TPTP Problem File: SLH0587^1.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    : Eval_FO/0005_Ailamazyan/prob_02187_083346__15810584_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1906 ( 591 unt; 622 typ;   0 def)
%            Number of atoms       : 3789 (1462 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 11560 ( 473   ~;  54   |; 320   &;9269   @)
%                                         (   0 <=>;1444  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :   95 (  94 usr)
%            Number of type conns  : 2793 (2793   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  531 ( 528 usr;  48 con; 0-4 aty)
%            Number of variables   : 3920 ( 587   ^;3229   !; 104   ?;3920   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:11:03.168
%------------------------------------------------------------------------------
% Could-be-implicit typings (94)
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_I_Eo_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__FO__Ofo____term_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__FO__Ofo____term_Itf__a_J,type,
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thf(ty_n_t__Product____Type__Ounit,type,
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thf(ty_n_t__Option__Ooption_I_Eo_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (528)
thf(sy_c_Ailamazyan_Oeval__eterm_001tf__a_001tf__c,type,
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thf(sy_c_Ailamazyan_Oeval__eterms_001tf__a_001tf__c,type,
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thf(sy_c_Ailamazyan_Oeval__table_001tf__a_001tf__c,type,
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thf(sy_c_Ailamazyan_Oextends__subst_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
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thf(sy_c_Ailamazyan_Oproj__vals_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
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thf(sy_c_member_001t__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member7495770370193619614um_a_c: option5508230710481627655um_a_c > set_op2049297850117519933um_a_c > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
    member_option_nat: option_nat > set_option_nat > $o ).

thf(sy_c_member_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J,type,
    member6131852849436380942um_a_c: option_Sum_sum_a_c > set_op4691527921008582829um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_M_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member1490001167679721142um_a_c: produc4683466728719161101um_a_c > set_Pr1218573636708433773um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member5403367102549231476um_a_c: produc4904670758419960541um_a_c > set_Pr8041419848822008851um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member7870376738782077333um_a_c: produc6487107448139085310um_a_c > set_Pr2999174882133146036um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member6694634564379471929at_nat: produc5146536252030154256at_nat > set_Pr2390076351701138800at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_M_Eo_J,type,
    member7213031410929038765_a_c_o: produc5829762120286046742_a_c_o > set_Pr2109328554291888588_a_c_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member4235642376205993464um_a_c: produc7878403009594063951um_a_c > set_Pr8580000064110529967um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_Mt__Nat__Onat_J,type,
    member6914143599624037265_c_nat: produc1537306021406423912_c_nat > set_Pr6185592159103593672_c_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member1253563076732355473um_a_c: produc5100097535369517928um_a_c > set_Pr2809021152199675080um_a_c > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
    member9116954335612470352_set_o: produc7369051934464679207_set_o > set_Pr4577759397028426247_set_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J_J,type,
    member1684655781224426852um_a_c: produc8122018529987161019um_a_c > set_Pr3153079062249398043um_a_c > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $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__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_J,type,
    member8087264379667941508um_a_c: set_list_Sum_sum_a_c > set_se1773346511508022051um_a_c > $o ).

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

thf(sy_v_X,type,
    x: set_list_Sum_sum_a_c ).

thf(sy_v__092_060sigma_062____,type,
    sigma: nat > option_Sum_sum_a_c ).

thf(sy_v_as____,type,
    as: list_Sum_sum_a_c ).

thf(sy_v_ts,type,
    ts: list_fo_term_a ).

thf(sy_v_vs____,type,
    vs: list_Sum_sum_a_c ).

% Relevant facts (1276)
thf(fact_0_as__def_I1_J,axiom,
    member7772695417316360142um_a_c @ as @ x ).

% as_def(1)
thf(fact_1__092_060open_062_Ithe_A_092_060circ_062_A_092_060sigma_062_J_A_092_060odot_062e_Ats_A_092_060in_062_AX_092_060close_062,axiom,
    member7772695417316360142um_a_c @ ( eval_eterms_a_c @ ( comp_o2608084742246830931_c_nat @ the_Sum_sum_a_c @ sigma ) @ ts ) @ x ).

% \<open>(the \<circ> \<sigma>) \<odot>e ts \<in> X\<close>
thf(fact_2__092_060open_062vs_A_092_060in_062_Aeval__table_Ats_AX_092_060close_062,axiom,
    member7772695417316360142um_a_c @ vs @ ( eval_table_a_c @ ts @ x ) ).

% \<open>vs \<in> eval_table ts X\<close>
thf(fact_3_as__def_I3_J,axiom,
    ( vs
    = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ the_Sum_sum_a_c @ sigma ) @ ( fv_fo_terms_list_a @ ts ) ) ) ).

% as_def(3)
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as_A_092_060sigma_062_O_A_092_060lbrakk_062as_A_092_060in_062_AX_059_Aunify__vals__terms_Aas_Ats_AMap_Oempty_A_061_ASome_A_092_060sigma_062_059_Avs_A_061_Amap_A_Ithe_A_092_060circ_062_A_092_060sigma_062_J_A_Ifv__fo__terms__list_Ats_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [As: list_Sum_sum_a_c] :
        ( ( member7772695417316360142um_a_c @ As @ x )
       => ! [Sigma: nat > option_Sum_sum_a_c] :
            ( ( ( unify_vals_terms_a_c @ As @ ts
                @ ^ [X: nat] : none_Sum_sum_a_c )
              = ( some_n5886337604800580545um_a_c @ Sigma ) )
           => ( vs
             != ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ the_Sum_sum_a_c @ Sigma ) @ ( fv_fo_terms_list_a @ ts ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>as \<sigma>. \<lbrakk>as \<in> X; unify_vals_terms as ts Map.empty = Some \<sigma>; vs = map (the \<circ> \<sigma>) (fv_fo_terms_list ts)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_eval__eterms__cong,axiom,
    ! [Ts: list_fo_term_a,Sigma2: nat > sum_sum_a_c,Sigma3: nat > sum_sum_a_c] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ ( fv_fo_terms_set_a @ Ts ) )
         => ( ( Sigma2 @ N )
            = ( Sigma3 @ N ) ) )
     => ( ( eval_eterms_a_c @ Sigma2 @ Ts )
        = ( eval_eterms_a_c @ Sigma3 @ Ts ) ) ) ).

% eval_eterms_cong
thf(fact_6_eval__eterms__fv__fo__terms__set,axiom,
    ! [Sigma2: nat > sum_sum_a_c,Ts: list_fo_term_a,Sigma3: nat > sum_sum_a_c,N2: nat] :
      ( ( ( eval_eterms_a_c @ Sigma2 @ Ts )
        = ( eval_eterms_a_c @ Sigma3 @ Ts ) )
     => ( ( member_nat @ N2 @ ( fv_fo_terms_set_a @ Ts ) )
       => ( ( Sigma2 @ N2 )
          = ( Sigma3 @ N2 ) ) ) ) ).

% eval_eterms_fv_fo_terms_set
thf(fact_7_proj__vals__def,axiom,
    ( proj_v4774688853359684992um_a_c
    = ( ^ [R: set_nat_Sum_sum_a_c,Ns: list_nat] :
          ( image_7318554134589591124um_a_c
          @ ^ [Tau: nat > sum_sum_a_c] : ( map_nat_Sum_sum_a_c @ Tau @ Ns )
          @ R ) ) ) ).

% proj_vals_def
thf(fact_8_as__def_I2_J,axiom,
    ( ( unify_vals_terms_a_c @ as @ ts
      @ ^ [X: nat] : none_Sum_sum_a_c )
    = ( some_n5886337604800580545um_a_c @ sigma ) ) ).

% as_def(2)
thf(fact_9_eval__eterms__def,axiom,
    ( eval_eterms_a_c
    = ( ^ [Sigma4: nat > sum_sum_a_c] : ( map_fo7965992548707740699um_a_c @ ( eval_eterm_a_c @ Sigma4 ) ) ) ) ).

% eval_eterms_def
thf(fact_10_eval__terms__eterms,axiom,
    ! [Sigma2: nat > a,Ts: list_fo_term_a] :
      ( ( map_a_Sum_sum_a_c @ sum_Inl_a_c @ ( eval_terms_a @ Sigma2 @ Ts ) )
      = ( eval_eterms_a_c @ ( comp_a1017791430566731092_c_nat @ sum_Inl_a_c @ Sigma2 ) @ Ts ) ) ).

% eval_terms_eterms
thf(fact_11_unify__vals__terms__sound,axiom,
    ! [Vs: list_Sum_sum_a_c,Ts: list_fo_term_a,Sigma2: nat > option_Sum_sum_a_c,Sigma3: nat > option_Sum_sum_a_c] :
      ( ( ( unify_vals_terms_a_c @ Vs @ Ts @ Sigma2 )
        = ( some_n5886337604800580545um_a_c @ Sigma3 ) )
     => ( ( eval_eterms_a_c @ ( comp_o2608084742246830931_c_nat @ the_Sum_sum_a_c @ Sigma3 ) @ Ts )
        = Vs ) ) ).

% unify_vals_terms_sound
thf(fact_12_List_Omap_Ocomp,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( comp_l3632815226211824003st_nat @ ( map_op5891491687147645028um_a_c @ F ) @ ( map_na1879090439989666701um_a_c @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_13_List_Omap_Ocomp,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: nat > sum_sum_a_c] :
      ( ( comp_l3095377238765006259st_nat @ ( map_Su9140647359041231892um_a_c @ F ) @ ( map_nat_Sum_sum_a_c @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_S8935214348162178051_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_14_List_Omap_Ocomp,axiom,
    ! [F: nat > sum_sum_a_c,G: nat > nat] :
      ( ( comp_l3643885024703143644st_nat @ ( map_nat_Sum_sum_a_c @ F ) @ ( map_nat_nat @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_n8637127618742830252_c_nat @ F @ G ) ) ) ).

% List.map.comp
thf(fact_15_list_Omap__comp,axiom,
    ! [G: option_Sum_sum_a_c > sum_sum_a_c,F: nat > option_Sum_sum_a_c,V: list_nat] :
      ( ( map_op5891491687147645028um_a_c @ G @ ( map_na1879090439989666701um_a_c @ F @ V ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_16_list_Omap__comp,axiom,
    ! [G: sum_sum_a_c > sum_sum_a_c,F: nat > sum_sum_a_c,V: list_nat] :
      ( ( map_Su9140647359041231892um_a_c @ G @ ( map_nat_Sum_sum_a_c @ F @ V ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_S8935214348162178051_c_nat @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_17_list_Omap__comp,axiom,
    ! [G: nat > sum_sum_a_c,F: nat > nat,V: list_nat] :
      ( ( map_nat_Sum_sum_a_c @ G @ ( map_nat_nat @ F @ V ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_n8637127618742830252_c_nat @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_18_List_Omap_Ocompositionality,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,List: list_nat] :
      ( ( map_op5891491687147645028um_a_c @ F @ ( map_na1879090439989666701um_a_c @ G @ List ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_19_List_Omap_Ocompositionality,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: nat > sum_sum_a_c,List: list_nat] :
      ( ( map_Su9140647359041231892um_a_c @ F @ ( map_nat_Sum_sum_a_c @ G @ List ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_S8935214348162178051_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_20_List_Omap_Ocompositionality,axiom,
    ! [F: nat > sum_sum_a_c,G: nat > nat,List: list_nat] :
      ( ( map_nat_Sum_sum_a_c @ F @ ( map_nat_nat @ G @ List ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_n8637127618742830252_c_nat @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_21_map__map,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,Xs: list_nat] :
      ( ( map_op5891491687147645028um_a_c @ F @ ( map_na1879090439989666701um_a_c @ G @ Xs ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_22_map__map,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: nat > sum_sum_a_c,Xs: list_nat] :
      ( ( map_Su9140647359041231892um_a_c @ F @ ( map_nat_Sum_sum_a_c @ G @ Xs ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_S8935214348162178051_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_23_map__map,axiom,
    ! [F: nat > sum_sum_a_c,G: nat > nat,Xs: list_nat] :
      ( ( map_nat_Sum_sum_a_c @ F @ ( map_nat_nat @ G @ Xs ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_n8637127618742830252_c_nat @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_24_map__comp__map,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( comp_l3632815226211824003st_nat @ ( map_op5891491687147645028um_a_c @ F ) @ ( map_na1879090439989666701um_a_c @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_25_map__comp__map,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: nat > sum_sum_a_c] :
      ( ( comp_l3095377238765006259st_nat @ ( map_Su9140647359041231892um_a_c @ F ) @ ( map_nat_Sum_sum_a_c @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_S8935214348162178051_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_26_map__comp__map,axiom,
    ! [F: nat > sum_sum_a_c,G: nat > nat] :
      ( ( comp_l3643885024703143644st_nat @ ( map_nat_Sum_sum_a_c @ F ) @ ( map_nat_nat @ G ) )
      = ( map_nat_Sum_sum_a_c @ ( comp_n8637127618742830252_c_nat @ F @ G ) ) ) ).

% map_comp_map
thf(fact_27_option_Ocollapse,axiom,
    ! [Option: option_Sum_sum_a_c] :
      ( ( Option != none_Sum_sum_a_c )
     => ( ( some_Sum_sum_a_c @ ( the_Sum_sum_a_c @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_28_option_Ocollapse,axiom,
    ! [Option: option6339742336662979638um_a_c] :
      ( ( Option != none_n2797051391425193029um_a_c )
     => ( ( some_n5886337604800580545um_a_c @ ( the_na7545698918922854738um_a_c @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_29_not__None__eq,axiom,
    ! [X2: option_Sum_sum_a_c] :
      ( ( X2 != none_Sum_sum_a_c )
      = ( ? [Y: sum_sum_a_c] :
            ( X2
            = ( some_Sum_sum_a_c @ Y ) ) ) ) ).

