o
          @ ^ [B6: set_o] : ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ B6 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_946_INF__SUP__set,axiom,
    ! [G: $o > set_o,A2: set_set_o] :
      ( ( comple3063163877087187839_set_o
        @ ( image_set_o_set_o
          @ ^ [B6: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ B6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o
        @ ( image_set_o_set_o
          @ ^ [B6: set_o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ B6 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_947_INF__SUP__set,axiom,
    ! [G: produc2669932201312201782od_c_a > $o,A2: set_se312849170327590134od_c_a] :
      ( ( complete_Inf_Inf_o
        @ ( image_3105936341704502279_c_a_o
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ G @ B6 ) )
          @ A2 ) )
      = ( complete_Sup_Sup_o
        @ ( image_3105936341704502279_c_a_o
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( complete_Inf_Inf_o @ ( image_2427823168983691879_c_a_o @ G @ B6 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_948_INF__SUP__set,axiom,
    ! [G: produc1399170498009216118od_c_d > $o,A2: set_se1756390050576932150od_c_d] :
      ( ( complete_Inf_Inf_o
        @ ( image_1321941650430489543_c_d_o
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ G @ B6 ) )
          @ A2 ) )
      = ( complete_Sup_Sup_o
        @ ( image_1321941650430489543_c_d_o
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( complete_Inf_Inf_o @ ( image_3816251319797307943_c_d_o @ G @ B6 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_949_INF__SUP__set,axiom,
    ! [G: produc2669932201312201782od_c_a > set_o,A2: set_se312849170327590134od_c_a] :
      ( ( comple3063163877087187839_set_o
        @ ( image_1108906476932298471_set_o
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ G @ B6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o
        @ ( image_1108906476932298471_set_o
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ G @ B6 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_950_INF__SUP__set,axiom,
    ! [G: produc1399170498009216118od_c_d > set_o,A2: set_se1756390050576932150od_c_d] :
      ( ( comple3063163877087187839_set_o
        @ ( image_2583072847733522087_set_o
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ G @ B6 ) )
          @ A2 ) )
      = ( comple90263536869209701_set_o
        @ ( image_2583072847733522087_set_o
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ G @ B6 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_951_INF__SUP__set,axiom,
    ! [G: $o > set_Pr224028844097186518od_c_d,A2: set_set_o] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [B6: set_o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ G @ B6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [B6: set_o] : ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ G @ B6 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_952_INF__SUP__set,axiom,
    ! [G: $o > set_Pr8357624050790843030od_c_a,A2: set_set_o] :
      ( ( comple2587467249947738909od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [B6: set_o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ G @ B6 ) )
          @ A2 ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [B6: set_o] : ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ G @ B6 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_953_INF__SUP__set,axiom,
    ! [G: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_se312849170327590134od_c_a] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_4444820494315463365od_c_d
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ G @ B6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_4444820494315463365od_c_d
          @ ^ [B6: set_Pr8357624050790843030od_c_a] : ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ G @ B6 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_954_INF__SUP__set,axiom,
    ! [G: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_se1756390050576932150od_c_d] :
      ( ( comple3677244080108858205od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ G @ B6 ) )
          @ A2 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [B6: set_Pr224028844097186518od_c_d] : ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ G @ B6 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% INF_SUP_set
thf(fact_955_SUP__INF__set,axiom,
    ! [G: $o > $o,A2: set_set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_set_o_o
          @ ^ [X3: set_o] : ( complete_Inf_Inf_o @ ( image_o_o2 @ G @ X3 ) )
          @ A2 ) )
      = ( complete_Inf_Inf_o
        @ ( image_set_o_o
          @ ^ [X3: set_o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ X3 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_956_SUP__INF__set,axiom,
    ! [G: $o > set_o,A2: set_set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_set_o_set_o
          @ ^ [X3: set_o] : ( comple3063163877087187839_set_o @ ( image_o_set_o @ G @ X3 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o
        @ ( image_set_o_set_o
          @ ^ [X3: set_o] : ( comple90263536869209701_set_o @ ( image_o_set_o @ G @ X3 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_957_SUP__INF__set,axiom,
    ! [G: produc2669932201312201782od_c_a > $o,A2: set_se312849170327590134od_c_a] :
      ( ( complete_Sup_Sup_o
        @ ( image_3105936341704502279_c_a_o
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( complete_Inf_Inf_o @ ( image_2427823168983691879_c_a_o @ G @ X3 ) )
          @ A2 ) )
      = ( complete_Inf_Inf_o
        @ ( image_3105936341704502279_c_a_o
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ G @ X3 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_958_SUP__INF__set,axiom,
    ! [G: produc1399170498009216118od_c_d > $o,A2: set_se1756390050576932150od_c_d] :
      ( ( complete_Sup_Sup_o
        @ ( image_1321941650430489543_c_d_o
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( complete_Inf_Inf_o @ ( image_3816251319797307943_c_d_o @ G @ X3 ) )
          @ A2 ) )
      = ( complete_Inf_Inf_o
        @ ( image_1321941650430489543_c_d_o
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ G @ X3 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_959_SUP__INF__set,axiom,
    ! [G: produc2669932201312201782od_c_a > set_o,A2: set_se312849170327590134od_c_a] :
      ( ( comple90263536869209701_set_o
        @ ( image_1108906476932298471_set_o
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( comple3063163877087187839_set_o @ ( image_2784846393202376007_set_o @ G @ X3 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o
        @ ( image_1108906476932298471_set_o
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ G @ X3 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_960_SUP__INF__set,axiom,
    ! [G: produc1399170498009216118od_c_d > set_o,A2: set_se1756390050576932150od_c_d] :
      ( ( comple90263536869209701_set_o
        @ ( image_2583072847733522087_set_o
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( comple3063163877087187839_set_o @ ( image_8574480529271940359_set_o @ G @ X3 ) )
          @ A2 ) )
      = ( comple3063163877087187839_set_o
        @ ( image_2583072847733522087_set_o
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ G @ X3 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_961_SUP__INF__set,axiom,
    ! [G: $o > set_Pr224028844097186518od_c_d,A2: set_set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [X3: set_o] : ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ G @ X3 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_564716939699073635od_c_d
          @ ^ [X3: set_o] : ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ G @ X3 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_962_SUP__INF__set,axiom,
    ! [G: $o > set_Pr8357624050790843030od_c_a,A2: set_set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [X3: set_o] : ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ G @ X3 ) )
          @ A2 ) )
      = ( comple2587467249947738909od_c_a
        @ ( image_8698312146392730147od_c_a
          @ ^ [X3: set_o] : ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ G @ X3 ) )
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_963_SUP__INF__set,axiom,
    ! [G: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,A2: set_se312849170327590134od_c_a] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_4444820494315463365od_c_d
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( comple3677244080108858205od_c_d @ ( image_5093316278612678949od_c_d @ G @ X3 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_4444820494315463365od_c_d
          @ ^ [X3: set_Pr8357624050790843030od_c_a] : ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ G @ X3 ) )
          @ ( collec3436578395599707393od_c_a
            @ ^ [Uu: set_Pr8357624050790843030od_c_a] :
              ? [F3: set_Pr8357624050790843030od_c_a > produc2669932201312201782od_c_a] :
                ( ( Uu
                  = ( image_5869546789040316453od_c_a @ F3 @ A2 ) )
                & ! [X3: set_Pr8357624050790843030od_c_a] :
                    ( ( member9073573073229046463od_c_a @ X3 @ A2 )
                   => ( member5502040268208861279od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_964_SUP__INF__set,axiom,
    ! [G: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,A2: set_se1756390050576932150od_c_d] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( comple3677244080108858205od_c_d @ ( image_3500265693000114405od_c_d @ G @ X3 ) )
          @ A2 ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_563476247564946053od_c_d
          @ ^ [X3: set_Pr224028844097186518od_c_d] : ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ G @ X3 ) )
          @ ( collec4526355225760826689od_c_d
            @ ^ [Uu: set_Pr224028844097186518od_c_d] :
              ? [F3: set_Pr224028844097186518od_c_d > produc1399170498009216118od_c_d] :
                ( ( Uu
                  = ( image_1837892232819626021od_c_d @ F3 @ A2 ) )
                & ! [X3: set_Pr224028844097186518od_c_d] :
                    ( ( member939977866535389951od_c_d @ X3 @ A2 )
                   => ( member4231278564905875615od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% SUP_INF_set
thf(fact_965_INT__constant,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo267491471446728962od_c_a )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bo267491471446728962od_c_a )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_966_INT__constant,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_967_INT__constant,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3677244080108858205od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to8800170863989631014od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3677244080108858205od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_968_INT__constant,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple2587467249947738909od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to7710394033828511718od_c_a ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple2587467249947738909od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_969_INT__constant,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_top_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_970_INT__constant,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo2787708447028879507od_c_d )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bo2787708447028879507od_c_d )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INT_constant
thf(fact_971_empty__Collect__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > $o] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( collec2789978228259155154od_c_d @ P ) )
      = ( ! [X3: produc2736867335369716327od_c_d] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_972_Collect__empty__eq,axiom,
    ! [P: produc2736867335369716327od_c_d > $o] :
      ( ( ( collec2789978228259155154od_c_d @ P )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: produc2736867335369716327od_c_d] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_973_all__not__in__conv,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( ! [X3: produc2669932201312201782od_c_a] :
            ~ ( member5502040268208861279od_c_a @ X3 @ A2 ) )
      = ( A2 = bot_bo267491471446728962od_c_a ) ) ).

% all_not_in_conv
thf(fact_974_all__not__in__conv,axiom,
    ! [A2: set_Pr224028844097186518od_c_d] :
      ( ( ! [X3: produc1399170498009216118od_c_d] :
            ~ ( member4231278564905875615od_c_d @ X3 @ A2 ) )
      = ( A2 = bot_bo1357268301607848258od_c_d ) ) ).

% all_not_in_conv
thf(fact_975_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_976_all__not__in__conv,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d] :
      ( ( ! [X3: produc2736867335369716327od_c_d] :
            ~ ( member4507666829407384208od_c_d @ X3 @ A2 ) )
      = ( A2 = bot_bo9018198289998281395od_c_d ) ) ).

% all_not_in_conv
thf(fact_977_empty__iff,axiom,
    ! [C3: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ C3 @ bot_bo267491471446728962od_c_a ) ).

% empty_iff
thf(fact_978_empty__iff,axiom,
    ! [C3: produc1399170498009216118od_c_d] :
      ~ ( member4231278564905875615od_c_d @ C3 @ bot_bo1357268301607848258od_c_d ) ).

% empty_iff
thf(fact_979_empty__iff,axiom,
    ! [C3: $o] :
      ~ ( member_o @ C3 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_980_empty__iff,axiom,
    ! [C3: produc2736867335369716327od_c_d] :
      ~ ( member4507666829407384208od_c_d @ C3 @ bot_bo9018198289998281395od_c_d ) ).

% empty_iff
thf(fact_981_image__is__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( ( image_8721290246026763926od_c_d @ F @ A2 )
        = bot_bo2787708447028879507od_c_d )
      = ( A2 = bot_bo267491471446728962od_c_a ) ) ).