% not_None_eq
thf(fact_30_not__None__eq,axiom,
    ! [X2: option6339742336662979638um_a_c] :
      ( ( X2 != none_n2797051391425193029um_a_c )
      = ( ? [Y: nat > option_Sum_sum_a_c] :
            ( X2
            = ( some_n5886337604800580545um_a_c @ Y ) ) ) ) ).

% not_None_eq
thf(fact_31_not__Some__eq,axiom,
    ! [X2: option_Sum_sum_a_c] :
      ( ( ! [Y: sum_sum_a_c] :
            ( X2
           != ( some_Sum_sum_a_c @ Y ) ) )
      = ( X2 = none_Sum_sum_a_c ) ) ).

% not_Some_eq
thf(fact_32_not__Some__eq,axiom,
    ! [X2: option6339742336662979638um_a_c] :
      ( ( ! [Y: nat > option_Sum_sum_a_c] :
            ( X2
           != ( some_n5886337604800580545um_a_c @ Y ) ) )
      = ( X2 = none_n2797051391425193029um_a_c ) ) ).

% not_Some_eq
thf(fact_33_unify__vals__terms__extends,axiom,
    ! [Vs: list_Sum_sum_a_c,Ts: list_fo_term_a,Sigma2: nat > option_Sum_sum_a_c,Sigma3: nat > option_Sum_sum_a_c] :
      ( ( ( unify_vals_terms_a_c @ Vs @ Ts @ Sigma2 )
        = ( some_n5886337604800580545um_a_c @ Sigma3 ) )
     => ( extend2099978610298133290um_a_c @ Sigma2 @ Sigma3 ) ) ).

% unify_vals_terms_extends
thf(fact_34_option_Oexhaust__sel,axiom,
    ! [Option: option_Sum_sum_a_c] :
      ( ( Option != none_Sum_sum_a_c )
     => ( Option
        = ( some_Sum_sum_a_c @ ( the_Sum_sum_a_c @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_35_option_Oexhaust__sel,axiom,
    ! [Option: option6339742336662979638um_a_c] :
      ( ( Option != none_n2797051391425193029um_a_c )
     => ( Option
        = ( some_n5886337604800580545um_a_c @ ( the_na7545698918922854738um_a_c @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_36_eval__terms__cong,axiom,
    ! [Ts: list_fo_term_a,Sigma2: nat > a,Sigma3: nat > a] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ ( fv_fo_terms_set_a @ Ts ) )
         => ( ( Sigma2 @ N )
            = ( Sigma3 @ N ) ) )
     => ( ( eval_terms_a @ Sigma2 @ Ts )
        = ( eval_terms_a @ Sigma3 @ Ts ) ) ) ).

% eval_terms_cong
thf(fact_37_eval__terms__fv__fo__terms__set,axiom,
    ! [Sigma2: nat > a,Ts: list_fo_term_a,Sigma3: nat > a,N2: nat] :
      ( ( ( eval_terms_a @ Sigma2 @ Ts )
        = ( eval_terms_a @ Sigma3 @ Ts ) )
     => ( ( member_nat @ N2 @ ( fv_fo_terms_set_a @ Ts ) )
       => ( ( Sigma2 @ N2 )
          = ( Sigma3 @ N2 ) ) ) ) ).

% eval_terms_fv_fo_terms_set
thf(fact_38_comp__apply,axiom,
    ( comp_o2608084742246830931_c_nat
    = ( ^ [F2: option_Sum_sum_a_c > sum_sum_a_c,G2: nat > option_Sum_sum_a_c,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_apply
thf(fact_39_image__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member4884986500679352621um_a_c @ X2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_7318554134589591124um_a_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_40_image__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c,X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_8132764781264612725um_a_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_41_image__eqI,axiom,
    ! [B: $o,F: list_Sum_sum_a_c > $o,X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( member_o @ B @ ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_42_image__eqI,axiom,
    ! [B: nat,F: list_Sum_sum_a_c > nat,X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( member_nat @ B @ ( image_7712311756382514062_c_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_43_image__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: $o > list_Sum_sum_a_c,X2: $o,A: set_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_o @ X2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_1489810555362023050um_a_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_44_image__eqI,axiom,
    ! [B: $o,F: $o > $o,X2: $o,A: set_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_o @ X2 @ A )
       => ( member_o @ B @ ( image_o_o2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_45_image__eqI,axiom,
    ! [B: nat,F: $o > nat,X2: $o,A: set_o] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_o @ X2 @ A )
       => ( member_nat @ B @ ( image_o_nat2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_46_image__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: nat > list_Sum_sum_a_c,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_1048630792592356622um_a_c @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_47_image__eqI,axiom,
    ! [B: $o,F: nat > $o,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_48_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_49_option_Oinject,axiom,
    ! [X22: nat > option_Sum_sum_a_c,Y2: nat > option_Sum_sum_a_c] :
      ( ( ( some_n5886337604800580545um_a_c @ X22 )
        = ( some_n5886337604800580545um_a_c @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_50_None__notin__image__Some,axiom,
    ! [A: set_Sum_sum_a_c] :
      ~ ( member6131852849436380942um_a_c @ none_Sum_sum_a_c @ ( image_152546304820467493um_a_c @ some_Sum_sum_a_c @ A ) ) ).

% None_notin_image_Some
thf(fact_51_None__notin__image__Some,axiom,
    ! [A: set_na2086515510486960284um_a_c] :
      ~ ( member2769028661778116429um_a_c @ none_n2797051391425193029um_a_c @ ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ A ) ) ).

% None_notin_image_Some
thf(fact_52_rev__image__eqI,axiom,
    ! [X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c,B: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( member4884986500679352621um_a_c @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( image_7318554134589591124um_a_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_53_rev__image__eqI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( image_8132764781264612725um_a_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_54_rev__image__eqI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: $o,F: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_o @ B @ ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_55_rev__image__eqI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: nat,F: list_Sum_sum_a_c > nat] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_7712311756382514062_c_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_56_rev__image__eqI,axiom,
    ! [X2: $o,A: set_o,B: list_Sum_sum_a_c,F: $o > list_Sum_sum_a_c] :
      ( ( member_o @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( image_1489810555362023050um_a_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_57_rev__image__eqI,axiom,
    ! [X2: $o,A: set_o,B: $o,F: $o > $o] :
      ( ( member_o @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_o @ B @ ( image_o_o2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_58_rev__image__eqI,axiom,
    ! [X2: $o,A: set_o,B: nat,F: $o > nat] :
      ( ( member_o @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_o_nat2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_59_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: list_Sum_sum_a_c,F: nat > list_Sum_sum_a_c] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( image_1048630792592356622um_a_c @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_60_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: $o,F: nat > $o] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_o @ B @ ( image_nat_o2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_61_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_62_ball__imageD,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: list_Sum_sum_a_c > $o] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ ( image_7318554134589591124um_a_c @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: nat > sum_sum_a_c] :
          ( ( member4884986500679352621um_a_c @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_63_image__cong,axiom,
    ! [M: set_nat_Sum_sum_a_c,N3: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( M = N3 )
     => ( ! [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_7318554134589591124um_a_c @ F @ M )
          = ( image_7318554134589591124um_a_c @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_64_bex__imageD,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: list_Sum_sum_a_c > $o] :
      ( ? [X4: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X4 @ ( image_7318554134589591124um_a_c @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: nat > sum_sum_a_c] :
          ( ( member4884986500679352621um_a_c @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_65_image__iff,axiom,
    ! [Z: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ Z @ ( image_7318554134589591124um_a_c @ F @ A ) )
      = ( ? [X: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X @ A )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_66_imageI,axiom,
    ! [X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( member4884986500679352621um_a_c @ X2 @ A )
     => ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_7318554134589591124um_a_c @ F @ A ) ) ) ).

% imageI
thf(fact_67_imageI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_8132764781264612725um_a_c @ F @ A ) ) ) ).

% imageI
thf(fact_68_imageI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( member_o @ ( F @ X2 ) @ ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ).

% imageI
thf(fact_69_imageI,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > nat] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_7712311756382514062_c_nat @ F @ A ) ) ) ).

% imageI
thf(fact_70_imageI,axiom,
    ! [X2: $o,A: set_o,F: $o > list_Sum_sum_a_c] :
      ( ( member_o @ X2 @ A )
     => ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_1489810555362023050um_a_c @ F @ A ) ) ) ).

% imageI
thf(fact_71_imageI,axiom,
    ! [X2: $o,A: set_o,F: $o > $o] :
      ( ( member_o @ X2 @ A )
     => ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A ) ) ) ).

% imageI
thf(fact_72_imageI,axiom,
    ! [X2: $o,A: set_o,F: $o > nat] :
      ( ( member_o @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_o_nat2 @ F @ A ) ) ) ).

% imageI
thf(fact_73_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > list_Sum_sum_a_c] :
      ( ( member_nat @ X2 @ A )
     => ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_1048630792592356622um_a_c @ F @ A ) ) ) ).

% imageI
thf(fact_74_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > $o] :
      ( ( member_nat @ X2 @ A )
     => ( member_o @ ( F @ X2 ) @ ( image_nat_o2 @ F @ A ) ) ) ).

% imageI
thf(fact_75_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_nat_nat2 @ F @ A ) ) ) ).

% imageI
thf(fact_76_comp__eq__dest__lhs,axiom,
    ! [A2: option_Sum_sum_a_c > sum_sum_a_c,B: nat > option_Sum_sum_a_c,C: nat > sum_sum_a_c,V: nat] :
      ( ( ( comp_o2608084742246830931_c_nat @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_77_mem__Collect__eq,axiom,
    ! [A2: list_Sum_sum_a_c,P: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ A2 @ ( collec8219452656984879116um_a_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( member_o @ A2 @ ( collect_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A2: nat > sum_sum_a_c,P: ( nat > sum_sum_a_c ) > $o] :
      ( ( member4884986500679352621um_a_c @ A2 @ ( collec5227572641185395563um_a_c @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ( collec8219452656984879116um_a_c
        @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A: set_o] :
      ( ( collect_o
        @ ^ [X: $o] : ( member_o @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A: set_nat_Sum_sum_a_c] :
      ( ( collec5227572641185395563um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_87_Collect__cong,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ! [X3: nat > sum_sum_a_c] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec5227572641185395563um_a_c @ P )
        = ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_cong
thf(fact_88_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_89_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_90_comp__eq__elim,axiom,
    ! [A2: option_Sum_sum_a_c > sum_sum_a_c,B: nat > option_Sum_sum_a_c,C: option_Sum_sum_a_c > sum_sum_a_c,D: nat > option_Sum_sum_a_c] :
      ( ( ( comp_o2608084742246830931_c_nat @ A2 @ B )
        = ( comp_o2608084742246830931_c_nat @ C @ D ) )
     => ! [V2: nat] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_91_comp__eq__dest,axiom,
    ! [A2: option_Sum_sum_a_c > sum_sum_a_c,B: nat > option_Sum_sum_a_c,C: option_Sum_sum_a_c > sum_sum_a_c,D: nat > option_Sum_sum_a_c,V: nat] :
      ( ( ( comp_o2608084742246830931_c_nat @ A2 @ B )
        = ( comp_o2608084742246830931_c_nat @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_92_comp__assoc,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,H: nat > nat] :
      ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ H )
      = ( comp_o2608084742246830931_c_nat @ F @ ( comp_n4859801332697915004_c_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_93_comp__assoc,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c] :
      ( ( comp_o2608084742246830931_c_nat @ ( comp_S7844377493806175402um_a_c @ F @ G ) @ H )
      = ( comp_S8935214348162178051_c_nat @ F @ ( comp_o2608084742246830931_c_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_94_comp__assoc,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > option_Sum_sum_a_c,H: nat > option_Sum_sum_a_c] :
      ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ G ) @ H )
      = ( comp_o2608084742246830931_c_nat @ F @ ( comp_o5720133263240570531_c_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_95_comp__def,axiom,
    ( comp_o2608084742246830931_c_nat
    = ( ^ [F2: option_Sum_sum_a_c > sum_sum_a_c,G2: nat > option_Sum_sum_a_c,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).

% comp_def
thf(fact_96_Compr__image__eq,axiom,
    ! [F: $o > $o,A: set_o,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_o_o2 @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_o_o2 @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_97_Compr__image__eq,axiom,
    ! [F: nat > $o,A: set_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_nat_o2 @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_o2 @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_98_Compr__image__eq,axiom,
    ! [F: $o > nat,A: set_o,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_o_nat2 @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_o_nat2 @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_99_Compr__image__eq,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat2 @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_nat2 @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_100_Compr__image__eq,axiom,
    ! [F: product_prod_nat_nat > $o,A: set_Pr1261947904930325089at_nat,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_3693632289388996572_nat_o @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_3693632289388996572_nat_o @ F
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_101_Compr__image__eq,axiom,
    ! [F: product_prod_nat_nat > nat,A: set_Pr1261947904930325089at_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_2486076414777270412at_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_2486076414777270412at_nat @ F
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_102_Compr__image__eq,axiom,
    ! [F: $o > product_prod_nat_nat,A: set_o,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( image_3855930084881510382at_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_3855930084881510382at_nat @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_103_Compr__image__eq,axiom,
    ! [F: nat > product_prod_nat_nat,A: set_nat,P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( image_5846123807819985514at_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_5846123807819985514at_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_104_Compr__image__eq,axiom,
    ! [F: $o > list_Sum_sum_a_c,A: set_o,P: list_Sum_sum_a_c > $o] :
      ( ( collec8219452656984879116um_a_c
        @ ^ [X: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X @ ( image_1489810555362023050um_a_c @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_1489810555362023050um_a_c @ F
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_105_Compr__image__eq,axiom,
    ! [F: list_Sum_sum_a_c > $o,A: set_list_Sum_sum_a_c,P: $o > $o] :
      ( ( collect_o
        @ ^ [X: $o] :
            ( ( member_o @ X @ ( image_1429702505460152410_a_c_o @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_1429702505460152410_a_c_o @ F
        @ ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_106_image__image,axiom,
    ! [F: list_Sum_sum_a_c > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( image_8132764781264612725um_a_c @ F @ ( image_7318554134589591124um_a_c @ G @ A ) )
      = ( image_7318554134589591124um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_107_image__image,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ ( image_5025405432202070451um_a_c @ G @ A ) )
      = ( image_7318554134589591124um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_108_imageE,axiom,
    ! [B: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_7318554134589591124um_a_c @ F @ A ) )
     => ~ ! [X3: nat > sum_sum_a_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member4884986500679352621um_a_c @ X3 @ A ) ) ) ).