% image_is_empty
thf(fact_982_image__is__empty,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( ( image_3436769108421412982od_c_d @ F @ A2 )
        = bot_bo9018198289998281395od_c_d )
      = ( A2 = bot_bo1357268301607848258od_c_d ) ) ).

% image_is_empty
thf(fact_983_image__is__empty,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( ( image_8975626525747676852od_c_d @ F @ A2 )
        = bot_bo2787708447028879507od_c_d )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_984_image__is__empty,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( ( image_728402167444666691od_c_d @ F @ A2 )
        = bot_bo5048930500500818850od_c_d )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_985_image__is__empty,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( ( image_8861997374138323203od_c_a @ F @ A2 )
        = bot_bo3605389620251476834od_c_a )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_986_image__is__empty,axiom,
    ! [F: $o > set_o,A2: set_o] :
      ( ( ( image_o_set_o @ F @ A2 )
        = bot_bot_set_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_987_image__is__empty,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( ( image_242321854666033093od_c_d @ F @ A2 )
        = bot_bo2787708447028879507od_c_d )
      = ( A2 = bot_bo2787708447028879507od_c_d ) ) ).

% image_is_empty
thf(fact_988_empty__is__image,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bo2787708447028879507od_c_d
        = ( image_8721290246026763926od_c_d @ F @ A2 ) )
      = ( A2 = bot_bo267491471446728962od_c_a ) ) ).

% empty_is_image
thf(fact_989_empty__is__image,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d,A2: set_Pr224028844097186518od_c_d] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( image_3436769108421412982od_c_d @ F @ A2 ) )
      = ( A2 = bot_bo1357268301607848258od_c_d ) ) ).

% empty_is_image
thf(fact_990_empty__is__image,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( bot_bo2787708447028879507od_c_d
        = ( image_8975626525747676852od_c_d @ F @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_991_empty__is__image,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( bot_bo5048930500500818850od_c_d
        = ( image_728402167444666691od_c_d @ F @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_992_empty__is__image,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( bot_bo3605389620251476834od_c_a
        = ( image_8861997374138323203od_c_a @ F @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_993_empty__is__image,axiom,
    ! [F: $o > set_o,A2: set_o] :
      ( ( bot_bot_set_set_o
        = ( image_o_set_o @ F @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_994_empty__is__image,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( bot_bo2787708447028879507od_c_d
        = ( image_242321854666033093od_c_d @ F @ A2 ) )
      = ( A2 = bot_bo2787708447028879507od_c_d ) ) ).

% empty_is_image
thf(fact_995_image__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( image_8721290246026763926od_c_d @ F @ bot_bo267491471446728962od_c_a )
      = bot_bo2787708447028879507od_c_d ) ).

% image_empty
thf(fact_996_image__empty,axiom,
    ! [F: produc1399170498009216118od_c_d > produc2736867335369716327od_c_d] :
      ( ( image_3436769108421412982od_c_d @ F @ bot_bo1357268301607848258od_c_d )
      = bot_bo9018198289998281395od_c_d ) ).

% image_empty
thf(fact_997_image__empty,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( image_8975626525747676852od_c_d @ F @ bot_bot_set_o )
      = bot_bo2787708447028879507od_c_d ) ).

% image_empty
thf(fact_998_image__empty,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d] :
      ( ( image_728402167444666691od_c_d @ F @ bot_bot_set_o )
      = bot_bo5048930500500818850od_c_d ) ).

% image_empty
thf(fact_999_image__empty,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( image_8861997374138323203od_c_a @ F @ bot_bot_set_o )
      = bot_bo3605389620251476834od_c_a ) ).

% image_empty
thf(fact_1000_image__empty,axiom,
    ! [F: $o > set_o] :
      ( ( image_o_set_o @ F @ bot_bot_set_o )
      = bot_bot_set_set_o ) ).

% image_empty
thf(fact_1001_image__empty,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( image_242321854666033093od_c_d @ F @ bot_bo2787708447028879507od_c_d )
      = bot_bo2787708447028879507od_c_d ) ).

% image_empty
thf(fact_1002_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple3966289243721909940od_c_d @ A2 )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1003_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_o] :
      ( ( ( complete_Sup_Sup_o @ A2 )
        = bot_bot_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( X3 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1004_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A2 )
        = bot_bot_set_o )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1005_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_se1756390050576932150od_c_d] :
      ( ( ( comple7812139117791525187od_c_d @ A2 )
        = bot_bo1357268301607848258od_c_d )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1006_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_se312849170327590134od_c_a] :
      ( ( ( comple6722362287630405891od_c_a @ A2 )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
           => ( X3 = bot_bo267491471446728962od_c_a ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_1007_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ A2 ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1008_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_o] :
      ( ( bot_bot_o
        = ( complete_Sup_Sup_o @ A2 ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( X3 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1009_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A2 ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1010_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_se1756390050576932150od_c_d] :
      ( ( bot_bo1357268301607848258od_c_d
        = ( comple7812139117791525187od_c_d @ A2 ) )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1011_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_se312849170327590134od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( comple6722362287630405891od_c_a @ A2 ) )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
           => ( X3 = bot_bo267491471446728962od_c_a ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_1012_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec2789978228259155154od_c_d
            @ ^ [S3: produc2736867335369716327od_c_d] : P )
          = top_to3206388957835446679od_c_d ) )
      & ( ~ P
       => ( ( collec2789978228259155154od_c_d
            @ ^ [S3: produc2736867335369716327od_c_d] : P )
          = bot_bo9018198289998281395od_c_d ) ) ) ).

% Collect_const
thf(fact_1013_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S3: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_1014_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec2789978228259155154od_c_d
            @ ( produc1656800403805021740_c_d_o
              @ ^ [A6: produc2669932201312201782od_c_a,B5: produc1399170498009216118od_c_d] : P ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ~ P
       => ( ( collec2789978228259155154od_c_d
            @ ( produc1656800403805021740_c_d_o
              @ ^ [A6: produc2669932201312201782od_c_a,B5: produc1399170498009216118od_c_d] : P ) )
          = bot_bo9018198289998281395od_c_d ) ) ) ).

% Collect_const_case_prod
thf(fact_1015_Sup__empty,axiom,
    ( ( comple3966289243721909940od_c_d @ bot_bo2787708447028879507od_c_d )
    = bot_bo9018198289998281395od_c_d ) ).

% Sup_empty
thf(fact_1016_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_1017_Sup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Sup_empty
thf(fact_1018_Sup__empty,axiom,
    ( ( comple7812139117791525187od_c_d @ bot_bo5048930500500818850od_c_d )
    = bot_bo1357268301607848258od_c_d ) ).

% Sup_empty
thf(fact_1019_Sup__empty,axiom,
    ( ( comple6722362287630405891od_c_a @ bot_bo3605389620251476834od_c_a )
    = bot_bo267491471446728962od_c_a ) ).

% Sup_empty
thf(fact_1020_Inf__UNIV,axiom,
    ( ( complete_Inf_Inf_o @ top_top_set_o )
    = bot_bot_o ) ).

% Inf_UNIV
thf(fact_1021_Inf__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inf_empty
thf(fact_1022_Inf__empty,axiom,
    ( ( complete_Inf_Inf_o @ bot_bot_set_o )
    = top_top_o ) ).

% Inf_empty
thf(fact_1023_SUP__bot,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [X3: produc2669932201312201782od_c_a] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_bot
thf(fact_1024_SUP__bot,axiom,
    ! [A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [X3: $o] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_bot
thf(fact_1025_SUP__bot,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_bot
thf(fact_1026_SUP__bot,axiom,
    ! [A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : bot_bot_set_o
          @ A2 ) )
      = bot_bot_set_o ) ).

% SUP_bot
thf(fact_1027_SUP__bot,axiom,
    ! [A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : bot_bo1357268301607848258od_c_d
          @ A2 ) )
      = bot_bo1357268301607848258od_c_d ) ).

% SUP_bot
thf(fact_1028_SUP__bot,axiom,
    ! [A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : bot_bo267491471446728962od_c_a
          @ A2 ) )
      = bot_bo267491471446728962od_c_a ) ).

% SUP_bot
thf(fact_1029_SUP__bot__conv_I1_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1030_SUP__bot__conv_I1_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1031_SUP__bot__conv_I1_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1032_SUP__bot__conv_I1_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bot_set_o ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1033_SUP__bot__conv_I1_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
        = bot_bo1357268301607848258od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1034_SUP__bot__conv_I1_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo267491471446728962od_c_a ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_1035_SUP__bot__conv_I2_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1036_SUP__bot__conv_I2_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1037_SUP__bot__conv_I2_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1038_SUP__bot__conv_I2_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bot_set_o ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1039_SUP__bot__conv_I2_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( bot_bo1357268301607848258od_c_d
        = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1040_SUP__bot__conv_I2_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( bot_bo267491471446728962od_c_a
        = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo267491471446728962od_c_a ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_1041_SUP__const,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo267491471446728962od_c_a )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_8721290246026763926od_c_d
            @ ^ [I: produc2669932201312201782od_c_a] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1042_SUP__const,axiom,
    ! [A2: set_o,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_8975626525747676852od_c_d
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1043_SUP__const,axiom,
    ! [A2: set_se7324893142979633703od_c_d,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo2787708447028879507od_c_d )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_242321854666033093od_c_d
            @ ^ [I: set_Pr2898135061646675015od_c_d] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1044_SUP__const,axiom,
    ! [A2: set_o,F: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1045_SUP__const,axiom,
    ! [A2: set_o,F: set_Pr224028844097186518od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple7812139117791525187od_c_d
          @ ( image_728402167444666691od_c_d
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1046_SUP__const,axiom,
    ! [A2: set_o,F: set_Pr8357624050790843030od_c_a] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple6722362287630405891od_c_a
          @ ( image_8861997374138323203od_c_a
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% SUP_const
thf(fact_1047_INF__const,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo267491471446728962od_c_a )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_8721290246026763926od_c_d
            @ ^ [I: produc2669932201312201782od_c_a] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1048_INF__const,axiom,
    ! [A2: set_o,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_8975626525747676852od_c_d
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1049_INF__const,axiom,
    ! [A2: set_o,F: set_Pr224028844097186518od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3677244080108858205od_c_d
          @ ( image_728402167444666691od_c_d
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1050_INF__const,axiom,
    ! [A2: set_o,F: set_Pr8357624050790843030od_c_a] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple2587467249947738909od_c_a
          @ ( image_8861997374138323203od_c_a
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1051_INF__const,axiom,
    ! [A2: set_o,F: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3063163877087187839_set_o
          @ ( image_o_set_o
            @ ^ [I: $o] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1052_INF__const,axiom,
    ! [A2: set_se7324893142979633703od_c_d,F: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo2787708447028879507od_c_d )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_242321854666033093od_c_d
            @ ^ [I: set_Pr2898135061646675015od_c_d] : F
            @ A2 ) )
        = F ) ) ).