% imageE
thf(fact_109_imageE,axiom,
    ! [B: list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_8132764781264612725um_a_c @ F @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% imageE
thf(fact_110_imageE,axiom,
    ! [B: list_Sum_sum_a_c,F: $o > list_Sum_sum_a_c,A: set_o] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_1489810555362023050um_a_c @ F @ A ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% imageE
thf(fact_111_imageE,axiom,
    ! [B: list_Sum_sum_a_c,F: nat > list_Sum_sum_a_c,A: set_nat] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_1048630792592356622um_a_c @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_112_imageE,axiom,
    ! [B: $o,F: list_Sum_sum_a_c > $o,A: set_list_Sum_sum_a_c] :
      ( ( member_o @ B @ ( image_1429702505460152410_a_c_o @ F @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% imageE
thf(fact_113_imageE,axiom,
    ! [B: $o,F: $o > $o,A: set_o] :
      ( ( member_o @ B @ ( image_o_o2 @ F @ A ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% imageE
thf(fact_114_imageE,axiom,
    ! [B: $o,F: nat > $o,A: set_nat] :
      ( ( member_o @ B @ ( image_nat_o2 @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_115_imageE,axiom,
    ! [B: nat,F: list_Sum_sum_a_c > nat,A: set_list_Sum_sum_a_c] :
      ( ( member_nat @ B @ ( image_7712311756382514062_c_nat @ F @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% imageE
thf(fact_116_imageE,axiom,
    ! [B: nat,F: $o > nat,A: set_o] :
      ( ( member_nat @ B @ ( image_o_nat2 @ F @ A ) )
     => ~ ! [X3: $o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% imageE
thf(fact_117_imageE,axiom,
    ! [B: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat2 @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_118_unify__vals__terms__complete,axiom,
    ! [Sigma5: nat > sum_sum_a_c,Ts: list_fo_term_a,Vs: list_Sum_sum_a_c,Sigma2: nat > option_Sum_sum_a_c] :
      ( ( ( eval_eterms_a_c @ Sigma5 @ Ts )
        = Vs )
     => ( ! [N: nat] :
            ( ( ( Sigma2 @ N )
             != none_Sum_sum_a_c )
           => ( ( Sigma2 @ N )
              = ( some_Sum_sum_a_c @ ( Sigma5 @ N ) ) ) )
       => ? [Sigma6: nat > option_Sum_sum_a_c] :
            ( ( unify_vals_terms_a_c @ Vs @ Ts @ Sigma2 )
            = ( some_n5886337604800580545um_a_c @ Sigma6 ) ) ) ) ).

% unify_vals_terms_complete
thf(fact_119_combine__options__cases,axiom,
    ! [X2: option_Sum_sum_a_c,P: option_Sum_sum_a_c > option_Sum_sum_a_c > $o,Y3: option_Sum_sum_a_c] :
      ( ( ( X2 = none_Sum_sum_a_c )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_Sum_sum_a_c )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A3: sum_sum_a_c,B2: sum_sum_a_c] :
              ( ( X2
                = ( some_Sum_sum_a_c @ A3 ) )
             => ( ( Y3
                  = ( some_Sum_sum_a_c @ B2 ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_120_combine__options__cases,axiom,
    ! [X2: option_Sum_sum_a_c,P: option_Sum_sum_a_c > option6339742336662979638um_a_c > $o,Y3: option6339742336662979638um_a_c] :
      ( ( ( X2 = none_Sum_sum_a_c )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_n2797051391425193029um_a_c )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A3: sum_sum_a_c,B2: nat > option_Sum_sum_a_c] :
              ( ( X2
                = ( some_Sum_sum_a_c @ A3 ) )
             => ( ( Y3
                  = ( some_n5886337604800580545um_a_c @ B2 ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_121_combine__options__cases,axiom,
    ! [X2: option6339742336662979638um_a_c,P: option6339742336662979638um_a_c > option_Sum_sum_a_c > $o,Y3: option_Sum_sum_a_c] :
      ( ( ( X2 = none_n2797051391425193029um_a_c )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_Sum_sum_a_c )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A3: nat > option_Sum_sum_a_c,B2: sum_sum_a_c] :
              ( ( X2
                = ( some_n5886337604800580545um_a_c @ A3 ) )
             => ( ( Y3
                  = ( some_Sum_sum_a_c @ B2 ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_122_combine__options__cases,axiom,
    ! [X2: option6339742336662979638um_a_c,P: option6339742336662979638um_a_c > option6339742336662979638um_a_c > $o,Y3: option6339742336662979638um_a_c] :
      ( ( ( X2 = none_n2797051391425193029um_a_c )
       => ( P @ X2 @ Y3 ) )
     => ( ( ( Y3 = none_n2797051391425193029um_a_c )
         => ( P @ X2 @ Y3 ) )
       => ( ! [A3: nat > option_Sum_sum_a_c,B2: nat > option_Sum_sum_a_c] :
              ( ( X2
                = ( some_n5886337604800580545um_a_c @ A3 ) )
             => ( ( Y3
                  = ( some_n5886337604800580545um_a_c @ B2 ) )
               => ( P @ X2 @ Y3 ) ) )
         => ( P @ X2 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_123_split__option__all,axiom,
    ( ( ^ [P2: option_Sum_sum_a_c > $o] :
        ! [X5: option_Sum_sum_a_c] : ( P2 @ X5 ) )
    = ( ^ [P3: option_Sum_sum_a_c > $o] :
          ( ( P3 @ none_Sum_sum_a_c )
          & ! [X: sum_sum_a_c] : ( P3 @ ( some_Sum_sum_a_c @ X ) ) ) ) ) ).

% split_option_all
thf(fact_124_split__option__all,axiom,
    ( ( ^ [P2: option6339742336662979638um_a_c > $o] :
        ! [X5: option6339742336662979638um_a_c] : ( P2 @ X5 ) )
    = ( ^ [P3: option6339742336662979638um_a_c > $o] :
          ( ( P3 @ none_n2797051391425193029um_a_c )
          & ! [X: nat > option_Sum_sum_a_c] : ( P3 @ ( some_n5886337604800580545um_a_c @ X ) ) ) ) ) ).

% split_option_all
thf(fact_125_split__option__ex,axiom,
    ( ( ^ [P2: option_Sum_sum_a_c > $o] :
        ? [X5: option_Sum_sum_a_c] : ( P2 @ X5 ) )
    = ( ^ [P3: option_Sum_sum_a_c > $o] :
          ( ( P3 @ none_Sum_sum_a_c )
          | ? [X: sum_sum_a_c] : ( P3 @ ( some_Sum_sum_a_c @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_126_split__option__ex,axiom,
    ( ( ^ [P2: option6339742336662979638um_a_c > $o] :
        ? [X5: option6339742336662979638um_a_c] : ( P2 @ X5 ) )
    = ( ^ [P3: option6339742336662979638um_a_c > $o] :
          ( ( P3 @ none_n2797051391425193029um_a_c )
          | ? [X: nat > option_Sum_sum_a_c] : ( P3 @ ( some_n5886337604800580545um_a_c @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_127_option_Oexhaust,axiom,
    ! [Y3: option_Sum_sum_a_c] :
      ( ( Y3 != none_Sum_sum_a_c )
     => ~ ! [X23: sum_sum_a_c] :
            ( Y3
           != ( some_Sum_sum_a_c @ X23 ) ) ) ).

% option.exhaust
thf(fact_128_option_Oexhaust,axiom,
    ! [Y3: option6339742336662979638um_a_c] :
      ( ( Y3 != none_n2797051391425193029um_a_c )
     => ~ ! [X23: nat > option_Sum_sum_a_c] :
            ( Y3
           != ( some_n5886337604800580545um_a_c @ X23 ) ) ) ).

% option.exhaust
thf(fact_129_option_OdiscI,axiom,
    ! [Option: option_Sum_sum_a_c,X22: sum_sum_a_c] :
      ( ( Option
        = ( some_Sum_sum_a_c @ X22 ) )
     => ( Option != none_Sum_sum_a_c ) ) ).

% option.discI
thf(fact_130_option_OdiscI,axiom,
    ! [Option: option6339742336662979638um_a_c,X22: nat > option_Sum_sum_a_c] :
      ( ( Option
        = ( some_n5886337604800580545um_a_c @ X22 ) )
     => ( Option != none_n2797051391425193029um_a_c ) ) ).

% option.discI
thf(fact_131_option_Odistinct_I1_J,axiom,
    ! [X22: sum_sum_a_c] :
      ( none_Sum_sum_a_c
     != ( some_Sum_sum_a_c @ X22 ) ) ).

% option.distinct(1)
thf(fact_132_option_Odistinct_I1_J,axiom,
    ! [X22: nat > option_Sum_sum_a_c] :
      ( none_n2797051391425193029um_a_c
     != ( some_n5886337604800580545um_a_c @ X22 ) ) ).

% option.distinct(1)
thf(fact_133_image__eq__imp__comp,axiom,
    ! [F: nat > option_Sum_sum_a_c,A: set_nat,G: nat > option_Sum_sum_a_c,B3: set_nat,H: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( image_8467336918603940942um_a_c @ F @ A )
        = ( image_8467336918603940942um_a_c @ G @ B3 ) )
     => ( ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ H @ F ) @ A )
        = ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ H @ G ) @ B3 ) ) ) ).

% image_eq_imp_comp
thf(fact_134_image__eq__imp__comp,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c,H: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ A )
        = ( image_7318554134589591124um_a_c @ G @ B3 ) )
     => ( ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ H @ F ) @ A )
        = ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ H @ G ) @ B3 ) ) ) ).

% image_eq_imp_comp
thf(fact_135_image__comp,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,R2: set_nat] :
      ( ( image_7911160186837173029um_a_c @ F @ ( image_8467336918603940942um_a_c @ G @ R2 ) )
      = ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_136_image__comp,axiom,
    ! [F: list_Sum_sum_a_c > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,R2: set_nat_Sum_sum_a_c] :
      ( ( image_8132764781264612725um_a_c @ F @ ( image_7318554134589591124um_a_c @ G @ R2 ) )
      = ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_137_image__comp,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c,R2: set_nat_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ ( image_5025405432202070451um_a_c @ G @ R2 ) )
      = ( image_7318554134589591124um_a_c @ ( comp_n57436397065578056um_a_c @ F @ G ) @ R2 ) ) ).

% image_comp
thf(fact_138_option_Osel,axiom,
    ! [X22: sum_sum_a_c] :
      ( ( the_Sum_sum_a_c @ ( some_Sum_sum_a_c @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_139_option_Osel,axiom,
    ! [X22: nat > option_Sum_sum_a_c] :
      ( ( the_na7545698918922854738um_a_c @ ( some_n5886337604800580545um_a_c @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_140_option_Oexpand,axiom,
    ! [Option: option_Sum_sum_a_c,Option2: option_Sum_sum_a_c] :
      ( ( ( Option = none_Sum_sum_a_c )
        = ( Option2 = none_Sum_sum_a_c ) )
     => ( ( ( Option != none_Sum_sum_a_c )
         => ( ( Option2 != none_Sum_sum_a_c )
           => ( ( the_Sum_sum_a_c @ Option )
              = ( the_Sum_sum_a_c @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_141_Sup_OSUP__image,axiom,
    ! [Sup: set_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c,F: nat > option_Sum_sum_a_c,A: set_nat] :
      ( ( Sup @ ( image_7911160186837173029um_a_c @ G @ ( image_8467336918603940942um_a_c @ F @ A ) ) )
      = ( Sup @ ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ G @ F ) @ A ) ) ) ).

% Sup.SUP_image
thf(fact_142_Sup_OSUP__image,axiom,
    ! [Sup: set_list_Sum_sum_a_c > list_Sum_sum_a_c,G: list_Sum_sum_a_c > list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( Sup @ ( image_8132764781264612725um_a_c @ G @ ( image_7318554134589591124um_a_c @ F @ A ) ) )
      = ( Sup @ ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ G @ F ) @ A ) ) ) ).

% Sup.SUP_image
thf(fact_143_Sup_OSUP__image,axiom,
    ! [Sup: set_list_Sum_sum_a_c > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( Sup @ ( image_7318554134589591124um_a_c @ G @ ( image_5025405432202070451um_a_c @ F @ A ) ) )
      = ( Sup @ ( image_7318554134589591124um_a_c @ ( comp_n57436397065578056um_a_c @ G @ F ) @ A ) ) ) ).

% Sup.SUP_image
thf(fact_144_Inf_OINF__image,axiom,
    ! [Inf: set_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c,F: nat > option_Sum_sum_a_c,A: set_nat] :
      ( ( Inf @ ( image_7911160186837173029um_a_c @ G @ ( image_8467336918603940942um_a_c @ F @ A ) ) )
      = ( Inf @ ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ G @ F ) @ A ) ) ) ).

% Inf.INF_image
thf(fact_145_Inf_OINF__image,axiom,
    ! [Inf: set_list_Sum_sum_a_c > list_Sum_sum_a_c,G: list_Sum_sum_a_c > list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( Inf @ ( image_8132764781264612725um_a_c @ G @ ( image_7318554134589591124um_a_c @ F @ A ) ) )
      = ( Inf @ ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ G @ F ) @ A ) ) ) ).

% Inf.INF_image
thf(fact_146_Inf_OINF__image,axiom,
    ! [Inf: set_list_Sum_sum_a_c > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( Inf @ ( image_7318554134589591124um_a_c @ G @ ( image_5025405432202070451um_a_c @ F @ A ) ) )
      = ( Inf @ ( image_7318554134589591124um_a_c @ ( comp_n57436397065578056um_a_c @ G @ F ) @ A ) ) ) ).

% Inf.INF_image
thf(fact_147_map__eq__map__tailrec,axiom,
    map_nat_Sum_sum_a_c = map_ta2267675918397559815um_a_c ).

% map_eq_map_tailrec
thf(fact_148_K__record__comp,axiom,
    ! [C: sum_sum_a_c,F: nat > option_Sum_sum_a_c] :
      ( ( comp_o2608084742246830931_c_nat
        @ ^ [X: option_Sum_sum_a_c] : C
        @ F )
      = ( ^ [X: nat] : C ) ) ).

% K_record_comp
thf(fact_149_Inf_OINF__cong,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c,C2: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,D2: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,Inf: set_list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( A = B3 )
     => ( ! [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ B3 )
           => ( ( C2 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( Inf @ ( image_7318554134589591124um_a_c @ C2 @ A ) )
          = ( Inf @ ( image_7318554134589591124um_a_c @ D2 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_150_Sup_OSUP__cong,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c,C2: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,D2: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,Sup: set_list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( A = B3 )
     => ( ! [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ B3 )
           => ( ( C2 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( Sup @ ( image_7318554134589591124um_a_c @ C2 @ A ) )
          = ( Sup @ ( image_7318554134589591124um_a_c @ D2 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_151_fun_Omap__comp,axiom,
    ! [G: sum_sum_a_c > sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c,V: nat > option_Sum_sum_a_c] :
      ( ( comp_S8935214348162178051_c_nat @ G @ ( comp_o2608084742246830931_c_nat @ F @ V ) )
      = ( comp_o2608084742246830931_c_nat @ ( comp_S7844377493806175402um_a_c @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_152_fun_Omap__comp,axiom,
    ! [G: option_Sum_sum_a_c > sum_sum_a_c,F: nat > option_Sum_sum_a_c,V: nat > nat] :
      ( ( comp_o2608084742246830931_c_nat @ G @ ( comp_n4859801332697915004_c_nat @ F @ V ) )
      = ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_153_fun_Omap__comp,axiom,
    ! [G: option_Sum_sum_a_c > sum_sum_a_c,F: option_Sum_sum_a_c > option_Sum_sum_a_c,V: nat > option_Sum_sum_a_c] :
      ( ( comp_o2608084742246830931_c_nat @ G @ ( comp_o5720133263240570531_c_nat @ F @ V ) )
      = ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ G @ F ) @ V ) ) ).

% fun.map_comp
thf(fact_154_Some__image__these__eq,axiom,
    ! [A: set_op4691527921008582829um_a_c] :
      ( ( image_152546304820467493um_a_c @ some_Sum_sum_a_c @ ( these_Sum_sum_a_c @ A ) )
      = ( collec6474438989942423884um_a_c
        @ ^ [X: option_Sum_sum_a_c] :
            ( ( member6131852849436380942um_a_c @ X @ A )
            & ( X != none_Sum_sum_a_c ) ) ) ) ).

% Some_image_these_eq
thf(fact_155_Some__image__these__eq,axiom,
    ! [A: set_op349181139376313708um_a_c] :
      ( ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ ( these_6788615929263401833um_a_c @ A ) )
      = ( collec3897716057508394891um_a_c
        @ ^ [X: option6339742336662979638um_a_c] :
            ( ( member2769028661778116429um_a_c @ X @ A )
            & ( X != none_n2797051391425193029um_a_c ) ) ) ) ).

% Some_image_these_eq
thf(fact_156_cproper__interval__option_Ocases,axiom,
    ! [X2: produc76160212281364077um_a_c] :
      ( ( X2
       != ( produc6724097468676373157um_a_c @ none_o5565254296229333781um_a_c @ none_o5565254296229333781um_a_c ) )
     => ( ! [X3: option6339742336662979638um_a_c] :
            ( X2
           != ( produc6724097468676373157um_a_c @ none_o5565254296229333781um_a_c @ ( some_o4321091564477925521um_a_c @ X3 ) ) )
       => ( ! [X3: option6339742336662979638um_a_c] :
              ( X2
             != ( produc6724097468676373157um_a_c @ ( some_o4321091564477925521um_a_c @ X3 ) @ none_o5565254296229333781um_a_c ) )
         => ( ! [X3: option6339742336662979638um_a_c] :
                ( X2
               != ( produc6724097468676373157um_a_c @ ( some_o4321091564477925521um_a_c @ X3 ) @ ( some_o4321091564477925521um_a_c @ none_n2797051391425193029um_a_c ) ) )
           => ~ ! [X3: option6339742336662979638um_a_c,Y4: nat > option_Sum_sum_a_c] :
                  ( X2
                 != ( produc6724097468676373157um_a_c @ ( some_o4321091564477925521um_a_c @ X3 ) @ ( some_o4321091564477925521um_a_c @ ( some_n5886337604800580545um_a_c @ Y4 ) ) ) ) ) ) ) ) ).

% cproper_interval_option.cases
thf(fact_157_these__image__Some__eq,axiom,
    ! [A: set_na2086515510486960284um_a_c] :
      ( ( these_6788615929263401833um_a_c @ ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ A ) )
      = A ) ).

% these_image_Some_eq
thf(fact_158_comp__the__Some,axiom,
    ( ( comp_o6733401286528806058um_a_c @ the_Sum_sum_a_c @ some_Sum_sum_a_c )
    = id_Sum_sum_a_c ) ).

% comp_the_Some
thf(fact_159_comp__the__Some,axiom,
    ( ( comp_o1906857025035928935um_a_c @ the_na7545698918922854738um_a_c @ some_n5886337604800580545um_a_c )
    = id_nat4568008704623681077um_a_c ) ).

% comp_the_Some
thf(fact_160_notin__range__Some,axiom,
    ! [X2: option_Sum_sum_a_c] :
      ( ( ~ ( member6131852849436380942um_a_c @ X2 @ ( image_152546304820467493um_a_c @ some_Sum_sum_a_c @ top_to8990974080698231821um_a_c ) ) )
      = ( X2 = none_Sum_sum_a_c ) ) ).

% notin_range_Some
thf(fact_161_notin__range__Some,axiom,
    ! [X2: option6339742336662979638um_a_c] :
      ( ( ~ ( member2769028661778116429um_a_c @ X2 @ ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ top_to5366858714144749388um_a_c ) ) )
      = ( X2 = none_n2797051391425193029um_a_c ) ) ).

% notin_range_Some
thf(fact_162_notin__range__Some,axiom,
    ! [X2: option_o] :
      ( ( ~ ( member_option_o @ X2 @ ( image_o_option_o @ some_o @ top_top_set_o ) ) )
      = ( X2 = none_o ) ) ).

% notin_range_Some
thf(fact_163_notin__range__Some,axiom,
    ! [X2: option_nat] :
      ( ( ~ ( member_option_nat @ X2 @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) )
      = ( X2 = none_nat ) ) ).

% notin_range_Some
thf(fact_164_Option_Othese__def,axiom,
    ( these_Sum_sum_a_c
    = ( ^ [A4: set_op4691527921008582829um_a_c] :
          ( image_7911160186837173029um_a_c @ the_Sum_sum_a_c
          @ ( collec6474438989942423884um_a_c
            @ ^ [X: option_Sum_sum_a_c] :
                ( ( member6131852849436380942um_a_c @ X @ A4 )
                & ( X != none_Sum_sum_a_c ) ) ) ) ) ) ).

% Option.these_def
thf(fact_165_comp__cong,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,X2: nat,F3: option_Sum_sum_a_c > sum_sum_a_c,G3: nat > option_Sum_sum_a_c,X6: nat] :
      ( ( ( F @ ( G @ X2 ) )
        = ( F3 @ ( G3 @ X6 ) ) )
     => ( ( comp_o2608084742246830931_c_nat @ F @ G @ X2 )
        = ( comp_o2608084742246830931_c_nat @ F3 @ G3 @ X6 ) ) ) ).

% comp_cong
thf(fact_166_iso__tuple__UNIV__I,axiom,
    ! [X2: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X2 @ top_to8424319039679220765um_a_c ) ).

% iso_tuple_UNIV_I
thf(fact_167_iso__tuple__UNIV__I,axiom,
    ! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_168_iso__tuple__UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_169_UNIV__I,axiom,
    ! [X2: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X2 @ top_to8424319039679220765um_a_c ) ).

% UNIV_I
thf(fact_170_UNIV__I,axiom,
    ! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).

% UNIV_I
thf(fact_171_UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_172_surj__id,axiom,
    ( ( image_o_o2 @ id_o @ top_top_set_o )
    = top_top_set_o ) ).

% surj_id
thf(fact_173_surj__id,axiom,
    ( ( image_nat_nat2 @ id_nat @ top_top_set_nat )
    = top_top_set_nat ) ).

% surj_id
thf(fact_174_UNIV__eq__I,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ! [X3: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X3 @ A )
     => ( top_to8424319039679220765um_a_c = A ) ) ).

% UNIV_eq_I
thf(fact_175_UNIV__eq__I,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] : ( member_o @ X3 @ A )
     => ( top_top_set_o = A ) ) ).

% UNIV_eq_I
thf(fact_176_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_177_UNIV__witness,axiom,
    ? [X3: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X3 @ top_to8424319039679220765um_a_c ) ).

% UNIV_witness
thf(fact_178_UNIV__witness,axiom,
    ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_179_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_180_UNIV__def,axiom,
    ( top_to3439567294830504444um_a_c
    = ( collec5227572641185395563um_a_c
      @ ^ [X: nat > sum_sum_a_c] : $true ) ) ).

% UNIV_def
thf(fact_181_UNIV__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $true ) ) ).

% UNIV_def
thf(fact_182_UNIV__def,axiom,
    ( top_top_set_o
    = ( collect_o
      @ ^ [X: $o] : $true ) ) ).

% UNIV_def
thf(fact_183_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $true ) ) ).

% UNIV_def
thf(fact_184_comp__eq__id__dest,axiom,
    ! [A2: option_Sum_sum_a_c > sum_sum_a_c,B: nat > option_Sum_sum_a_c,C: nat > sum_sum_a_c,V: nat] :
      ( ( ( comp_o2608084742246830931_c_nat @ A2 @ B )
        = ( comp_S8935214348162178051_c_nat @ id_Sum_sum_a_c @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_185_range__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,X2: nat > sum_sum_a_c] :
      ( ( B
        = ( F @ X2 ) )
     => ( member7772695417316360142um_a_c @ B @ ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c ) ) ) ).

% range_eqI
thf(fact_186_range__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: $o > list_Sum_sum_a_c,X2: $o] :
      ( ( B
        = ( F @ X2 ) )
     => ( member7772695417316360142um_a_c @ B @ ( image_1489810555362023050um_a_c @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_187_range__eqI,axiom,
    ! [B: $o,F: $o > $o,X2: $o] :
      ( ( B
        = ( F @ X2 ) )
     => ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_188_range__eqI,axiom,
    ! [B: nat,F: $o > nat,X2: $o] :
      ( ( B
        = ( F @ X2 ) )
     => ( member_nat @ B @ ( image_o_nat2 @ F @ top_top_set_o ) ) ) ).

% range_eqI
thf(fact_189_range__eqI,axiom,
    ! [B: list_Sum_sum_a_c,F: nat > list_Sum_sum_a_c,X2: nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( member7772695417316360142um_a_c @ B @ ( image_1048630792592356622um_a_c @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_190_range__eqI,axiom,
    ! [B: $o,F: nat > $o,X2: nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( member_o @ B @ ( image_nat_o2 @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_191_range__eqI,axiom,
    ! [B: nat,F: nat > nat,X2: nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( member_nat @ B @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ) ).

% range_eqI
thf(fact_192_surj__def,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c )
        = top_to8424319039679220765um_a_c )
      = ( ! [Y: list_Sum_sum_a_c] :
          ? [X: nat > sum_sum_a_c] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_193_surj__def,axiom,
    ! [F: $o > $o] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = top_top_set_o )
      = ( ! [Y: $o] :
          ? [X: $o] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_194_surj__def,axiom,
    ! [F: $o > nat] :
      ( ( ( image_o_nat2 @ F @ top_top_set_o )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: $o] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_195_surj__def,axiom,
    ! [F: nat > $o] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = top_top_set_o )
      = ( ! [Y: $o] :
          ? [X: nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_196_surj__def,axiom,
    ! [F: nat > nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
      = ( ! [Y: nat] :
          ? [X: nat] :
            ( Y
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_197_rangeI,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,X2: nat > sum_sum_a_c] : ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c ) ) ).

% rangeI
thf(fact_198_rangeI,axiom,
    ! [F: $o > list_Sum_sum_a_c,X2: $o] : ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_1489810555362023050um_a_c @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_199_rangeI,axiom,
    ! [F: $o > $o,X2: $o] : ( member_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_200_rangeI,axiom,
    ! [F: $o > nat,X2: $o] : ( member_nat @ ( F @ X2 ) @ ( image_o_nat2 @ F @ top_top_set_o ) ) ).