% INF_const
thf(fact_1053_UN__constant,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo267491471446728962od_c_a )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bo267491471446728962od_c_a )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1054_UN__constant,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1055_UN__constant,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo2787708447028879507od_c_d )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bo2787708447028879507od_c_d )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1056_UN__constant,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1057_UN__constant,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple7812139117791525187od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo1357268301607848258od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple7812139117791525187od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1058_UN__constant,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple6722362287630405891od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo267491471446728962od_c_a ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple6722362287630405891od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% UN_constant
thf(fact_1059_Inf__bool__def,axiom,
    ( complete_Inf_Inf_o
    = ( ^ [A5: set_o] :
          ~ ( member_o @ $false @ A5 ) ) ) ).

% Inf_bool_def
thf(fact_1060_ball__reg,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,P: produc2669932201312201782od_c_a > $o,Q2: produc2669932201312201782od_c_a > $o] :
      ( ! [X2: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ! [X2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X2 @ R2 )
           => ( P @ X2 ) )
       => ! [X6: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X6 @ R2 )
           => ( Q2 @ X6 ) ) ) ) ).

% ball_reg
thf(fact_1061_ball__reg,axiom,
    ! [R2: set_Pr224028844097186518od_c_d,P: produc1399170498009216118od_c_d > $o,Q2: produc1399170498009216118od_c_d > $o] :
      ( ! [X2: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ! [X2: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X2 @ R2 )
           => ( P @ X2 ) )
       => ! [X6: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ X6 @ R2 )
           => ( Q2 @ X6 ) ) ) ) ).

% ball_reg
thf(fact_1062_ball__reg,axiom,
    ! [R2: set_o,P: $o > $o,Q2: $o > $o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ R2 )
           => ( P @ X2 ) )
       => ! [X6: $o] :
            ( ( member_o @ X6 @ R2 )
           => ( Q2 @ X6 ) ) ) ) ).

% ball_reg
thf(fact_1063_ball__reg,axiom,
    ! [R2: set_Pr2898135061646675015od_c_d,P: produc2736867335369716327od_c_d > $o,Q2: produc2736867335369716327od_c_d > $o] :
      ( ! [X2: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X2 @ R2 )
         => ( ( P @ X2 )
           => ( Q2 @ X2 ) ) )
     => ( ! [X2: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X2 @ R2 )
           => ( P @ X2 ) )
       => ! [X6: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ X6 @ R2 )
           => ( Q2 @ X6 ) ) ) ) ).

% ball_reg
thf(fact_1064_Union__empty,axiom,
    ( ( comple3966289243721909940od_c_d @ bot_bo2787708447028879507od_c_d )
    = bot_bo9018198289998281395od_c_d ) ).

% Union_empty
thf(fact_1065_Union__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Union_empty
thf(fact_1066_Union__empty,axiom,
    ( ( comple7812139117791525187od_c_d @ bot_bo5048930500500818850od_c_d )
    = bot_bo1357268301607848258od_c_d ) ).

% Union_empty
thf(fact_1067_Union__empty,axiom,
    ( ( comple6722362287630405891od_c_a @ bot_bo3605389620251476834od_c_a )
    = bot_bo267491471446728962od_c_a ) ).

% Union_empty
thf(fact_1068_Set_Oempty__def,axiom,
    ( bot_bo9018198289998281395od_c_d
    = ( collec2789978228259155154od_c_d
      @ ^ [X3: produc2736867335369716327od_c_d] : $false ) ) ).

% Set.empty_def
thf(fact_1069_ex__in__conv,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( ? [X3: produc2669932201312201782od_c_a] : ( member5502040268208861279od_c_a @ X3 @ A2 ) )
      = ( A2 != bot_bo267491471446728962od_c_a ) ) ).

% ex_in_conv
thf(fact_1070_ex__in__conv,axiom,
    ! [A2: set_Pr224028844097186518od_c_d] :
      ( ( ? [X3: produc1399170498009216118od_c_d] : ( member4231278564905875615od_c_d @ X3 @ A2 ) )
      = ( A2 != bot_bo1357268301607848258od_c_d ) ) ).

% ex_in_conv
thf(fact_1071_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ? [X3: $o] : ( member_o @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1072_ex__in__conv,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d] :
      ( ( ? [X3: produc2736867335369716327od_c_d] : ( member4507666829407384208od_c_d @ X3 @ A2 ) )
      = ( A2 != bot_bo9018198289998281395od_c_d ) ) ).

% ex_in_conv
thf(fact_1073_equals0I,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ! [Y: produc2669932201312201782od_c_a] :
          ~ ( member5502040268208861279od_c_a @ Y @ A2 )
     => ( A2 = bot_bo267491471446728962od_c_a ) ) ).

% equals0I
thf(fact_1074_equals0I,axiom,
    ! [A2: set_Pr224028844097186518od_c_d] :
      ( ! [Y: produc1399170498009216118od_c_d] :
          ~ ( member4231278564905875615od_c_d @ Y @ A2 )
     => ( A2 = bot_bo1357268301607848258od_c_d ) ) ).

% equals0I
thf(fact_1075_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y: $o] :
          ~ ( member_o @ Y @ A2 )
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1076_equals0I,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d] :
      ( ! [Y: produc2736867335369716327od_c_d] :
          ~ ( member4507666829407384208od_c_d @ Y @ A2 )
     => ( A2 = bot_bo9018198289998281395od_c_d ) ) ).

% equals0I
thf(fact_1077_equals0D,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,A: produc2669932201312201782od_c_a] :
      ( ( A2 = bot_bo267491471446728962od_c_a )
     => ~ ( member5502040268208861279od_c_a @ A @ A2 ) ) ).

% equals0D
thf(fact_1078_equals0D,axiom,
    ! [A2: set_Pr224028844097186518od_c_d,A: produc1399170498009216118od_c_d] :
      ( ( A2 = bot_bo1357268301607848258od_c_d )
     => ~ ( member4231278564905875615od_c_d @ A @ A2 ) ) ).

% equals0D
thf(fact_1079_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ ( member_o @ A @ A2 ) ) ).

% equals0D
thf(fact_1080_equals0D,axiom,
    ! [A2: set_Pr2898135061646675015od_c_d,A: produc2736867335369716327od_c_d] :
      ( ( A2 = bot_bo9018198289998281395od_c_d )
     => ~ ( member4507666829407384208od_c_d @ A @ A2 ) ) ).

% equals0D
thf(fact_1081_Ball__def,axiom,
    ( ball_P9076889553231183840od_c_a
    = ( ^ [A5: set_Pr8357624050790843030od_c_a,P3: produc2669932201312201782od_c_a > $o] :
        ! [X3: produc2669932201312201782od_c_a] :
          ( ( member5502040268208861279od_c_a @ X3 @ A5 )
         => ( P3 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_1082_Ball__def,axiom,
    ( ball_P7806127849928198176od_c_d
    = ( ^ [A5: set_Pr224028844097186518od_c_d,P3: produc1399170498009216118od_c_d > $o] :
        ! [X3: produc1399170498009216118od_c_d] :
          ( ( member4231278564905875615od_c_d @ X3 @ A5 )
         => ( P3 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_1083_Ball__def,axiom,
    ( ball_o
    = ( ^ [A5: set_o,P3: $o > $o] :
        ! [X3: $o] :
          ( ( member_o @ X3 @ A5 )
         => ( P3 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_1084_Ball__def,axiom,
    ( ball_P4037256200083562001od_c_d
    = ( ^ [A5: set_Pr2898135061646675015od_c_d,P3: produc2736867335369716327od_c_d > $o] :
        ! [X3: produc2736867335369716327od_c_d] :
          ( ( member4507666829407384208od_c_d @ X3 @ A5 )
         => ( P3 @ X3 ) ) ) ) ).

% Ball_def
thf(fact_1085_emptyE,axiom,
    ! [A: produc2669932201312201782od_c_a] :
      ~ ( member5502040268208861279od_c_a @ A @ bot_bo267491471446728962od_c_a ) ).

% emptyE
thf(fact_1086_emptyE,axiom,
    ! [A: produc1399170498009216118od_c_d] :
      ~ ( member4231278564905875615od_c_d @ A @ bot_bo1357268301607848258od_c_d ) ).

% emptyE
thf(fact_1087_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_1088_emptyE,axiom,
    ! [A: produc2736867335369716327od_c_d] :
      ~ ( member4507666829407384208od_c_d @ A @ bot_bo9018198289998281395od_c_d ) ).

% emptyE
thf(fact_1089_SUP__constant,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo267491471446728962od_c_a )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bo267491471446728962od_c_a )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1090_SUP__constant,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1091_SUP__constant,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo2787708447028879507od_c_d )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = bot_bo9018198289998281395od_c_d ) )
      & ( ( A2 != bot_bo2787708447028879507od_c_d )
       => ( ( comple3966289243721909940od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1092_SUP__constant,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bot_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1093_SUP__constant,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple7812139117791525187od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo1357268301607848258od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple7812139117791525187od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1094_SUP__constant,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple6722362287630405891od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = bot_bo267491471446728962od_c_a ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple6722362287630405891od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% SUP_constant
thf(fact_1095_SUP__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ F @ bot_bo267491471446728962od_c_a ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_empty
thf(fact_1096_SUP__empty,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ F @ bot_bot_set_o ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_empty
thf(fact_1097_SUP__empty,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ F @ bot_bo2787708447028879507od_c_d ) )
      = bot_bo9018198289998281395od_c_d ) ).

% SUP_empty
thf(fact_1098_SUP__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% SUP_empty
thf(fact_1099_SUP__empty,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ F @ bot_bot_set_o ) )
      = bot_bo1357268301607848258od_c_d ) ).

% SUP_empty
thf(fact_1100_SUP__empty,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ F @ bot_bot_set_o ) )
      = bot_bo267491471446728962od_c_a ) ).

% SUP_empty
thf(fact_1101_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_1102_empty__Union__conv,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ A2 ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% empty_Union_conv
thf(fact_1103_empty__Union__conv,axiom,
    ! [A2: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A2 ) )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% empty_Union_conv
thf(fact_1104_empty__Union__conv,axiom,
    ! [A2: set_se1756390050576932150od_c_d] :
      ( ( bot_bo1357268301607848258od_c_d
        = ( comple7812139117791525187od_c_d @ A2 ) )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% empty_Union_conv
thf(fact_1105_empty__Union__conv,axiom,
    ! [A2: set_se312849170327590134od_c_a] :
      ( ( bot_bo267491471446728962od_c_a
        = ( comple6722362287630405891od_c_a @ A2 ) )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
           => ( X3 = bot_bo267491471446728962od_c_a ) ) ) ) ).

% empty_Union_conv
thf(fact_1106_Union__empty__conv,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple3966289243721909940od_c_d @ A2 )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% Union_empty_conv
thf(fact_1107_Union__empty__conv,axiom,
    ! [A2: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A2 )
        = bot_bot_set_o )
      = ( ! [X3: set_o] :
            ( ( member_set_o @ X3 @ A2 )
           => ( X3 = bot_bot_set_o ) ) ) ) ).

% Union_empty_conv
thf(fact_1108_Union__empty__conv,axiom,
    ! [A2: set_se1756390050576932150od_c_d] :
      ( ( ( comple7812139117791525187od_c_d @ A2 )
        = bot_bo1357268301607848258od_c_d )
      = ( ! [X3: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X3 @ A2 )
           => ( X3 = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% Union_empty_conv
thf(fact_1109_Union__empty__conv,axiom,
    ! [A2: set_se312849170327590134od_c_a] :
      ( ( ( comple6722362287630405891od_c_a @ A2 )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X3: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X3 @ A2 )
           => ( X3 = bot_bo267491471446728962od_c_a ) ) ) ) ).

% Union_empty_conv
thf(fact_1110_Inter__eq,axiom,
    ( comple2587467249947738909od_c_a
    = ( ^ [A5: set_se312849170327590134od_c_a] :
          ( collec8405073525749625505od_c_a
          @ ^ [X3: produc2669932201312201782od_c_a] :
            ! [Y2: set_Pr8357624050790843030od_c_a] :
              ( ( member9073573073229046463od_c_a @ Y2 @ A5 )
             => ( member5502040268208861279od_c_a @ X3 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1111_Inter__eq,axiom,
    ( comple3677244080108858205od_c_d
    = ( ^ [A5: set_se1756390050576932150od_c_d] :
          ( collec7134311822446639841od_c_d
          @ ^ [X3: produc1399170498009216118od_c_d] :
            ! [Y2: set_Pr224028844097186518od_c_d] :
              ( ( member939977866535389951od_c_d @ Y2 @ A5 )
             => ( member4231278564905875615od_c_d @ X3 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1112_Inter__eq,axiom,
    ( comple3063163877087187839_set_o
    = ( ^ [A5: set_set_o] :
          ( collect_o
          @ ^ [X3: $o] :
            ! [Y2: set_o] :
              ( ( member_set_o @ Y2 @ A5 )
             => ( member_o @ X3 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1113_Inter__eq,axiom,
    ( comple5098958804508528846od_c_d
    = ( ^ [A5: set_se7324893142979633703od_c_d] :
          ( collec2789978228259155154od_c_d
          @ ^ [X3: produc2736867335369716327od_c_d] :
            ! [Y2: set_Pr2898135061646675015od_c_d] :
              ( ( member4652649041329350768od_c_d @ Y2 @ A5 )
             => ( member4507666829407384208od_c_d @ X3 @ Y2 ) ) ) ) ) ).

% Inter_eq
thf(fact_1114_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > $o,X: $o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1115_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > set_o,X: set_o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple90263536869209701_set_o @ ( image_o_set_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1116_SUP__eq__const,axiom,
    ! [I2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > $o,X: $o] :
      ( ( I2 != bot_bo267491471446728962od_c_a )
     => ( ! [I3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_2427823168983691879_c_a_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1117_SUP__eq__const,axiom,
    ! [I2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > $o,X: $o] :
      ( ( I2 != bot_bo1357268301607848258od_c_d )
     => ( ! [I3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_3816251319797307943_c_d_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1118_SUP__eq__const,axiom,
    ! [I2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_o,X: set_o] :
      ( ( I2 != bot_bo267491471446728962od_c_a )
     => ( ! [I3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple90263536869209701_set_o @ ( image_2784846393202376007_set_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1119_SUP__eq__const,axiom,
    ! [I2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > set_o,X: set_o] :
      ( ( I2 != bot_bo1357268301607848258od_c_d )
     => ( ! [I3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple90263536869209701_set_o @ ( image_8574480529271940359_set_o @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1120_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > set_Pr224028844097186518od_c_d,X: set_Pr224028844097186518od_c_d] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1121_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > set_Pr8357624050790843030od_c_a,X: set_Pr8357624050790843030od_c_a] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1122_SUP__eq__const,axiom,
    ! [I2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_Pr224028844097186518od_c_d,X: set_Pr224028844097186518od_c_d] :
      ( ( I2 != bot_bo267491471446728962od_c_a )
     => ( ! [I3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_5093316278612678949od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1123_SUP__eq__const,axiom,
    ! [I2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > set_Pr224028844097186518od_c_d,X: set_Pr224028844097186518od_c_d] :
      ( ( I2 != bot_bo1357268301607848258od_c_d )
     => ( ! [I3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7812139117791525187od_c_d @ ( image_3500265693000114405od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_1124_INF__eq__const,axiom,
    ! [I2: set_se7324893142979633703od_c_d,F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,X: set_Pr2898135061646675015od_c_d] :
      ( ( I2 != bot_bo2787708447028879507od_c_d )
     => ( ! [I3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1125_INF__eq__const,axiom,
    ! [I2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,X: set_Pr2898135061646675015od_c_d] :
      ( ( I2 != bot_bo267491471446728962od_c_a )
     => ( ! [I3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1126_INF__eq__const,axiom,
    ! [I2: set_o,F: $o > set_Pr2898135061646675015od_c_d,X: set_Pr2898135061646675015od_c_d] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1127_INF__eq__const,axiom,
    ! [I2: set_o,F: $o > set_Pr224028844097186518od_c_d,X: set_Pr224028844097186518od_c_d] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1128_INF__eq__const,axiom,
    ! [I2: set_o,F: $o > set_Pr8357624050790843030od_c_a,X: set_Pr8357624050790843030od_c_a] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1129_INF__eq__const,axiom,
    ! [I2: set_o,F: $o > set_o,X: set_o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1130_INF__eq__const,axiom,
    ! [I2: set_Pr8357624050790843030od_c_a,F: produc2669932201312201782od_c_a > $o,X: $o] :
      ( ( I2 != bot_bo267491471446728962od_c_a )
     => ( ! [I3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Inf_Inf_o @ ( image_2427823168983691879_c_a_o @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1131_INF__eq__const,axiom,
    ! [I2: set_Pr224028844097186518od_c_d,F: produc1399170498009216118od_c_d > $o,X: $o] :
      ( ( I2 != bot_bo1357268301607848258od_c_d )
     => ( ! [I3: produc1399170498009216118od_c_d] :
            ( ( member4231278564905875615od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Inf_Inf_o @ ( image_3816251319797307943_c_d_o @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1132_INF__eq__const,axiom,
    ! [I2: set_o,F: $o > $o,X: $o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Inf_Inf_o @ ( image_o_o2 @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1133_INF__eq__const,axiom,
    ! [I2: set_Pr2898135061646675015od_c_d,F: produc2736867335369716327od_c_d > $o,X: $o] :
      ( ( I2 != bot_bo9018198289998281395od_c_d )
     => ( ! [I3: produc2736867335369716327od_c_d] :
            ( ( member4507666829407384208od_c_d @ I3 @ I2 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Inf_Inf_o @ ( image_1680233356757228022_c_d_o @ F @ I2 ) )
          = X ) ) ) ).

% INF_eq_const
thf(fact_1134_Inter__empty,axiom,
    ( ( comple7806235888213564991et_nat @ bot_bot_set_set_nat )
    = top_top_set_nat ) ).

% Inter_empty
thf(fact_1135_Collect__ball__eq,axiom,
    ! [A2: set_o,P: produc1399170498009216118od_c_d > $o > $o] :
      ( ( collec7134311822446639841od_c_d
        @ ^ [X3: produc1399170498009216118od_c_d] :
          ! [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [Y2: $o] :
              ( collec7134311822446639841od_c_d
              @ ^ [X3: produc1399170498009216118od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1136_Collect__ball__eq,axiom,
    ! [A2: set_o,P: produc2669932201312201782od_c_a > $o > $o] :
      ( ( collec8405073525749625505od_c_a
        @ ^ [X3: produc2669932201312201782od_c_a] :
          ! [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple2587467249947738909od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [Y2: $o] :
              ( collec8405073525749625505od_c_a
              @ ^ [X3: produc2669932201312201782od_c_a] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1137_Collect__ball__eq,axiom,
    ! [A2: set_o,P: $o > $o > $o] :
      ( ( collect_o
        @ ^ [X3: $o] :
          ! [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple3063163877087187839_set_o
        @ ( image_o_set_o
          @ ^ [Y2: $o] :
              ( collect_o
              @ ^ [X3: $o] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1138_Collect__ball__eq,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,P: produc2736867335369716327od_c_d > produc2669932201312201782od_c_a > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ! [Y2: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [Y2: produc2669932201312201782od_c_a] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1139_Collect__ball__eq,axiom,
    ! [A2: set_o,P: produc2736867335369716327od_c_d > $o > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ! [Y2: $o] :
            ( ( member_o @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [Y2: $o] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1140_Collect__ball__eq,axiom,
    ! [A2: set_se7324893142979633703od_c_d,P: produc2736867335369716327od_c_d > set_Pr2898135061646675015od_c_d > $o] :
      ( ( collec2789978228259155154od_c_d
        @ ^ [X3: produc2736867335369716327od_c_d] :
          ! [Y2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ Y2 @ A2 )
           => ( P @ X3 @ Y2 ) ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [Y2: set_Pr2898135061646675015od_c_d] :
              ( collec2789978228259155154od_c_d
              @ ^ [X3: produc2736867335369716327od_c_d] : ( P @ X3 @ Y2 ) )
          @ A2 ) ) ) ).

% Collect_ball_eq
thf(fact_1141_INTER__eq,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
      = ( collec8405073525749625505od_c_a
        @ ^ [Y2: produc2669932201312201782od_c_a] :
          ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member5502040268208861279od_c_a @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1142_INTER__eq,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
      = ( collec7134311822446639841od_c_d
        @ ^ [Y2: produc1399170498009216118od_c_d] :
          ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member4231278564905875615od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1143_INTER__eq,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ A2 ) )
      = ( collect_o
        @ ^ [Y2: $o] :
          ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member_o @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1144_INTER__eq,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1145_INTER__eq,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1146_INTER__eq,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
      = ( collec2789978228259155154od_c_d
        @ ^ [Y2: produc2736867335369716327od_c_d] :
          ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( member4507666829407384208od_c_d @ Y2 @ ( B2 @ X3 ) ) ) ) ) ).

% INTER_eq
thf(fact_1147_UN__empty2,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8721290246026763926od_c_d
          @ ^ [X3: produc2669932201312201782od_c_a] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty2
thf(fact_1148_UN__empty2,axiom,
    ! [A2: set_o] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_8975626525747676852od_c_d
          @ ^ [X3: $o] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty2
thf(fact_1149_UN__empty2,axiom,
    ! [A2: set_se7324893142979633703od_c_d] :
      ( ( comple3966289243721909940od_c_d
        @ ( image_242321854666033093od_c_d
          @ ^ [X3: set_Pr2898135061646675015od_c_d] : bot_bo9018198289998281395od_c_d
          @ A2 ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty2
thf(fact_1150_UN__empty2,axiom,
    ! [A2: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X3: $o] : bot_bot_set_o
          @ A2 ) )
      = bot_bot_set_o ) ).

% UN_empty2
thf(fact_1151_UN__empty2,axiom,
    ! [A2: set_o] :
      ( ( comple7812139117791525187od_c_d
        @ ( image_728402167444666691od_c_d
          @ ^ [X3: $o] : bot_bo1357268301607848258od_c_d
          @ A2 ) )
      = bot_bo1357268301607848258od_c_d ) ).