% rangeI
thf(fact_201_rangeI,axiom,
    ! [F: nat > list_Sum_sum_a_c,X2: nat] : ( member7772695417316360142um_a_c @ ( F @ X2 ) @ ( image_1048630792592356622um_a_c @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_202_rangeI,axiom,
    ! [F: nat > $o,X2: nat] : ( member_o @ ( F @ X2 ) @ ( image_nat_o2 @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_203_rangeI,axiom,
    ! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat2 @ F @ top_top_set_nat ) ) ).

% rangeI
thf(fact_204_surjI,axiom,
    ! [G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: list_Sum_sum_a_c > nat > sum_sum_a_c] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_7318554134589591124um_a_c @ G @ top_to3439567294830504444um_a_c )
        = top_to8424319039679220765um_a_c ) ) ).

% surjI
thf(fact_205_surjI,axiom,
    ! [G: $o > $o,F: $o > $o] :
      ( ! [X3: $o] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_o_o2 @ G @ top_top_set_o )
        = top_top_set_o ) ) ).

% surjI
thf(fact_206_surjI,axiom,
    ! [G: $o > nat,F: nat > $o] :
      ( ! [X3: nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_o_nat2 @ G @ top_top_set_o )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_207_surjI,axiom,
    ! [G: nat > $o,F: $o > nat] :
      ( ! [X3: $o] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_nat_o2 @ G @ top_top_set_nat )
        = top_top_set_o ) ) ).

% surjI
thf(fact_208_surjI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_nat_nat2 @ G @ top_top_set_nat )
        = top_top_set_nat ) ) ).

% surjI
thf(fact_209_surjE,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c )
        = top_to8424319039679220765um_a_c )
     => ~ ! [X3: nat > sum_sum_a_c] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_210_surjE,axiom,
    ! [F: $o > $o,Y3: $o] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = top_top_set_o )
     => ~ ! [X3: $o] :
            ( Y3
            = ( ~ ( F @ X3 ) ) ) ) ).

% surjE
thf(fact_211_surjE,axiom,
    ! [F: $o > nat,Y3: nat] :
      ( ( ( image_o_nat2 @ F @ top_top_set_o )
        = top_top_set_nat )
     => ~ ! [X3: $o] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_212_surjE,axiom,
    ! [F: nat > $o,Y3: $o] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = top_top_set_o )
     => ~ ! [X3: nat] :
            ( Y3
            = ( ~ ( F @ X3 ) ) ) ) ).

% surjE
thf(fact_213_surjE,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ~ ! [X3: nat] :
            ( Y3
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_214_surjD,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c )
        = top_to8424319039679220765um_a_c )
     => ? [X3: nat > sum_sum_a_c] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_215_surjD,axiom,
    ! [F: $o > $o,Y3: $o] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = top_top_set_o )
     => ? [X3: $o] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_216_surjD,axiom,
    ! [F: $o > nat,Y3: nat] :
      ( ( ( image_o_nat2 @ F @ top_top_set_o )
        = top_top_set_nat )
     => ? [X3: $o] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_217_surjD,axiom,
    ! [F: nat > $o,Y3: $o] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = top_top_set_o )
     => ? [X3: nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_218_surjD,axiom,
    ! [F: nat > nat,Y3: nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ? [X3: nat] :
          ( Y3
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_219_in__these__eq,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_op2049297850117519933um_a_c] :
      ( ( member7772695417316360142um_a_c @ X2 @ ( these_1556098360075300026um_a_c @ A ) )
      = ( member7495770370193619614um_a_c @ ( some_l2604840475217348882um_a_c @ X2 ) @ A ) ) ).

% in_these_eq
thf(fact_220_in__these__eq,axiom,
    ! [X2: $o,A: set_option_o] :
      ( ( member_o @ X2 @ ( these_o @ A ) )
      = ( member_option_o @ ( some_o @ X2 ) @ A ) ) ).

% in_these_eq
thf(fact_221_in__these__eq,axiom,
    ! [X2: nat,A: set_option_nat] :
      ( ( member_nat @ X2 @ ( these_nat @ A ) )
      = ( member_option_nat @ ( some_nat @ X2 ) @ A ) ) ).

% in_these_eq
thf(fact_222_in__these__eq,axiom,
    ! [X2: nat > option_Sum_sum_a_c,A: set_op349181139376313708um_a_c] :
      ( ( member6370862702656978045um_a_c @ X2 @ ( these_6788615929263401833um_a_c @ A ) )
      = ( member2769028661778116429um_a_c @ ( some_n5886337604800580545um_a_c @ X2 ) @ A ) ) ).

% in_these_eq
thf(fact_223_range__composition,axiom,
    ! [F: list_Sum_sum_a_c > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
        @ top_to3439567294830504444um_a_c )
      = ( image_8132764781264612725um_a_c @ F @ ( image_7318554134589591124um_a_c @ G @ top_to3439567294830504444um_a_c ) ) ) ).

% range_composition
thf(fact_224_range__composition,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
        @ top_to3439567294830504444um_a_c )
      = ( image_7318554134589591124um_a_c @ F @ ( image_5025405432202070451um_a_c @ G @ top_to3439567294830504444um_a_c ) ) ) ).

% range_composition
thf(fact_225_range__composition,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: $o > nat > sum_sum_a_c] :
      ( ( image_1489810555362023050um_a_c
        @ ^ [X: $o] : ( F @ ( G @ X ) )
        @ top_top_set_o )
      = ( image_7318554134589591124um_a_c @ F @ ( image_2419171341444471785um_a_c @ G @ top_top_set_o ) ) ) ).

% range_composition
thf(fact_226_range__composition,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: nat > nat > sum_sum_a_c] :
      ( ( image_1048630792592356622um_a_c
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ top_top_set_nat )
      = ( image_7318554134589591124um_a_c @ F @ ( image_7220470569846912621um_a_c @ G @ top_top_set_nat ) ) ) ).

% range_composition
thf(fact_227_rangeE,axiom,
    ! [B: list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c ) )
     => ~ ! [X3: nat > sum_sum_a_c] :
            ( B
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_228_rangeE,axiom,
    ! [B: list_Sum_sum_a_c,F: $o > list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_1489810555362023050um_a_c @ F @ top_top_set_o ) )
     => ~ ! [X3: $o] :
            ( B
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_229_rangeE,axiom,
    ! [B: $o,F: $o > $o] :
      ( ( member_o @ B @ ( image_o_o2 @ F @ top_top_set_o ) )
     => ~ ! [X3: $o] :
            ( B
            = ( ~ ( F @ X3 ) ) ) ) ).

% rangeE
thf(fact_230_rangeE,axiom,
    ! [B: nat,F: $o > nat] :
      ( ( member_nat @ B @ ( image_o_nat2 @ F @ top_top_set_o ) )
     => ~ ! [X3: $o] :
            ( B
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_231_rangeE,axiom,
    ! [B: list_Sum_sum_a_c,F: nat > list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_1048630792592356622um_a_c @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_232_rangeE,axiom,
    ! [B: $o,F: nat > $o] :
      ( ( member_o @ B @ ( image_nat_o2 @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B
            = ( ~ ( F @ X3 ) ) ) ) ).

% rangeE
thf(fact_233_rangeE,axiom,
    ! [B: nat,F: nat > nat] :
      ( ( member_nat @ B @ ( image_nat_nat2 @ F @ top_top_set_nat ) )
     => ~ ! [X3: nat] :
            ( B
           != ( F @ X3 ) ) ) ).

% rangeE
thf(fact_234_comp__surj,axiom,
    ! [F: $o > $o,G: $o > $o] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = top_top_set_o )
     => ( ( ( image_o_o2 @ G @ top_top_set_o )
          = top_top_set_o )
       => ( ( image_o_o2 @ ( comp_o_o_o @ G @ F ) @ top_top_set_o )
          = top_top_set_o ) ) ) ).

% comp_surj
thf(fact_235_comp__surj,axiom,
    ! [F: $o > $o,G: $o > nat] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = top_top_set_o )
     => ( ( ( image_o_nat2 @ G @ top_top_set_o )
          = top_top_set_nat )
       => ( ( image_o_nat2 @ ( comp_o_nat_o @ G @ F ) @ top_top_set_o )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_236_comp__surj,axiom,
    ! [F: $o > nat,G: nat > $o] :
      ( ( ( image_o_nat2 @ F @ top_top_set_o )
        = top_top_set_nat )
     => ( ( ( image_nat_o2 @ G @ top_top_set_nat )
          = top_top_set_o )
       => ( ( image_o_o2 @ ( comp_nat_o_o @ G @ F ) @ top_top_set_o )
          = top_top_set_o ) ) ) ).

% comp_surj
thf(fact_237_comp__surj,axiom,
    ! [F: $o > nat,G: nat > nat] :
      ( ( ( image_o_nat2 @ F @ top_top_set_o )
        = top_top_set_nat )
     => ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_o_nat2 @ ( comp_nat_nat_o @ G @ F ) @ top_top_set_o )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_238_comp__surj,axiom,
    ! [F: nat > $o,G: $o > $o] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = top_top_set_o )
     => ( ( ( image_o_o2 @ G @ top_top_set_o )
          = top_top_set_o )
       => ( ( image_nat_o2 @ ( comp_o_o_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_o ) ) ) ).

% comp_surj
thf(fact_239_comp__surj,axiom,
    ! [F: nat > $o,G: $o > nat] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = top_top_set_o )
     => ( ( ( image_o_nat2 @ G @ top_top_set_o )
          = top_top_set_nat )
       => ( ( image_nat_nat2 @ ( comp_o_nat_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_240_comp__surj,axiom,
    ! [F: nat > nat,G: nat > $o] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_o2 @ G @ top_top_set_nat )
          = top_top_set_o )
       => ( ( image_nat_o2 @ ( comp_nat_o_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_o ) ) ) ).

% comp_surj
thf(fact_241_comp__surj,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( ( image_nat_nat2 @ F @ top_top_set_nat )
        = top_top_set_nat )
     => ( ( ( image_nat_nat2 @ G @ top_top_set_nat )
          = top_top_set_nat )
       => ( ( image_nat_nat2 @ ( comp_nat_nat_nat @ G @ F ) @ top_top_set_nat )
          = top_top_set_nat ) ) ) ).

% comp_surj
thf(fact_242_comp__surj,axiom,
    ! [F: nat > option_Sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( image_8467336918603940942um_a_c @ F @ top_top_set_nat )
        = top_to2662137674714748765um_a_c )
     => ( ( ( image_7911160186837173029um_a_c @ G @ top_to2662137674714748765um_a_c )
          = top_to8990974080698231821um_a_c )
       => ( ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ G @ F ) @ top_top_set_nat )
          = top_to8990974080698231821um_a_c ) ) ) ).

% comp_surj
thf(fact_243_comp__surj,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: list_Sum_sum_a_c > $o] :
      ( ( ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c )
        = top_to8424319039679220765um_a_c )
     => ( ( ( image_1429702505460152410_a_c_o @ G @ top_to8424319039679220765um_a_c )
          = top_top_set_o )
       => ( ( image_3393300230255336507_a_c_o @ ( comp_l8081135279519614228um_a_c @ G @ F ) @ top_to3439567294830504444um_a_c )
          = top_top_set_o ) ) ) ).

% comp_surj
thf(fact_244_fun_Omap__ident__strong,axiom,
    ! [T: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ! [Z2: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ Z2 @ ( image_7318554134589591124um_a_c @ T @ top_to3439567294830504444um_a_c ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_l1055203015525864169um_a_c @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_245_fun_Omap__ident__strong,axiom,
    ! [T: $o > list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ! [Z2: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ Z2 @ ( image_1489810555362023050um_a_c @ T @ top_top_set_o ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_l709488137209556101_a_c_o @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_246_fun_Omap__ident__strong,axiom,
    ! [T: $o > $o,F: $o > $o] :
      ( ! [Z2: $o] :
          ( ( member_o @ Z2 @ ( image_o_o2 @ T @ top_top_set_o ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_o_o_o @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_247_fun_Omap__ident__strong,axiom,
    ! [T: $o > nat,F: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( image_o_nat2 @ T @ top_top_set_o ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_nat_nat_o @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_248_fun_Omap__ident__strong,axiom,
    ! [T: nat > list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ! [Z2: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ Z2 @ ( image_1048630792592356622um_a_c @ T @ top_top_set_nat ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_l5454522811287394339_c_nat @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_249_fun_Omap__ident__strong,axiom,
    ! [T: nat > $o,F: $o > $o] :
      ( ! [Z2: $o] :
          ( ( member_o @ Z2 @ ( image_nat_o2 @ T @ top_top_set_nat ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_o_o_nat @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_250_fun_Omap__ident__strong,axiom,
    ! [T: nat > nat,F: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( image_nat_nat2 @ T @ top_top_set_nat ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( comp_nat_nat_nat @ F @ T )
        = T ) ) ).

% fun.map_ident_strong
thf(fact_251_fun_Oinj__map__strong,axiom,
    ! [X2: nat > option_Sum_sum_a_c,Xa: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c,Fa: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ! [Z2: option_Sum_sum_a_c,Za: option_Sum_sum_a_c] :
          ( ( member6131852849436380942um_a_c @ Z2 @ ( image_8467336918603940942um_a_c @ X2 @ top_top_set_nat ) )
         => ( ( member6131852849436380942um_a_c @ Za @ ( image_8467336918603940942um_a_c @ Xa @ top_top_set_nat ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( comp_o2608084742246830931_c_nat @ F @ X2 )
          = ( comp_o2608084742246830931_c_nat @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% fun.inj_map_strong
thf(fact_252_fun_Omap__cong0,axiom,
    ! [X2: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ! [Z2: option_Sum_sum_a_c] :
          ( ( member6131852849436380942um_a_c @ Z2 @ ( image_8467336918603940942um_a_c @ X2 @ top_top_set_nat ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( comp_o2608084742246830931_c_nat @ F @ X2 )
        = ( comp_o2608084742246830931_c_nat @ G @ X2 ) ) ) ).