% UN_empty2
thf(fact_1152_UN__empty2,axiom,
    ! [A2: set_o] :
      ( ( comple6722362287630405891od_c_a
        @ ( image_8861997374138323203od_c_a
          @ ^ [X3: $o] : bot_bo267491471446728962od_c_a
          @ A2 ) )
      = bot_bo267491471446728962od_c_a ) ).

% UN_empty2
thf(fact_1153_UN__empty,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ bot_bo267491471446728962od_c_a ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty
thf(fact_1154_UN__empty,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ bot_bot_set_o ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty
thf(fact_1155_UN__empty,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ bot_bo2787708447028879507od_c_d ) )
      = bot_bo9018198289998281395od_c_d ) ).

% UN_empty
thf(fact_1156_UN__empty,axiom,
    ! [B2: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_1157_UN__empty,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d] :
      ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ bot_bot_set_o ) )
      = bot_bo1357268301607848258od_c_d ) ).

% UN_empty
thf(fact_1158_UN__empty,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ bot_bot_set_o ) )
      = bot_bo267491471446728962od_c_a ) ).

% UN_empty
thf(fact_1159_UNION__empty__conv_I1_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1160_UNION__empty__conv_I1_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1161_UNION__empty__conv_I1_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( bot_bo9018198289998281395od_c_d
        = ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1162_UNION__empty__conv_I1_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bot_set_o ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1163_UNION__empty__conv_I1_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( bot_bo1357268301607848258od_c_d
        = ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1164_UNION__empty__conv_I1_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( bot_bo267491471446728962od_c_a
        = ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) ) )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo267491471446728962od_c_a ) ) ) ) ).

% UNION_empty_conv(1)
thf(fact_1165_UNION__empty__conv_I2_J,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d,A2: set_Pr8357624050790843030od_c_a] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: produc2669932201312201782od_c_a] :
            ( ( member5502040268208861279od_c_a @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1166_UNION__empty__conv_I2_J,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d,A2: set_o] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1167_UNION__empty__conv_I2_J,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d,A2: set_se7324893142979633703od_c_d] :
      ( ( ( comple3966289243721909940od_c_d @ ( image_242321854666033093od_c_d @ B2 @ A2 ) )
        = bot_bo9018198289998281395od_c_d )
      = ( ! [X3: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo9018198289998281395od_c_d ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1168_UNION__empty__conv_I2_J,axiom,
    ! [B2: $o > set_o,A2: set_o] :
      ( ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B2 @ A2 ) )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bot_set_o ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1169_UNION__empty__conv_I2_J,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d,A2: set_o] :
      ( ( ( comple7812139117791525187od_c_d @ ( image_728402167444666691od_c_d @ B2 @ A2 ) )
        = bot_bo1357268301607848258od_c_d )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo1357268301607848258od_c_d ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1170_UNION__empty__conv_I2_J,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a,A2: set_o] :
      ( ( ( comple6722362287630405891od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ A2 ) )
        = bot_bo267491471446728962od_c_a )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A2 )
           => ( ( B2 @ X3 )
              = bot_bo267491471446728962od_c_a ) ) ) ) ).

% UNION_empty_conv(2)
thf(fact_1171_INF__empty,axiom,
    ! [F: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ F @ bot_bo267491471446728962od_c_a ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_empty
thf(fact_1172_INF__empty,axiom,
    ! [F: $o > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ F @ bot_bot_set_o ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_empty
thf(fact_1173_INF__empty,axiom,
    ! [F: $o > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ F @ bot_bot_set_o ) )
      = top_to8800170863989631014od_c_d ) ).

% INF_empty
thf(fact_1174_INF__empty,axiom,
    ! [F: $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ F @ bot_bot_set_o ) )
      = top_to7710394033828511718od_c_a ) ).

% INF_empty
thf(fact_1175_INF__empty,axiom,
    ! [F: $o > set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ F @ bot_bot_set_o ) )
      = top_top_set_o ) ).

% INF_empty
thf(fact_1176_INF__empty,axiom,
    ! [F: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ F @ bot_bo2787708447028879507od_c_d ) )
      = top_to3206388957835446679od_c_d ) ).

% INF_empty
thf(fact_1177_INF__constant,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo267491471446728962od_c_a )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bo267491471446728962od_c_a )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8721290246026763926od_c_d
              @ ^ [Y2: produc2669932201312201782od_c_a] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1178_INF__constant,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_8975626525747676852od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1179_INF__constant,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3677244080108858205od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to8800170863989631014od_c_d ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3677244080108858205od_c_d
            @ ( image_728402167444666691od_c_d
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1180_INF__constant,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple2587467249947738909od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_to7710394033828511718od_c_a ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple2587467249947738909od_c_a
            @ ( image_8861997374138323203od_c_a
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1181_INF__constant,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( ( A2 = bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = top_top_set_o ) )
      & ( ( A2 != bot_bot_set_o )
       => ( ( comple3063163877087187839_set_o
            @ ( image_o_set_o
              @ ^ [Y2: $o] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1182_INF__constant,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( ( A2 = bot_bo2787708447028879507od_c_d )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = top_to3206388957835446679od_c_d ) )
      & ( ( A2 != bot_bo2787708447028879507od_c_d )
       => ( ( comple5098958804508528846od_c_d
            @ ( image_242321854666033093od_c_d
              @ ^ [Y2: set_Pr2898135061646675015od_c_d] : C3
              @ A2 ) )
          = C3 ) ) ) ).

% INF_constant
thf(fact_1183_Inf__Sup,axiom,
    ! [A2: set_se3083084249005913095od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_1688364829223432677od_c_d @ comple3966289243721909940od_c_d @ A2 ) )
      = ( comple3966289243721909940od_c_d
        @ ( image_1688364829223432677od_c_d @ comple5098958804508528846od_c_d
          @ ( collec259608718228948114od_c_d
            @ ^ [Uu: set_se7324893142979633703od_c_d] :
              ? [F3: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
                ( ( Uu
                  = ( image_1688364829223432677od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se7324893142979633703od_c_d] :
                    ( ( member1954756648797149776od_c_d @ X3 @ A2 )
                   => ( member4652649041329350768od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_1184_Inf__Sup,axiom,
    ! [A2: set_set_o] :
      ( ( complete_Inf_Inf_o @ ( image_set_o_o @ complete_Sup_Sup_o @ A2 ) )
      = ( complete_Sup_Sup_o
        @ ( image_set_o_o @ complete_Inf_Inf_o
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_1185_Inf__Sup,axiom,
    ! [A2: set_set_set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_1148209357520620709_set_o @ comple90263536869209701_set_o @ A2 ) )
      = ( comple90263536869209701_set_o
        @ ( image_1148209357520620709_set_o @ comple3063163877087187839_set_o
          @ ( collect_set_set_o
            @ ^ [Uu: set_set_o] :
              ? [F3: set_set_o > set_o] :
                ( ( Uu
                  = ( image_1148209357520620709_set_o @ F3 @ A2 ) )
                & ! [X3: set_set_o] :
                    ( ( member_set_set_o @ X3 @ A2 )
                   => ( member_set_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_1186_Inf__Sup,axiom,
    ! [A2: set_se3236662274807499670od_c_d] :
      ( ( comple3677244080108858205od_c_d @ ( image_7166828263236204581od_c_d @ comple7812139117791525187od_c_d @ A2 ) )
      = ( comple7812139117791525187od_c_d
        @ ( image_7166828263236204581od_c_d @ comple3677244080108858205od_c_d
          @ ( collec6855837246311779233od_c_d
            @ ^ [Uu: set_se1756390050576932150od_c_d] :
              ? [F3: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
                ( ( Uu
                  = ( image_7166828263236204581od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se1756390050576932150od_c_d] :
                    ( ( member7900373713009613151od_c_d @ X3 @ A2 )
                   => ( member939977866535389951od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_1187_Inf__Sup,axiom,
    ! [A2: set_se6970335898041508694od_c_a] :
      ( ( comple2587467249947738909od_c_a @ ( image_1048133724346124325od_c_a @ comple6722362287630405891od_c_a @ A2 ) )
      = ( comple6722362287630405891od_c_a
        @ ( image_1048133724346124325od_c_a @ comple2587467249947738909od_c_a
          @ ( collec5412296366062437217od_c_a
            @ ^ [Uu: set_se312849170327590134od_c_a] :
              ? [F3: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
                ( ( Uu
                  = ( image_1048133724346124325od_c_a @ F3 @ A2 ) )
                & ! [X3: set_se312849170327590134od_c_a] :
                    ( ( member6456832832760271135od_c_a @ X3 @ A2 )
                   => ( member9073573073229046463od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup
thf(fact_1188_Sup__Inf,axiom,
    ! [A2: set_se3083084249005913095od_c_d] :
      ( ( comple3966289243721909940od_c_d @ ( image_1688364829223432677od_c_d @ comple5098958804508528846od_c_d @ A2 ) )
      = ( comple5098958804508528846od_c_d
        @ ( image_1688364829223432677od_c_d @ comple3966289243721909940od_c_d
          @ ( collec259608718228948114od_c_d
            @ ^ [Uu: set_se7324893142979633703od_c_d] :
              ? [F3: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
                ( ( Uu
                  = ( image_1688364829223432677od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se7324893142979633703od_c_d] :
                    ( ( member1954756648797149776od_c_d @ X3 @ A2 )
                   => ( member4652649041329350768od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_1189_Sup__Inf,axiom,
    ! [A2: set_set_o] :
      ( ( complete_Sup_Sup_o @ ( image_set_o_o @ complete_Inf_Inf_o @ A2 ) )
      = ( complete_Inf_Inf_o
        @ ( image_set_o_o @ complete_Sup_Sup_o
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_1190_Sup__Inf,axiom,
    ! [A2: set_set_set_o] :
      ( ( comple90263536869209701_set_o @ ( image_1148209357520620709_set_o @ comple3063163877087187839_set_o @ A2 ) )
      = ( comple3063163877087187839_set_o
        @ ( image_1148209357520620709_set_o @ comple90263536869209701_set_o
          @ ( collect_set_set_o
            @ ^ [Uu: set_set_o] :
              ? [F3: set_set_o > set_o] :
                ( ( Uu
                  = ( image_1148209357520620709_set_o @ F3 @ A2 ) )
                & ! [X3: set_set_o] :
                    ( ( member_set_set_o @ X3 @ A2 )
                   => ( member_set_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_1191_Sup__Inf,axiom,
    ! [A2: set_se3236662274807499670od_c_d] :
      ( ( comple7812139117791525187od_c_d @ ( image_7166828263236204581od_c_d @ comple3677244080108858205od_c_d @ A2 ) )
      = ( comple3677244080108858205od_c_d
        @ ( image_7166828263236204581od_c_d @ comple7812139117791525187od_c_d
          @ ( collec6855837246311779233od_c_d
            @ ^ [Uu: set_se1756390050576932150od_c_d] :
              ? [F3: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
                ( ( Uu
                  = ( image_7166828263236204581od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se1756390050576932150od_c_d] :
                    ( ( member7900373713009613151od_c_d @ X3 @ A2 )
                   => ( member939977866535389951od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_1192_Sup__Inf,axiom,
    ! [A2: set_se6970335898041508694od_c_a] :
      ( ( comple6722362287630405891od_c_a @ ( image_1048133724346124325od_c_a @ comple2587467249947738909od_c_a @ A2 ) )
      = ( comple2587467249947738909od_c_a
        @ ( image_1048133724346124325od_c_a @ comple6722362287630405891od_c_a
          @ ( collec5412296366062437217od_c_a
            @ ^ [Uu: set_se312849170327590134od_c_a] :
              ? [F3: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
                ( ( Uu
                  = ( image_1048133724346124325od_c_a @ F3 @ A2 ) )
                & ! [X3: set_se312849170327590134od_c_a] :
                    ( ( member6456832832760271135od_c_a @ X3 @ A2 )
                   => ( member9073573073229046463od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Sup_Inf
thf(fact_1193_INT__empty,axiom,
    ! [B2: produc2669932201312201782od_c_a > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_8721290246026763926od_c_d @ B2 @ bot_bo267491471446728962od_c_a ) )
      = top_to3206388957835446679od_c_d ) ).

% INT_empty
thf(fact_1194_INT__empty,axiom,
    ! [B2: $o > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_8975626525747676852od_c_d @ B2 @ bot_bot_set_o ) )
      = top_to3206388957835446679od_c_d ) ).

% INT_empty
thf(fact_1195_INT__empty,axiom,
    ! [B2: $o > set_Pr224028844097186518od_c_d] :
      ( ( comple3677244080108858205od_c_d @ ( image_728402167444666691od_c_d @ B2 @ bot_bot_set_o ) )
      = top_to8800170863989631014od_c_d ) ).

% INT_empty
thf(fact_1196_INT__empty,axiom,
    ! [B2: $o > set_Pr8357624050790843030od_c_a] :
      ( ( comple2587467249947738909od_c_a @ ( image_8861997374138323203od_c_a @ B2 @ bot_bot_set_o ) )
      = top_to7710394033828511718od_c_a ) ).

% INT_empty
thf(fact_1197_INT__empty,axiom,
    ! [B2: $o > set_o] :
      ( ( comple3063163877087187839_set_o @ ( image_o_set_o @ B2 @ bot_bot_set_o ) )
      = top_top_set_o ) ).

% INT_empty
thf(fact_1198_INT__empty,axiom,
    ! [B2: set_Pr2898135061646675015od_c_d > set_Pr2898135061646675015od_c_d] :
      ( ( comple5098958804508528846od_c_d @ ( image_242321854666033093od_c_d @ B2 @ bot_bo2787708447028879507od_c_d ) )
      = top_to3206388957835446679od_c_d ) ).

% INT_empty
thf(fact_1199_cINF__const,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo267491471446728962od_c_a )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_8721290246026763926od_c_d
            @ ^ [X3: produc2669932201312201782od_c_a] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1200_cINF__const,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_8975626525747676852od_c_d
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1201_cINF__const,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3677244080108858205od_c_d
          @ ( image_728402167444666691od_c_d
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1202_cINF__const,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple2587467249947738909od_c_a
          @ ( image_8861997374138323203od_c_a
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1203_cINF__const,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3063163877087187839_set_o
          @ ( image_o_set_o
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1204_cINF__const,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo2787708447028879507od_c_d )
     => ( ( comple5098958804508528846od_c_d
          @ ( image_242321854666033093od_c_d
            @ ^ [X3: set_Pr2898135061646675015od_c_d] : C3
            @ A2 ) )
        = C3 ) ) ).

% cINF_const
thf(fact_1205_cSUP__const,axiom,
    ! [A2: set_Pr8357624050790843030od_c_a,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo267491471446728962od_c_a )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_8721290246026763926od_c_d
            @ ^ [X3: produc2669932201312201782od_c_a] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1206_cSUP__const,axiom,
    ! [A2: set_o,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_8975626525747676852od_c_d
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1207_cSUP__const,axiom,
    ! [A2: set_se7324893142979633703od_c_d,C3: set_Pr2898135061646675015od_c_d] :
      ( ( A2 != bot_bo2787708447028879507od_c_d )
     => ( ( comple3966289243721909940od_c_d
          @ ( image_242321854666033093od_c_d
            @ ^ [X3: set_Pr2898135061646675015od_c_d] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1208_cSUP__const,axiom,
    ! [A2: set_o,C3: set_o] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1209_cSUP__const,axiom,
    ! [A2: set_o,C3: set_Pr224028844097186518od_c_d] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple7812139117791525187od_c_d
          @ ( image_728402167444666691od_c_d
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1210_cSUP__const,axiom,
    ! [A2: set_o,C3: set_Pr8357624050790843030od_c_a] :
      ( ( A2 != bot_bot_set_o )
     => ( ( comple6722362287630405891od_c_a
          @ ( image_8861997374138323203od_c_a
            @ ^ [X3: $o] : C3
            @ A2 ) )
        = C3 ) ) ).

% cSUP_const
thf(fact_1211_case__prod__Pair__iden,axiom,
    ! [P2: produc2736867335369716327od_c_d] :
      ( ( produc446450147452880635od_c_d @ produc2733640664908071511od_c_d @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1212_case__prod__Pair__iden,axiom,
    ! [P2: produc3360334827542460560od_c_d] :
      ( ( produc1947021571192102911od_c_d @ produc5940215602367092546od_c_d @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1213_case__prod__Pair__iden,axiom,
    ! [P2: produc1399170498009216118od_c_d] :
      ( ( produc8181551862507722939od_c_d @ produc3045323491883059558od_c_d @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_1214_transitions__from_H_Opelims,axiom,
    ! [X: produc8383368745218977101od_c_a > set_Pr224028844097186518od_c_d,Xa: set_Pr8357624050790843030od_c_a,Xb: produc2346461404597154448od_c_d,Y3: set_Pr8765919954514477662od_c_d] :
      ( ( ( transi3152886439771060669_c_a_d @ X @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P3947645159334878623od_c_d @ transi2637542945977006142_c_a_d @ ( produc8674100842660647322od_c_d @ X @ ( produc8893270258774895377od_c_d @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple3601077897935609291od_c_d
                @ ( image_878321926469689261od_c_d
                  @ ^ [X3: produc2669932201312201782od_c_a] :
                      ( image_2084376701796949005od_c_d
                      @ ( produc8862148518236521539od_c_d
                        @ ^ [Y2: d,Q3: produc2346461404597154448od_c_d] : ( produc4256147637767519854od_c_d @ Xb @ ( produc2733640664908071511od_c_d @ X3 @ ( produc3045323491883059558od_c_d @ Y2 @ Q3 ) ) ) )
                      @ ( X @ ( produc8589540272964826173od_c_a @ Xb @ X3 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P3947645159334878623od_c_d @ transi2637542945977006142_c_a_d @ ( produc8674100842660647322od_c_d @ X @ ( produc8893270258774895377od_c_d @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1215_transitions__from_H_Opelims,axiom,
    ! [X: produc3653416647907290097od_c_a > set_Pr2898135061646675015od_c_d,Xa: set_ma4406981705840051330od_c_a,Xb: produc1399170498009216118od_c_d,Y3: set_Pr6226829810622818539od_c_d] :
      ( ( ( transi7632266320214356496od_c_a @ X @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P7578589733015440028od_c_d @ transi1188201017268736591od_c_a @ ( produc226380280554009183od_c_d @ X @ ( produc280078620671471387od_c_d @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple221672675422206526od_c_d
                @ ( image_3565131711254137278od_c_d
                  @ ^ [X3: mappin2657711750964880076od_c_a] :
                      ( image_2731808243408159959od_c_d
                      @ ( produc4688940134641226273od_c_d
                        @ ^ [Y2: produc2669932201312201782od_c_a,Q3: produc1399170498009216118od_c_d] : ( produc7045801625143393711od_c_d @ Xb @ ( produc5940215602367092546od_c_d @ X3 @ ( produc2733640664908071511od_c_d @ Y2 @ Q3 ) ) ) )
                      @ ( X @ ( produc7298481920129392619od_c_a @ Xb @ X3 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P7578589733015440028od_c_d @ transi1188201017268736591od_c_a @ ( produc226380280554009183od_c_d @ X @ ( produc280078620671471387od_c_d @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1216_transitions__from_H_Opelims,axiom,
    ! [X: produc3865010254569102336_c_d_d > set_Pr1931820589448984816od_c_d,Xa: set_d,Xb: product_prod_c_d,Y3: set_Pr8911588288622154709od_c_d] :
      ( ( ( transi1060146855332110408_d_d_b @ X @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P2806420562413756613od_c_d @ transi7885823061173256905_d_d_b @ ( produc8316072011672839488od_c_d @ X @ ( produc6033585261367860510od_c_d @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple3178548425920542786od_c_d
                @ ( image_738897005216940825od_c_d
                  @ ^ [X3: d] :
                      ( image_4957440889986854762od_c_d
                      @ ( produc2490630522695257248od_c_d
                        @ ^ [Y2: b,Q3: product_prod_c_d] : ( produc860619188946429157od_c_d @ Xb @ ( produc3045323491883059558od_c_d @ X3 @ ( produc3440865926910568064od_c_d @ Y2 @ Q3 ) ) ) )
                      @ ( X @ ( produc5754655040294037616_c_d_d @ Xb @ X3 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P2806420562413756613od_c_d @ transi7885823061173256905_d_d_b @ ( produc8316072011672839488od_c_d @ X @ ( produc6033585261367860510od_c_d @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1217_transitions__from_H_Opelims,axiom,
    ! [X: product_prod_d_b > set_Product_prod_c_d,Xa: set_b,Xb: d,Y3: set_Pr224028844097186518od_c_d] :
      ( ( ( transi6114789321072267300_d_b_c @ X @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6977691515319939181et_b_d @ transi6534242587438537637_d_b_c @ ( produc4290703804598205544et_b_d @ X @ ( product_Pair_set_b_d @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple7812139117791525187od_c_d
                @ ( image_1627772815154405276od_c_d
                  @ ^ [X3: b] :
                      ( image_5617561763891002707od_c_d
                      @ ( produc8093752837941898505od_c_d
                        @ ^ [Y2: c,Q3: d] : ( produc3045323491883059558od_c_d @ Xb @ ( produc3440865926910568064od_c_d @ X3 @ ( product_Pair_c_d @ Y2 @ Q3 ) ) ) )
                      @ ( X @ ( product_Pair_d_b @ Xb @ X3 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P6977691515319939181et_b_d @ transi6534242587438537637_d_b_c @ ( produc4290703804598205544et_b_d @ X @ ( product_Pair_set_b_d @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1218_transitions__from_H_Opelims,axiom,
    ! [X: product_prod_a_b > set_Product_prod_c_a,Xa: set_b,Xb: a,Y3: set_Pr8357624050790843030od_c_a] :
      ( ( ( transi7772877549217426849_a_b_c @ X @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6074164402210365168et_b_a @ transi8192330815583697186_a_b_c @ ( produc3600278798717983083et_b_a @ X @ ( product_Pair_set_b_a @ Xa @ Xb ) ) )
       => ~ ( ( Y3
              = ( comple6722362287630405891od_c_a
                @ ( image_537995984993285980od_c_a
                  @ ^ [X3: b] :
                      ( image_3444923613702048726od_c_a
                      @ ( produc3316798848510373132od_c_a
                        @ ^ [Y2: c,Q3: a] : ( produc5762801557034676134od_c_a @ Xb @ ( produc3440865914000881661od_c_a @ X3 @ ( product_Pair_c_a @ Y2 @ Q3 ) ) ) )
                      @ ( X @ ( product_Pair_a_b @ Xb @ X3 ) ) )
                  @ Xa ) ) )
           => ~ ( accp_P6074164402210365168et_b_a @ transi8192330815583697186_a_b_c @ ( produc3600278798717983083et_b_a @ X @ ( product_Pair_set_b_a @ Xa @ Xb ) ) ) ) ) ) ).