% fun.map_cong0
thf(fact_253_fun_Omap__cong,axiom,
    ! [X2: nat > option_Sum_sum_a_c,Ya: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( X2 = Ya )
     => ( ! [Z2: option_Sum_sum_a_c] :
            ( ( member6131852849436380942um_a_c @ Z2 @ ( image_8467336918603940942um_a_c @ Ya @ top_top_set_nat ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( comp_o2608084742246830931_c_nat @ F @ X2 )
          = ( comp_o2608084742246830931_c_nat @ G @ Ya ) ) ) ) ).

% fun.map_cong
thf(fact_254_fun_Oset__map,axiom,
    ! [F: list_Sum_sum_a_c > list_Sum_sum_a_c,V: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ ( comp_l1055203015525864169um_a_c @ F @ V ) @ top_to3439567294830504444um_a_c )
      = ( image_8132764781264612725um_a_c @ F @ ( image_7318554134589591124um_a_c @ V @ top_to3439567294830504444um_a_c ) ) ) ).

% fun.set_map
thf(fact_255_fun_Oset__map,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,V: ( nat > sum_sum_a_c ) > nat > sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ ( comp_n57436397065578056um_a_c @ F @ V ) @ top_to3439567294830504444um_a_c )
      = ( image_7318554134589591124um_a_c @ F @ ( image_5025405432202070451um_a_c @ V @ top_to3439567294830504444um_a_c ) ) ) ).

% fun.set_map
thf(fact_256_fun_Oset__map,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,V: $o > nat > sum_sum_a_c] :
      ( ( image_1489810555362023050um_a_c @ ( comp_n6914878987079581414_a_c_o @ F @ V ) @ top_top_set_o )
      = ( image_7318554134589591124um_a_c @ F @ ( image_2419171341444471785um_a_c @ V @ top_top_set_o ) ) ) ).

% fun.set_map
thf(fact_257_fun_Oset__map,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,V: nat > option_Sum_sum_a_c] :
      ( ( image_95481340843707134um_a_c @ ( comp_o2608084742246830931_c_nat @ F @ V ) @ top_top_set_nat )
      = ( image_7911160186837173029um_a_c @ F @ ( image_8467336918603940942um_a_c @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_258_fun_Oset__map,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,V: nat > nat > sum_sum_a_c] :
      ( ( image_1048630792592356622um_a_c @ ( comp_n3741280631081836546_c_nat @ F @ V ) @ top_top_set_nat )
      = ( image_7318554134589591124um_a_c @ F @ ( image_7220470569846912621um_a_c @ V @ top_top_set_nat ) ) ) ).

% fun.set_map
thf(fact_259_these__imageI,axiom,
    ! [F: $o > option_o,X2: $o,Y3: $o,X7: set_o] :
      ( ( ( F @ X2 )
        = ( some_o @ Y3 ) )
     => ( ( member_o @ X2 @ X7 )
       => ( member_o @ Y3 @ ( these_o @ ( image_o_option_o @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_260_these__imageI,axiom,
    ! [F: $o > option_nat,X2: $o,Y3: nat,X7: set_o] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( member_o @ X2 @ X7 )
       => ( member_nat @ Y3 @ ( these_nat @ ( image_o_option_nat @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_261_these__imageI,axiom,
    ! [F: nat > option_o,X2: nat,Y3: $o,X7: set_nat] :
      ( ( ( F @ X2 )
        = ( some_o @ Y3 ) )
     => ( ( member_nat @ X2 @ X7 )
       => ( member_o @ Y3 @ ( these_o @ ( image_nat_option_o @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_262_these__imageI,axiom,
    ! [F: nat > option_nat,X2: nat,Y3: nat,X7: set_nat] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( member_nat @ X2 @ X7 )
       => ( member_nat @ Y3 @ ( these_nat @ ( image_nat_option_nat @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_263_these__imageI,axiom,
    ! [F: list_Sum_sum_a_c > option_o,X2: list_Sum_sum_a_c,Y3: $o,X7: set_list_Sum_sum_a_c] :
      ( ( ( F @ X2 )
        = ( some_o @ Y3 ) )
     => ( ( member7772695417316360142um_a_c @ X2 @ X7 )
       => ( member_o @ Y3 @ ( these_o @ ( image_1428308667371930656tion_o @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_264_these__imageI,axiom,
    ! [F: list_Sum_sum_a_c > option_nat,X2: list_Sum_sum_a_c,Y3: nat,X7: set_list_Sum_sum_a_c] :
      ( ( ( F @ X2 )
        = ( some_nat @ Y3 ) )
     => ( ( member7772695417316360142um_a_c @ X2 @ X7 )
       => ( member_nat @ Y3 @ ( these_nat @ ( image_8404841799247632350on_nat @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_265_these__imageI,axiom,
    ! [F: $o > option5508230710481627655um_a_c,X2: $o,Y3: list_Sum_sum_a_c,X7: set_o] :
      ( ( ( F @ X2 )
        = ( some_l2604840475217348882um_a_c @ Y3 ) )
     => ( ( member_o @ X2 @ X7 )
       => ( member7772695417316360142um_a_c @ Y3 @ ( these_1556098360075300026um_a_c @ ( image_8208725098656925530um_a_c @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_266_these__imageI,axiom,
    ! [F: nat > option5508230710481627655um_a_c,X2: nat,Y3: list_Sum_sum_a_c,X7: set_nat] :
      ( ( ( F @ X2 )
        = ( some_l2604840475217348882um_a_c @ Y3 ) )
     => ( ( member_nat @ X2 @ X7 )
       => ( member7772695417316360142um_a_c @ Y3 @ ( these_1556098360075300026um_a_c @ ( image_2199710440538197982um_a_c @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_267_these__imageI,axiom,
    ! [F: $o > option6339742336662979638um_a_c,X2: $o,Y3: nat > option_Sum_sum_a_c,X7: set_o] :
      ( ( ( F @ X2 )
        = ( some_n5886337604800580545um_a_c @ Y3 ) )
     => ( ( member_o @ X2 @ X7 )
       => ( member6370862702656978045um_a_c @ Y3 @ ( these_6788615929263401833um_a_c @ ( image_8234478139391505929um_a_c @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_268_these__imageI,axiom,
    ! [F: nat > option6339742336662979638um_a_c,X2: nat,Y3: nat > option_Sum_sum_a_c,X7: set_nat] :
      ( ( ( F @ X2 )
        = ( some_n5886337604800580545um_a_c @ Y3 ) )
     => ( ( member_nat @ X2 @ X7 )
       => ( member6370862702656978045um_a_c @ Y3 @ ( these_6788615929263401833um_a_c @ ( image_830678609308427405um_a_c @ F @ X7 ) ) ) ) ) ).

% these_imageI
thf(fact_269_surj__fun__eq,axiom,
    ! [F: nat > option_Sum_sum_a_c,X7: set_nat,G1: option_Sum_sum_a_c > sum_sum_a_c,G22: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( image_8467336918603940942um_a_c @ F @ X7 )
        = top_to2662137674714748765um_a_c )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X7 )
           => ( ( comp_o2608084742246830931_c_nat @ G1 @ F @ X3 )
              = ( comp_o2608084742246830931_c_nat @ G22 @ F @ X3 ) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
thf(fact_270_DEADID_Oin__rel,axiom,
    ( ( ^ [Y5: list_Sum_sum_a_c,Z3: list_Sum_sum_a_c] : ( Y5 = Z3 ) )
    = ( ^ [A5: list_Sum_sum_a_c,B4: list_Sum_sum_a_c] :
        ? [Z4: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ Z4 @ top_to8424319039679220765um_a_c )
          & ( ( id_list_Sum_sum_a_c @ Z4 )
            = A5 )
          & ( ( id_list_Sum_sum_a_c @ Z4 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_271_DEADID_Oin__rel,axiom,
    ( ( ^ [Y5: $o,Z3: $o] : ( Y5 = Z3 ) )
    = ( ^ [A5: $o,B4: $o] :
        ? [Z4: $o] :
          ( ( member_o @ Z4 @ top_top_set_o )
          & ( ( id_o @ Z4 )
            = A5 )
          & ( ( id_o @ Z4 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_272_DEADID_Oin__rel,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
    = ( ^ [A5: nat,B4: nat] :
        ? [Z4: nat] :
          ( ( member_nat @ Z4 @ top_top_set_nat )
          & ( ( id_nat @ Z4 )
            = A5 )
          & ( ( id_nat @ Z4 )
            = B4 ) ) ) ) ).

% DEADID.in_rel
thf(fact_273_top__empty__eq,axiom,
    ( top_to720311448976641704_a_c_o
    = ( ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ top_to8424319039679220765um_a_c ) ) ) ).

% top_empty_eq
thf(fact_274_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X: $o] : ( member_o @ X @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_275_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_276_top__set__def,axiom,
    ( top_to3439567294830504444um_a_c
    = ( collec5227572641185395563um_a_c @ top_to2236515607677555593_a_c_o ) ) ).

% top_set_def
thf(fact_277_top__set__def,axiom,
    ( top_to4669805908274784177at_nat
    = ( collec3392354462482085612at_nat @ top_to3137496036531963500_nat_o ) ) ).

% top_set_def
thf(fact_278_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_279_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_280_type__copy__map__cong0,axiom,
    ! [M: nat > option_Sum_sum_a_c,G: nat > nat,X2: nat,N3: option_Sum_sum_a_c > option_Sum_sum_a_c,H: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N3 @ ( H @ X2 ) ) )
     => ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ M ) @ G @ X2 )
        = ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ N3 ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_281_type__copy__map__cong0,axiom,
    ! [M: option_Sum_sum_a_c > option_Sum_sum_a_c,G: nat > option_Sum_sum_a_c,X2: nat,N3: nat > option_Sum_sum_a_c,H: nat > nat,F: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N3 @ ( H @ X2 ) ) )
     => ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ M ) @ G @ X2 )
        = ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ N3 ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_282_rewriteL__comp__comp,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c,L: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c] :
      ( ( ( comp_S7844377493806175402um_a_c @ F @ G )
        = L )
     => ( ( comp_S8935214348162178051_c_nat @ F @ ( comp_o2608084742246830931_c_nat @ G @ H ) )
        = ( comp_o2608084742246830931_c_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_283_rewriteL__comp__comp,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > option_Sum_sum_a_c,L: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c] :
      ( ( ( comp_o5865246759373812730um_a_c @ F @ G )
        = L )
     => ( ( comp_o2608084742246830931_c_nat @ F @ ( comp_o5720133263240570531_c_nat @ G @ H ) )
        = ( comp_o2608084742246830931_c_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_284_rewriteL__comp__comp,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,L: nat > sum_sum_a_c,H: nat > nat] :
      ( ( ( comp_o2608084742246830931_c_nat @ F @ G )
        = L )
     => ( ( comp_o2608084742246830931_c_nat @ F @ ( comp_n4859801332697915004_c_nat @ G @ H ) )
        = ( comp_n8637127618742830252_c_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_285_rewriteR__comp__comp,axiom,
    ! [G: nat > option_Sum_sum_a_c,H: nat > nat,R2: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( comp_n4859801332697915004_c_nat @ G @ H )
        = R2 )
     => ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ H )
        = ( comp_o2608084742246830931_c_nat @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_286_rewriteR__comp__comp,axiom,
    ! [G: option_Sum_sum_a_c > option_Sum_sum_a_c,H: nat > option_Sum_sum_a_c,R2: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( comp_o5720133263240570531_c_nat @ G @ H )
        = R2 )
     => ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ G ) @ H )
        = ( comp_o2608084742246830931_c_nat @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_287_rewriteR__comp__comp,axiom,
    ! [G: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c,R2: nat > sum_sum_a_c,F: sum_sum_a_c > sum_sum_a_c] :
      ( ( ( comp_o2608084742246830931_c_nat @ G @ H )
        = R2 )
     => ( ( comp_o2608084742246830931_c_nat @ ( comp_S7844377493806175402um_a_c @ F @ G ) @ H )
        = ( comp_S8935214348162178051_c_nat @ F @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_288_rewriteL__comp__comp2,axiom,
    ! [F: sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > sum_sum_a_c,L1: option_Sum_sum_a_c > sum_sum_a_c,L2: option_Sum_sum_a_c > option_Sum_sum_a_c,H: nat > option_Sum_sum_a_c,R2: nat > option_Sum_sum_a_c] :
      ( ( ( comp_S7844377493806175402um_a_c @ F @ G )
        = ( comp_o5865246759373812730um_a_c @ L1 @ L2 ) )
     => ( ( ( comp_o5720133263240570531_c_nat @ L2 @ H )
          = R2 )
       => ( ( comp_S8935214348162178051_c_nat @ F @ ( comp_o2608084742246830931_c_nat @ G @ H ) )
          = ( comp_o2608084742246830931_c_nat @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_289_rewriteL__comp__comp2,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: option_Sum_sum_a_c > option_Sum_sum_a_c,L1: sum_sum_a_c > sum_sum_a_c,L2: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c,R2: nat > sum_sum_a_c] :
      ( ( ( comp_o5865246759373812730um_a_c @ F @ G )
        = ( comp_S7844377493806175402um_a_c @ L1 @ L2 ) )
     => ( ( ( comp_o2608084742246830931_c_nat @ L2 @ H )
          = R2 )
       => ( ( comp_o2608084742246830931_c_nat @ F @ ( comp_o5720133263240570531_c_nat @ G @ H ) )
          = ( comp_S8935214348162178051_c_nat @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_290_rewriteL__comp__comp2,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,L1: option_Sum_sum_a_c > sum_sum_a_c,L2: nat > option_Sum_sum_a_c,H: nat > nat,R2: nat > option_Sum_sum_a_c] :
      ( ( ( comp_o2608084742246830931_c_nat @ F @ G )
        = ( comp_o2608084742246830931_c_nat @ L1 @ L2 ) )
     => ( ( ( comp_n4859801332697915004_c_nat @ L2 @ H )
          = R2 )
       => ( ( comp_o2608084742246830931_c_nat @ F @ ( comp_n4859801332697915004_c_nat @ G @ H ) )
          = ( comp_o2608084742246830931_c_nat @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_291_rewriteR__comp__comp2,axiom,
    ! [G: nat > option_Sum_sum_a_c,H: nat > nat,R1: option_Sum_sum_a_c > option_Sum_sum_a_c,R22: nat > option_Sum_sum_a_c,F: option_Sum_sum_a_c > sum_sum_a_c,L: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( comp_n4859801332697915004_c_nat @ G @ H )
        = ( comp_o5720133263240570531_c_nat @ R1 @ R22 ) )
     => ( ( ( comp_o5865246759373812730um_a_c @ F @ R1 )
          = L )
       => ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ G ) @ H )
          = ( comp_o2608084742246830931_c_nat @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_292_rewriteR__comp__comp2,axiom,
    ! [G: option_Sum_sum_a_c > option_Sum_sum_a_c,H: nat > option_Sum_sum_a_c,R1: nat > option_Sum_sum_a_c,R22: nat > nat,F: option_Sum_sum_a_c > sum_sum_a_c,L: nat > sum_sum_a_c] :
      ( ( ( comp_o5720133263240570531_c_nat @ G @ H )
        = ( comp_n4859801332697915004_c_nat @ R1 @ R22 ) )
     => ( ( ( comp_o2608084742246830931_c_nat @ F @ R1 )
          = L )
       => ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ G ) @ H )
          = ( comp_n8637127618742830252_c_nat @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_293_rewriteR__comp__comp2,axiom,
    ! [G: option_Sum_sum_a_c > sum_sum_a_c,H: nat > option_Sum_sum_a_c,R1: option_Sum_sum_a_c > sum_sum_a_c,R22: nat > option_Sum_sum_a_c,F: sum_sum_a_c > sum_sum_a_c,L: option_Sum_sum_a_c > sum_sum_a_c] :
      ( ( ( comp_o2608084742246830931_c_nat @ G @ H )
        = ( comp_o2608084742246830931_c_nat @ R1 @ R22 ) )
     => ( ( ( comp_S7844377493806175402um_a_c @ F @ R1 )
          = L )
       => ( ( comp_o2608084742246830931_c_nat @ ( comp_S7844377493806175402um_a_c @ F @ G ) @ H )
          = ( comp_o2608084742246830931_c_nat @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_294_UNIV__option__conv,axiom,
    ( top_to2662137674714748765um_a_c
    = ( insert8830009938712558887um_a_c @ none_Sum_sum_a_c @ ( image_152546304820467493um_a_c @ some_Sum_sum_a_c @ top_to8990974080698231821um_a_c ) ) ) ).

% UNIV_option_conv
thf(fact_295_UNIV__option__conv,axiom,
    ( top_to7076492143176088604um_a_c
    = ( insert1117320595054930534um_a_c @ none_n2797051391425193029um_a_c @ ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ top_to5366858714144749388um_a_c ) ) ) ).

% UNIV_option_conv
thf(fact_296_UNIV__option__conv,axiom,
    ( top_top_set_option_o
    = ( insert_option_o @ none_o @ ( image_o_option_o @ some_o @ top_top_set_o ) ) ) ).

% UNIV_option_conv
thf(fact_297_UNIV__option__conv,axiom,
    ( top_to8920198386146353926on_nat
    = ( insert_option_nat @ none_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) ) ) ).

% UNIV_option_conv
thf(fact_298_asymI,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R2 )
         => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R2 ) )
     => ( asym_on_o @ top_top_set_o @ R2 ) ) ).

% asymI
thf(fact_299_asymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
         => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 ) )
     => ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymI
thf(fact_300_comp__apply__eq,axiom,
    ! [F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c,X2: nat,H: option_Sum_sum_a_c > sum_sum_a_c,K: nat > option_Sum_sum_a_c] :
      ( ( ( F @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_o2608084742246830931_c_nat @ F @ G @ X2 )
        = ( comp_o2608084742246830931_c_nat @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_301_type__copy__Rep__o__Abs,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( comp_P5156358013004227690it_o_o @ Rep @ Abs )
        = id_o ) ) ).

% type_copy_Rep_o_Abs
thf(fact_302_insertCI,axiom,
    ! [A2: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c] :
      ( ( ~ ( member7772695417316360142um_a_c @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ B @ B3 ) ) ) ).

% insertCI
thf(fact_303_insertCI,axiom,
    ! [A2: $o,B3: set_o,B: $o] :
      ( ( ~ ( member_o @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_o @ A2 @ ( insert_o @ B @ B3 ) ) ) ).

% insertCI
thf(fact_304_insertCI,axiom,
    ! [A2: nat,B3: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A2 @ B3 )
       => ( A2 = B ) )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).

% insertCI
thf(fact_305_insert__iff,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ B @ A ) )
      = ( ( A2 = B )
        | ( member7772695417316360142um_a_c @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_306_insert__iff,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
      = ( ( A2 = B )
        | ( member_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_307_insert__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
      = ( ( A2 = B )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_308_insert__absorb2,axiom,
    ! [X2: $o,A: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ X2 @ A ) )
      = ( insert_o @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_309_insert__image,axiom,
    ! [X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( member4884986500679352621um_a_c @ X2 @ A )
     => ( ( insert7520310754158225127um_a_c @ ( F @ X2 ) @ ( image_7318554134589591124um_a_c @ F @ A ) )
        = ( image_7318554134589591124um_a_c @ F @ A ) ) ) ).

% insert_image
thf(fact_310_insert__image,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ( insert_o @ ( F @ X2 ) @ ( image_1429702505460152410_a_c_o @ F @ A ) )
        = ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ).

% insert_image
thf(fact_311_insert__image,axiom,
    ! [X2: $o,A: set_o,F: $o > $o] :
      ( ( member_o @ X2 @ A )
     => ( ( insert_o @ ( F @ X2 ) @ ( image_o_o2 @ F @ A ) )
        = ( image_o_o2 @ F @ A ) ) ) ).

% insert_image
thf(fact_312_insert__image,axiom,
    ! [X2: nat,A: set_nat,F: nat > $o] :
      ( ( member_nat @ X2 @ A )
     => ( ( insert_o @ ( F @ X2 ) @ ( image_nat_o2 @ F @ A ) )
        = ( image_nat_o2 @ F @ A ) ) ) ).

% insert_image
thf(fact_313_image__insert,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A2: nat > sum_sum_a_c,B3: set_nat_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ ( insert7583143589955530566um_a_c @ A2 @ B3 ) )
      = ( insert7520310754158225127um_a_c @ ( F @ A2 ) @ ( image_7318554134589591124um_a_c @ F @ B3 ) ) ) ).

% image_insert
thf(fact_314_image__insert,axiom,
    ! [F: $o > $o,A2: $o,B3: set_o] :
      ( ( image_o_o2 @ F @ ( insert_o @ A2 @ B3 ) )
      = ( insert_o @ ( F @ A2 ) @ ( image_o_o2 @ F @ B3 ) ) ) ).

% image_insert
thf(fact_315_asym__onI,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c] :
      ( ! [X3: list_Sum_sum_a_c,Y4: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( ( member7772695417316360142um_a_c @ Y4 @ A )
           => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X3 @ Y4 ) @ R2 )
             => ~ ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ Y4 @ X3 ) @ R2 ) ) ) )
     => ( asym_o1853620279808084485um_a_c @ A @ R2 ) ) ).

% asym_onI
thf(fact_316_asym__onI,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( member_o @ X3 @ A )
         => ( ( member_o @ Y4 @ A )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R2 )
             => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_o @ A @ R2 ) ) ).

% asym_onI
thf(fact_317_asym__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ( member_nat @ Y4 @ A )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
             => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 ) ) ) )
     => ( asym_on_nat @ A @ R2 ) ) ).

% asym_onI
thf(fact_318_these__insert__None,axiom,
    ! [A: set_op4691527921008582829um_a_c] :
      ( ( these_Sum_sum_a_c @ ( insert8830009938712558887um_a_c @ none_Sum_sum_a_c @ A ) )
      = ( these_Sum_sum_a_c @ A ) ) ).

% these_insert_None
thf(fact_319_these__insert__Some,axiom,
    ! [X2: $o,A: set_option_o] :
      ( ( these_o @ ( insert_option_o @ ( some_o @ X2 ) @ A ) )
      = ( insert_o @ X2 @ ( these_o @ A ) ) ) ).

% these_insert_Some
thf(fact_320_these__insert__Some,axiom,
    ! [X2: nat > option_Sum_sum_a_c,A: set_op349181139376313708um_a_c] :
      ( ( these_6788615929263401833um_a_c @ ( insert1117320595054930534um_a_c @ ( some_n5886337604800580545um_a_c @ X2 ) @ A ) )
      = ( insert6226696471693955350um_a_c @ X2 @ ( these_6788615929263401833um_a_c @ A ) ) ) ).

% these_insert_Some
thf(fact_321_insert__Collect,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( insert_o @ A2 @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_322_insert__Collect,axiom,
    ! [A2: nat > sum_sum_a_c,P: ( nat > sum_sum_a_c ) > $o] :
      ( ( insert7583143589955530566um_a_c @ A2 @ ( collec5227572641185395563um_a_c @ P ) )
      = ( collec5227572641185395563um_a_c
        @ ^ [U: nat > sum_sum_a_c] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_323_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_324_insert__Collect,axiom,
    ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U: product_prod_nat_nat] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_325_insert__compr,axiom,
    ( insert7520310754158225127um_a_c
    = ( ^ [A5: list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( X = A5 )
              | ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_326_insert__compr,axiom,
    ( insert_o
    = ( ^ [A5: $o,B5: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( X = A5 )
              | ( member_o @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_327_insert__compr,axiom,
    ( insert7583143589955530566um_a_c
    = ( ^ [A5: nat > sum_sum_a_c,B5: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( X = A5 )
              | ( member4884986500679352621um_a_c @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_328_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A5: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A5 )
              | ( member_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_329_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A5: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( X = A5 )
              | ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_330_insertE,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ B @ A ) )
     => ( ( A2 != B )
       => ( member7772695417316360142um_a_c @ A2 @ A ) ) ) ).

% insertE
thf(fact_331_insertE,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
     => ( ( A2 = (~ B) )
       => ( member_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_332_insertE,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
     => ( ( A2 != B )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_333_insertI1,axiom,
    ! [A2: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) ) ).

% insertI1
thf(fact_334_insertI1,axiom,
    ! [A2: $o,B3: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B3 ) ) ).

% insertI1
thf(fact_335_insertI1,axiom,
    ! [A2: nat,B3: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B3 ) ) ).

% insertI1
thf(fact_336_insertI2,axiom,
    ! [A2: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ B3 )
     => ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ B @ B3 ) ) ) ).

% insertI2
thf(fact_337_insertI2,axiom,
    ! [A2: $o,B3: set_o,B: $o] :
      ( ( member_o @ A2 @ B3 )
     => ( member_o @ A2 @ ( insert_o @ B @ B3 ) ) ) ).

% insertI2
thf(fact_338_insertI2,axiom,
    ! [A2: nat,B3: set_nat,B: nat] :
      ( ( member_nat @ A2 @ B3 )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).

% insertI2
thf(fact_339_Set_Oset__insert,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ~ ! [B6: set_list_Sum_sum_a_c] :
            ( ( A
              = ( insert7520310754158225127um_a_c @ X2 @ B6 ) )
           => ( member7772695417316360142um_a_c @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_340_Set_Oset__insert,axiom,
    ! [X2: $o,A: set_o] :
      ( ( member_o @ X2 @ A )
     => ~ ! [B6: set_o] :
            ( ( A
              = ( insert_o @ X2 @ B6 ) )
           => ( member_o @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_341_Set_Oset__insert,axiom,
    ! [X2: nat,A: set_nat] :
      ( ( member_nat @ X2 @ A )
     => ~ ! [B6: set_nat] :
            ( ( A
              = ( insert_nat @ X2 @ B6 ) )
           => ( member_nat @ X2 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_342_insert__ident,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ~ ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ~ ( member7772695417316360142um_a_c @ X2 @ B3 )
       => ( ( ( insert7520310754158225127um_a_c @ X2 @ A )
            = ( insert7520310754158225127um_a_c @ X2 @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_343_insert__ident,axiom,
    ! [X2: $o,A: set_o,B3: set_o] :
      ( ~ ( member_o @ X2 @ A )
     => ( ~ ( member_o @ X2 @ B3 )
       => ( ( ( insert_o @ X2 @ A )
            = ( insert_o @ X2 @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_344_insert__ident,axiom,
    ! [X2: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X2 @ A )
     => ( ~ ( member_nat @ X2 @ B3 )
       => ( ( ( insert_nat @ X2 @ A )
            = ( insert_nat @ X2 @ B3 ) )
          = ( A = B3 ) ) ) ) ).