% transitions_from'.pelims
thf(fact_1219_Inf__Sup__le,axiom,
    ! [A2: set_se3083084249005913095od_c_d] :
      ( ord_le6996245393998823od_c_d @ ( comple5098958804508528846od_c_d @ ( image_1688364829223432677od_c_d @ comple3966289243721909940od_c_d @ A2 ) )
      @ ( comple3966289243721909940od_c_d
        @ ( image_1688364829223432677od_c_d @ comple5098958804508528846od_c_d
          @ ( collec259608718228948114od_c_d
            @ ^ [Uu: set_se7324893142979633703od_c_d] :
              ? [F3: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
                ( ( Uu
                  = ( image_1688364829223432677od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se7324893142979633703od_c_d] :
                    ( ( member1954756648797149776od_c_d @ X3 @ A2 )
                   => ( member4652649041329350768od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_1220_Inf__Sup__le,axiom,
    ! [A2: set_set_o] :
      ( ord_less_eq_o @ ( complete_Inf_Inf_o @ ( image_set_o_o @ complete_Sup_Sup_o @ A2 ) )
      @ ( complete_Sup_Sup_o
        @ ( image_set_o_o @ complete_Inf_Inf_o
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_1221_Inf__Sup__le,axiom,
    ! [A2: set_set_set_o] :
      ( ord_less_eq_set_o @ ( comple3063163877087187839_set_o @ ( image_1148209357520620709_set_o @ comple90263536869209701_set_o @ A2 ) )
      @ ( comple90263536869209701_set_o
        @ ( image_1148209357520620709_set_o @ comple3063163877087187839_set_o
          @ ( collect_set_set_o
            @ ^ [Uu: set_set_o] :
              ? [F3: set_set_o > set_o] :
                ( ( Uu
                  = ( image_1148209357520620709_set_o @ F3 @ A2 ) )
                & ! [X3: set_set_o] :
                    ( ( member_set_set_o @ X3 @ A2 )
                   => ( member_set_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_1222_Inf__Sup__le,axiom,
    ! [A2: set_se3236662274807499670od_c_d] :
      ( ord_le4316114996076978806od_c_d @ ( comple3677244080108858205od_c_d @ ( image_7166828263236204581od_c_d @ comple7812139117791525187od_c_d @ A2 ) )
      @ ( comple7812139117791525187od_c_d
        @ ( image_7166828263236204581od_c_d @ comple3677244080108858205od_c_d
          @ ( collec6855837246311779233od_c_d
            @ ^ [Uu: set_se1756390050576932150od_c_d] :
              ? [F3: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
                ( ( Uu
                  = ( image_7166828263236204581od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se1756390050576932150od_c_d] :
                    ( ( member7900373713009613151od_c_d @ X3 @ A2 )
                   => ( member939977866535389951od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_1223_Inf__Sup__le,axiom,
    ! [A2: set_se6970335898041508694od_c_a] :
      ( ord_le3226338165915859510od_c_a @ ( comple2587467249947738909od_c_a @ ( image_1048133724346124325od_c_a @ comple6722362287630405891od_c_a @ A2 ) )
      @ ( comple6722362287630405891od_c_a
        @ ( image_1048133724346124325od_c_a @ comple2587467249947738909od_c_a
          @ ( collec5412296366062437217od_c_a
            @ ^ [Uu: set_se312849170327590134od_c_a] :
              ? [F3: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
                ( ( Uu
                  = ( image_1048133724346124325od_c_a @ F3 @ A2 ) )
                & ! [X3: set_se312849170327590134od_c_a] :
                    ( ( member6456832832760271135od_c_a @ X3 @ A2 )
                   => ( member9073573073229046463od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) ) ) ).

% Inf_Sup_le
thf(fact_1224_Sup__Inf__le,axiom,
    ! [A2: set_se3083084249005913095od_c_d] :
      ( ord_le6996245393998823od_c_d
      @ ( comple3966289243721909940od_c_d
        @ ( image_1688364829223432677od_c_d @ comple5098958804508528846od_c_d
          @ ( collec259608718228948114od_c_d
            @ ^ [Uu: set_se7324893142979633703od_c_d] :
              ? [F3: set_se7324893142979633703od_c_d > set_Pr2898135061646675015od_c_d] :
                ( ( Uu
                  = ( image_1688364829223432677od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se7324893142979633703od_c_d] :
                    ( ( member1954756648797149776od_c_d @ X3 @ A2 )
                   => ( member4652649041329350768od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) )
      @ ( comple5098958804508528846od_c_d @ ( image_1688364829223432677od_c_d @ comple3966289243721909940od_c_d @ A2 ) ) ) ).

% Sup_Inf_le
thf(fact_1225_Sup__Inf__le,axiom,
    ! [A2: set_set_o] :
      ( ord_less_eq_o
      @ ( complete_Sup_Sup_o
        @ ( image_set_o_o @ complete_Inf_Inf_o
          @ ( collect_set_o
            @ ^ [Uu: set_o] :
              ? [F3: set_o > $o] :
                ( ( Uu
                  = ( image_set_o_o @ F3 @ A2 ) )
                & ! [X3: set_o] :
                    ( ( member_set_o @ X3 @ A2 )
                   => ( member_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) )
      @ ( complete_Inf_Inf_o @ ( image_set_o_o @ complete_Sup_Sup_o @ A2 ) ) ) ).

% Sup_Inf_le
thf(fact_1226_Sup__Inf__le,axiom,
    ! [A2: set_set_set_o] :
      ( ord_less_eq_set_o
      @ ( comple90263536869209701_set_o
        @ ( image_1148209357520620709_set_o @ comple3063163877087187839_set_o
          @ ( collect_set_set_o
            @ ^ [Uu: set_set_o] :
              ? [F3: set_set_o > set_o] :
                ( ( Uu
                  = ( image_1148209357520620709_set_o @ F3 @ A2 ) )
                & ! [X3: set_set_o] :
                    ( ( member_set_set_o @ X3 @ A2 )
                   => ( member_set_o @ ( F3 @ X3 ) @ X3 ) ) ) ) ) )
      @ ( comple3063163877087187839_set_o @ ( image_1148209357520620709_set_o @ comple90263536869209701_set_o @ A2 ) ) ) ).

% Sup_Inf_le
thf(fact_1227_Sup__Inf__le,axiom,
    ! [A2: set_se3236662274807499670od_c_d] :
      ( ord_le4316114996076978806od_c_d
      @ ( comple7812139117791525187od_c_d
        @ ( image_7166828263236204581od_c_d @ comple3677244080108858205od_c_d
          @ ( collec6855837246311779233od_c_d
            @ ^ [Uu: set_se1756390050576932150od_c_d] :
              ? [F3: set_se1756390050576932150od_c_d > set_Pr224028844097186518od_c_d] :
                ( ( Uu
                  = ( image_7166828263236204581od_c_d @ F3 @ A2 ) )
                & ! [X3: set_se1756390050576932150od_c_d] :
                    ( ( member7900373713009613151od_c_d @ X3 @ A2 )
                   => ( member939977866535389951od_c_d @ ( F3 @ X3 ) @ X3 ) ) ) ) ) )
      @ ( comple3677244080108858205od_c_d @ ( image_7166828263236204581od_c_d @ comple7812139117791525187od_c_d @ A2 ) ) ) ).

% Sup_Inf_le
thf(fact_1228_Sup__Inf__le,axiom,
    ! [A2: set_se6970335898041508694od_c_a] :
      ( ord_le3226338165915859510od_c_a
      @ ( comple6722362287630405891od_c_a
        @ ( image_1048133724346124325od_c_a @ comple2587467249947738909od_c_a
          @ ( collec5412296366062437217od_c_a
            @ ^ [Uu: set_se312849170327590134od_c_a] :
              ? [F3: set_se312849170327590134od_c_a > set_Pr8357624050790843030od_c_a] :
                ( ( Uu
                  = ( image_1048133724346124325od_c_a @ F3 @ A2 ) )
                & ! [X3: set_se312849170327590134od_c_a] :
                    ( ( member6456832832760271135od_c_a @ X3 @ A2 )
                   => ( member9073573073229046463od_c_a @ ( F3 @ X3 ) @ X3 ) ) ) ) ) )
      @ ( comple2587467249947738909od_c_a @ ( image_1048133724346124325od_c_a @ comple6722362287630405891od_c_a @ A2 ) ) ) ).

% Sup_Inf_le
thf(fact_1229_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1230_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1231_Collect__case__prod__mono,axiom,
    ! [A2: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o,B2: produc2669932201312201782od_c_a > produc1399170498009216118od_c_d > $o] :
      ( ( ord_le4749349702084388302_c_d_o @ A2 @ B2 )
     => ( ord_le6996245393998823od_c_d @ ( collec2789978228259155154od_c_d @ ( produc1656800403805021740_c_d_o @ A2 ) ) @ ( collec2789978228259155154od_c_d @ ( produc1656800403805021740_c_d_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_1232_Inf__superset__mono,axiom,
    ! [B2: set_o,A2: set_o] :
      ( ( ord_less_eq_set_o @ B2 @ A2 )
     => ( ord_less_eq_o @ ( complete_Inf_Inf_o @ A2 ) @ ( complete_Inf_Inf_o @ B2 ) ) ) ).

% Inf_superset_mono
thf(fact_1233_cSup__eq__maximum,axiom,
    ! [Z: nat,X4: set_nat] :
      ( ( member_nat @ Z @ X4 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ X4 )
           => ( ord_less_eq_nat @ X2 @ Z ) )
       => ( ( complete_Sup_Sup_nat @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1234_cSup__eq__maximum,axiom,
    ! [Z: set_Pr2898135061646675015od_c_d,X4: set_se7324893142979633703od_c_d] :
      ( ( member4652649041329350768od_c_d @ Z @ X4 )
     => ( ! [X2: set_Pr2898135061646675015od_c_d] :
            ( ( member4652649041329350768od_c_d @ X2 @ X4 )
           => ( ord_le6996245393998823od_c_d @ X2 @ Z ) )
       => ( ( comple3966289243721909940od_c_d @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1235_cSup__eq__maximum,axiom,
    ! [Z: $o,X4: set_o] :
      ( ( member_o @ Z @ X4 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ X4 )
           => ( ord_less_eq_o @ X2 @ Z ) )
       => ( ( complete_Sup_Sup_o @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1236_cSup__eq__maximum,axiom,
    ! [Z: set_o,X4: set_set_o] :
      ( ( member_set_o @ Z @ X4 )
     => ( ! [X2: set_o] :
            ( ( member_set_o @ X2 @ X4 )
           => ( ord_less_eq_set_o @ X2 @ Z ) )
       => ( ( comple90263536869209701_set_o @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1237_cSup__eq__maximum,axiom,
    ! [Z: set_Pr224028844097186518od_c_d,X4: set_se1756390050576932150od_c_d] :
      ( ( member939977866535389951od_c_d @ Z @ X4 )
     => ( ! [X2: set_Pr224028844097186518od_c_d] :
            ( ( member939977866535389951od_c_d @ X2 @ X4 )
           => ( ord_le4316114996076978806od_c_d @ X2 @ Z ) )
       => ( ( comple7812139117791525187od_c_d @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1238_cSup__eq__maximum,axiom,
    ! [Z: set_Pr8357624050790843030od_c_a,X4: set_se312849170327590134od_c_a] :
      ( ( member9073573073229046463od_c_a @ Z @ X4 )
     => ( ! [X2: set_Pr8357624050790843030od_c_a] :
            ( ( member9073573073229046463od_c_a @ X2 @ X4 )
           => ( ord_le3226338165915859510od_c_a @ X2 @ Z ) )
       => ( ( comple6722362287630405891od_c_a @ X4 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1239_wellorder__Inf__le1,axiom,
    ! [K: nat,A2: set_nat] :
      ( ( member_nat @ K @ A2 )
     => ( ord_less_eq_nat @ ( complete_Inf_Inf_nat @ A2 ) @ K ) ) ).

% wellorder_Inf_le1
thf(fact_1240_cInf__eq,axiom,
    ! [X4: set_nat,A: nat] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ X4 )
         => ( ord_less_eq_nat @ A @ X2 ) )
     => ( ! [Y: nat] :
            ( ! [X6: nat] :
                ( ( member_nat @ X6 @ X4 )
               => ( ord_less_eq_nat @ Y @ X6 ) )
           => ( ord_less_eq_nat @ Y @ A ) )
       => ( ( complete_Inf_Inf_nat @ X4 )
          = A ) ) ) ).

% cInf_eq
thf(fact_1241_cInf__eq__minimum,axiom,
    ! [Z: nat,X4: set_nat] :
      ( ( member_nat @ Z @ X4 )
     => ( ! [X2: nat] :
            ( ( member_nat @ X2 @ X4 )
           => ( ord_less_eq_nat @ Z @ X2 ) )
       => ( ( complete_Inf_Inf_nat @ X4 )
          = Z ) ) ) ).

% cInf_eq_minimum
thf(fact_1242_cInf__eq__minimum,axiom,
    ! [Z: $o,X4: set_o] :
      ( ( member_o @ Z @ X4 )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ X4 )
           => ( ord_less_eq_o @ Z @ X2 ) )
       => ( ( complete_Inf_Inf_o @ X4 )
          = Z ) ) ) ).

% cInf_eq_minimum
thf(fact_1243_Union__mono,axiom,
    ! [A2: set_se7324893142979633703od_c_d,B2: set_se7324893142979633703od_c_d] :
      ( ( ord_le2993186199222016455od_c_d @ A2 @ B2 )
     => ( ord_le6996245393998823od_c_d @ ( comple3966289243721909940od_c_d @ A2 ) @ ( comple3966289243721909940od_c_d @ B2 ) ) ) ).

% Union_mono
thf(fact_1244_Union__mono,axiom,
    ! [A2: set_set_o,B2: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
     => ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).

% Union_mono
thf(fact_1245_Union__mono,axiom,
    ! [A2: set_se1756390050576932150od_c_d,B2: set_se1756390050576932150od_c_d] :
      ( ( ord_le189894988900439254od_c_d @ A2 @ B2 )
     => ( ord_le4316114996076978806od_c_d @ ( comple7812139117791525187od_c_d @ A2 ) @ ( comple7812139117791525187od_c_d @ B2 ) ) ) ).

% Union_mono
thf(fact_1246_Union__mono,axiom,
    ! [A2: set_se312849170327590134od_c_a,B2: set_se312849170327590134od_c_a] :
      ( ( ord_le7969726145505873046od_c_a @ A2 @ B2 )
     => ( ord_le3226338165915859510od_c_a @ ( comple6722362287630405891od_c_a @ A2 ) @ ( comple6722362287630405891od_c_a @ B2 ) ) ) ).

% Union_mono
thf(fact_1247_Sup__subset__mono,axiom,
    ! [A2: set_se7324893142979633703od_c_d,B2: set_se7324893142979633703od_c_d] :
      ( ( ord_le2993186199222016455od_c_d @ A2 @ B2 )
     => ( ord_le6996245393998823od_c_d @ ( comple3966289243721909940od_c_d @ A2 ) @ ( comple3966289243721909940od_c_d @ B2 ) ) ) ).

% Sup_subset_mono
thf(fact_1248_Sup__subset__mono,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A2 ) @ ( complete_Sup_Sup_o @ B2 ) ) ) ).

% Sup_subset_mono
thf(fact_1249_Sup__subset__mono,axiom,
    ! [A2: set_set_o,B2: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ A2 @ B2 )
     => ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A2 ) @ ( comple90263536869209701_set_o @ B2 ) ) ) ).

% Sup_subset_mono
thf(fact_1250_Sup__subset__mono,axiom,
    ! [A2: set_se1756390050576932150od_c_d,B2: set_se1756390050576932150od_c_d] :
      ( ( ord_le189894988900439254od_c_d @ A2 @ B2 )
     => ( ord_le4316114996076978806od_c_d @ ( comple7812139117791525187od_c_d @ A2 ) @ ( comple7812139117791525187od_c_d @ B2 ) ) ) ).

% Sup_subset_mono
thf(fact_1251_Sup__subset__mono,axiom,
    ! [A2: set_se312849170327590134od_c_a,B2: set_se312849170327590134od_c_a] :
      ( ( ord_le7969726145505873046od_c_a @ A2 @ B2 )
     => ( ord_le3226338165915859510od_c_a @ ( comple6722362287630405891od_c_a @ A2 ) @ ( comple6722362287630405891od_c_a @ B2 ) ) ) ).

% Sup_subset_mono
thf(fact_1252_pred__subset__eq2,axiom,
    ! [R2: set_Pr8357624050790843030od_c_a,S: set_Pr8357624050790843030od_c_a] :
      ( ( ord_le3047557917845655871_c_a_o
        @ ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ R2 )
        @ ^ [X3: a,Y2: produc2133359284458116493od_c_a] : ( member5502040268208861279od_c_a @ ( produc5762801557034676134od_c_a @ X3 @ Y2 ) @ S ) )
      = ( ord_le3226338165915859510od_c_a @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1253_pred__subset__eq2,axiom,
    ! [R2: set_Pr2898135061646675015od_c_d,S: set_Pr2898135061646675015od_c_d] :
      ( ( ord_le4749349702084388302_c_d_o
        @ ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ R2 )
        @ ^ [X3: produc2669932201312201782od_c_a,Y2: produc1399170498009216118od_c_d] : ( member4507666829407384208od_c_d @ ( produc2733640664908071511od_c_d @ X3 @ Y2 ) @ S ) )
      = ( ord_le6996245393998823od_c_d @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1254_pred__subset__eq2,axiom,
    ! [R2: set_Pr5052118491147477062od_c_d,S: set_Pr5052118491147477062od_c_d] :
      ( ( ord_le5801467283177209121_c_d_o
        @ ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ R2 )
        @ ^ [X3: mappin2657711750964880076od_c_a,Y2: produc2736867335369716327od_c_d] : ( member5308189881312773415od_c_d @ ( produc5940215602367092546od_c_d @ X3 @ Y2 ) @ S ) )
      = ( ord_le622602621904429222od_c_d @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1255_pred__subset__eq2,axiom,
    ! [R2: set_Pr224028844097186518od_c_d,S: set_Pr224028844097186518od_c_d] :
      ( ( ord_le6449920949082408063_c_d_o
        @ ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ R2 )
        @ ^ [X3: d,Y2: produc2346461404597154448od_c_d] : ( member4231278564905875615od_c_d @ ( produc3045323491883059558od_c_d @ X3 @ Y2 ) @ S ) )
      = ( ord_le4316114996076978806od_c_d @ R2 @ S ) ) ).

% pred_subset_eq2
thf(fact_1256_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1257_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1258_le__cases3,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1259_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1260_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% ord_eq_le_trans
thf(fact_1261_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% ord_le_eq_trans
thf(fact_1262_order__antisym,axiom,
    ! [X: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X )
       => ( X = Y3 ) ) ) ).

% order_antisym
thf(fact_1263_order_Otrans,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ord_less_eq_nat @ A @ C3 ) ) ) ).

% order.trans
thf(fact_1264_order__trans,axiom,
    ! [X: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1265_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1266_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1267_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1268_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C3: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C3 @ B )
       => ( ord_less_eq_nat @ C3 @ A ) ) ) ).

% dual_order.trans
thf(fact_1269_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1270_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A6 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1271_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C3 ) ) ) ) ) ).

% order_subst1
thf(fact_1272_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C3: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C3 )
       => ( ! [X2: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C3 ) ) ) ) ).

% order_subst2
thf(fact_1273_order__eq__refl,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X = Y3 )
     => ( ord_less_eq_nat @ X @ Y3 ) ) ).

% order_eq_refl
thf(fact_1274_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1275_mono__Suc,axiom,
    monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).

% mono_Suc

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( comple3966289243721909940od_c_d
      @ ( image_8721290246026763926od_c_d
        @ ^ [TA: produc2669932201312201782od_c_a] : ( image_3436769108421412982od_c_d @ ( produc2733640664908071511od_c_d @ TA ) @ ( fSM_Co3663583175821284817_d_b_c @ b2 ) )
        @ ( fSM_Co5321671403966444366_a_b_c @ a2 ) ) )
    = ( collec2789978228259155154od_c_d
      @ ^ [Uu: produc2736867335369716327od_c_d] :
        ? [TA: produc2669932201312201782od_c_a,TB: produc1399170498009216118od_c_d] :
          ( ( Uu
            = ( produc2733640664908071511od_c_d @ TA @ TB ) )
          & ( member5502040268208861279od_c_a @ TA @ ( fSM_Co5321671403966444366_a_b_c @ a2 ) )
          & ( member4231278564905875615od_c_d @ TB @ ( fSM_Co3663583175821284817_d_b_c @ b2 ) ) ) ) ) ).

%------------------------------------------------------------------------------