% insert_ident
thf(fact_345_insert__absorb,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( insert7520310754158225127um_a_c @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_346_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_347_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_348_insert__eq__iff,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ~ ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ~ ( member7772695417316360142um_a_c @ B @ B3 )
       => ( ( ( insert7520310754158225127um_a_c @ A2 @ A )
            = ( insert7520310754158225127um_a_c @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_list_Sum_sum_a_c] :
                  ( ( A
                    = ( insert7520310754158225127um_a_c @ B @ C3 ) )
                  & ~ ( member7772695417316360142um_a_c @ B @ C3 )
                  & ( B3
                    = ( insert7520310754158225127um_a_c @ A2 @ C3 ) )
                  & ~ ( member7772695417316360142um_a_c @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_349_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B: $o,B3: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ~ ( member_o @ B @ B3 )
       => ( ( ( insert_o @ A2 @ A )
            = ( insert_o @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 = (~ B) )
             => ? [C3: set_o] :
                  ( ( A
                    = ( insert_o @ B @ C3 ) )
                  & ~ ( member_o @ B @ C3 )
                  & ( B3
                    = ( insert_o @ A2 @ C3 ) )
                  & ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_350_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B: nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B @ B3 )
       => ( ( ( insert_nat @ A2 @ A )
            = ( insert_nat @ B @ B3 ) )
          = ( ( ( A2 = B )
             => ( A = B3 ) )
            & ( ( A2 != B )
             => ? [C3: set_nat] :
                  ( ( A
                    = ( insert_nat @ B @ C3 ) )
                  & ~ ( member_nat @ B @ C3 )
                  & ( B3
                    = ( insert_nat @ A2 @ C3 ) )
                  & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_351_insert__commute,axiom,
    ! [X2: $o,Y3: $o,A: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ Y3 @ A ) )
      = ( insert_o @ Y3 @ ( insert_o @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_352_mk__disjoint__insert,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ? [B6: set_list_Sum_sum_a_c] :
          ( ( A
            = ( insert7520310754158225127um_a_c @ A2 @ B6 ) )
          & ~ ( member7772695417316360142um_a_c @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_353_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B6: set_o] :
          ( ( A
            = ( insert_o @ A2 @ B6 ) )
          & ~ ( member_o @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_354_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B6: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B6 ) )
          & ~ ( member_nat @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_355_insert__UNIV,axiom,
    ! [X2: $o] :
      ( ( insert_o @ X2 @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_356_insert__UNIV,axiom,
    ! [X2: nat] :
      ( ( insert_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_357_type__copy__wit,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,X2: list_Sum_sum_a_c,S: $o > set_list_Sum_sum_a_c,Y3: $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( member7772695417316360142um_a_c @ X2 @ ( comp_o4311291469340968745t_unit @ S @ Rep @ ( Abs @ Y3 ) ) )
       => ( member7772695417316360142um_a_c @ X2 @ ( S @ Y3 ) ) ) ) ).

% type_copy_wit
thf(fact_358_type__copy__wit,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,X2: $o,S: $o > set_o,Y3: $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( member_o @ X2 @ ( comp_o6895741221744486154t_unit @ S @ Rep @ ( Abs @ Y3 ) ) )
       => ( member_o @ X2 @ ( S @ Y3 ) ) ) ) ).

% type_copy_wit
thf(fact_359_type__copy__wit,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,X2: nat,S: $o > set_nat,Y3: $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( member_nat @ X2 @ ( comp_o9190160103024335696t_unit @ S @ Rep @ ( Abs @ Y3 ) ) )
       => ( member_nat @ X2 @ ( S @ Y3 ) ) ) ) ).

% type_copy_wit
thf(fact_360_type__copy__obj__one__point__absE,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,S2: product_unit] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ~ ! [X3: $o] :
            ( S2
           != ( Abs @ X3 ) ) ) ).

% type_copy_obj_one_point_absE
thf(fact_361_type__copy__ex__RepI,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,F4: $o > $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( ? [X8: $o] : ( F4 @ X8 ) )
        = ( ? [B4: product_unit] : ( F4 @ ( Rep @ B4 ) ) ) ) ) ).

% type_copy_ex_RepI
thf(fact_362_asym__onD,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( asym_o1853620279808084485um_a_c @ A @ R2 )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( ( member7772695417316360142um_a_c @ Y3 @ A )
         => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X2 @ Y3 ) @ R2 )
           => ~ ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ Y3 @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_363_asym__onD,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( asym_on_o @ A @ R2 )
     => ( ( member_o @ X2 @ A )
       => ( ( member_o @ Y3 @ A )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_364_asym__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( asym_on_nat @ A @ R2 )
     => ( ( member_nat @ X2 @ A )
       => ( ( member_nat @ Y3 @ A )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 ) ) ) ) ) ).

% asym_onD
thf(fact_365_type__copy__map__comp0,axiom,
    ! [Rep: option_Sum_sum_a_c > option_Sum_sum_a_c,Abs: option_Sum_sum_a_c > option_Sum_sum_a_c,M: nat > sum_sum_a_c,M1: option_Sum_sum_a_c > sum_sum_a_c,M2: nat > option_Sum_sum_a_c,F: sum_sum_a_c > sum_sum_a_c,G: nat > nat] :
      ( ( type_d4090562224118678556um_a_c @ Rep @ Abs @ top_to2662137674714748765um_a_c )
     => ( ( M
          = ( comp_o2608084742246830931_c_nat @ M1 @ M2 ) )
       => ( ( comp_n8637127618742830252_c_nat @ ( comp_S8935214348162178051_c_nat @ F @ M ) @ G )
          = ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ ( comp_S7844377493806175402um_a_c @ F @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_o5720133263240570531_c_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_366_type__copy__map__comp0,axiom,
    ! [Rep: option_Sum_sum_a_c > $o,Abs: $o > option_Sum_sum_a_c,M: nat > option_Sum_sum_a_c,M1: $o > option_Sum_sum_a_c,M2: nat > $o,F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > nat] :
      ( ( type_d4428802878543290803_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( M
          = ( comp_o5261706199817336638_c_nat @ M1 @ M2 ) )
       => ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ M ) @ G )
          = ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ ( comp_o8941615319879664981_a_c_o @ F @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_o5261706199817336638_c_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_367_type__copy__map__comp0,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,M: nat > option_Sum_sum_a_c,M1: nat > option_Sum_sum_a_c,M2: nat > nat,F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > nat] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( M
          = ( comp_n4859801332697915004_c_nat @ M1 @ M2 ) )
       => ( ( comp_n8637127618742830252_c_nat @ ( comp_o2608084742246830931_c_nat @ F @ M ) @ G )
          = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_o2608084742246830931_c_nat @ F @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_n4859801332697915004_c_nat @ Abs @ M2 ) @ G ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_368_type__copy__map__comp0,axiom,
    ! [Rep: sum_sum_a_c > nat,Abs: nat > sum_sum_a_c,M: option_Sum_sum_a_c > option_Sum_sum_a_c,M1: nat > option_Sum_sum_a_c,M2: option_Sum_sum_a_c > nat,F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( type_d4811220683359377957_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( M
          = ( comp_n6384127074061997987um_a_c @ M1 @ M2 ) )
       => ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ M ) @ G )
          = ( comp_S8935214348162178051_c_nat @ ( comp_n8699188142952482819um_a_c @ ( comp_o2608084742246830931_c_nat @ F @ M1 ) @ Rep ) @ ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ Abs @ M2 ) @ G ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_369_type__copy__map__comp0,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,M: option_Sum_sum_a_c > option_Sum_sum_a_c,M1: nat > option_Sum_sum_a_c,M2: option_Sum_sum_a_c > nat,F: option_Sum_sum_a_c > sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( M
          = ( comp_n6384127074061997987um_a_c @ M1 @ M2 ) )
       => ( ( comp_o2608084742246830931_c_nat @ ( comp_o5865246759373812730um_a_c @ F @ M ) @ G )
          = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_o2608084742246830931_c_nat @ F @ M1 ) @ Rep ) @ ( comp_o5720133263240570531_c_nat @ ( comp_n6384127074061997987um_a_c @ Abs @ M2 ) @ G ) ) ) ) ) ).

% type_copy_map_comp0
thf(fact_370_type__copy__map__comp0__undo,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,Rep2: product_unit > $o,Abs2: $o > product_unit,Rep3: product_unit > $o,Abs3: $o > product_unit,M: $o > $o,M1: $o > $o,M2: $o > $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( type_d6188575255521822967unit_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_P7645380973975430442t_unit @ ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o_o_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_371_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > $o,Abs: $o > option_Sum_sum_a_c,Rep2: sum_sum_a_c > $o,Abs2: $o > sum_sum_a_c,Rep3: nat > $o,Abs3: $o > nat,M: $o > $o,M1: $o > $o,M2: $o > $o] :
      ( ( type_d4428802878543290803_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( type_d6698783210362209795_a_c_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d1000680970699258650_nat_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o8384877226014337134_c_nat @ ( comp_o_Sum_sum_a_c_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ ( comp_o_Sum_sum_a_c_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_o5261706199817336638_c_nat @ ( comp_o7497603823055233194_a_c_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o_o_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_372_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > $o,Abs: $o > option_Sum_sum_a_c,Rep2: sum_sum_a_c > $o,Abs2: $o > sum_sum_a_c,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: $o > $o,M2: nat > $o] :
      ( ( type_d4428802878543290803_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( type_d6698783210362209795_a_c_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
         => ( ( ( comp_n8637127618742830252_c_nat @ ( comp_o8384877226014337134_c_nat @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ ( comp_o_Sum_sum_a_c_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_o5261706199817336638_c_nat @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o_o_nat @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_373_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > $o,Abs: $o > option_Sum_sum_a_c,Rep2: sum_sum_a_c > nat,Abs2: nat > sum_sum_a_c,Rep3: nat > $o,Abs3: $o > nat,M: $o > nat,M1: $o > nat,M2: $o > $o] :
      ( ( type_d4428802878543290803_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( type_d4811220683359377957_c_nat @ Rep2 @ Abs2 @ top_top_set_nat )
       => ( ( type_d1000680970699258650_nat_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o8384877226014337134_c_nat @ ( comp_n1305631613094935484_a_c_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ ( comp_n1305631613094935484_a_c_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_o5261706199817336638_c_nat @ ( comp_o7497603823055233194_a_c_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o_nat_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_374_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > $o,Abs: $o > option_Sum_sum_a_c,Rep2: sum_sum_a_c > nat,Abs2: nat > sum_sum_a_c,Rep3: nat > nat,Abs3: nat > nat,M: nat > nat,M1: $o > nat,M2: nat > $o] :
      ( ( type_d4428802878543290803_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( type_d4811220683359377957_c_nat @ Rep2 @ Abs2 @ top_top_set_nat )
       => ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
         => ( ( ( comp_n8637127618742830252_c_nat @ ( comp_n8637127618742830252_c_nat @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ ( comp_n1305631613094935484_a_c_o @ Abs2 @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_o5261706199817336638_c_nat @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o_nat_nat @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_375_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,Rep2: sum_sum_a_c > $o,Abs2: $o > sum_sum_a_c,Rep3: nat > $o,Abs3: $o > nat,M: $o > $o,M1: nat > $o,M2: $o > nat] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( type_d6698783210362209795_a_c_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d1000680970699258650_nat_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o8384877226014337134_c_nat @ ( comp_o_Sum_sum_a_c_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_o8384877226014337134_c_nat @ Abs2 @ M1 ) @ Rep ) @ ( comp_o5261706199817336638_c_nat @ ( comp_n2455873569345506796_a_c_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_nat_o_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_376_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,Rep2: sum_sum_a_c > $o,Abs2: $o > sum_sum_a_c,Rep3: nat > nat,Abs3: nat > nat,M: nat > $o,M1: nat > $o,M2: nat > nat] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( type_d6698783210362209795_a_c_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
         => ( ( ( comp_n8637127618742830252_c_nat @ ( comp_o8384877226014337134_c_nat @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_o8384877226014337134_c_nat @ Abs2 @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_n4859801332697915004_c_nat @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_nat_o_nat @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_377_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,Rep2: sum_sum_a_c > nat,Abs2: nat > sum_sum_a_c,Rep3: nat > $o,Abs3: $o > nat,M: $o > nat,M1: nat > nat,M2: $o > nat] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( type_d4811220683359377957_c_nat @ Rep2 @ Abs2 @ top_top_set_nat )
       => ( ( type_d1000680970699258650_nat_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o8384877226014337134_c_nat @ ( comp_n1305631613094935484_a_c_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_n8637127618742830252_c_nat @ Abs2 @ M1 ) @ Rep ) @ ( comp_o5261706199817336638_c_nat @ ( comp_n2455873569345506796_a_c_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_nat_nat_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_378_type__copy__map__comp0__undo,axiom,
    ! [Rep: option_Sum_sum_a_c > nat,Abs: nat > option_Sum_sum_a_c,Rep2: sum_sum_a_c > nat,Abs2: nat > sum_sum_a_c,Rep3: nat > nat,Abs3: nat > nat,M: nat > nat,M1: nat > nat,M2: nat > nat] :
      ( ( type_d4649310127517125237_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( type_d4811220683359377957_c_nat @ Rep2 @ Abs2 @ top_top_set_nat )
       => ( ( type_d6250493948777748686at_nat @ Rep3 @ Abs3 @ top_top_set_nat )
         => ( ( ( comp_n8637127618742830252_c_nat @ ( comp_n8637127618742830252_c_nat @ Abs2 @ M ) @ Rep3 )
              = ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ ( comp_n8637127618742830252_c_nat @ Abs2 @ M1 ) @ Rep ) @ ( comp_n4859801332697915004_c_nat @ ( comp_n4859801332697915004_c_nat @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_nat_nat_nat @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_379_type__copy__map__comp0__undo,axiom,
    ! [Rep: nat > option_Sum_sum_a_c,Abs: option_Sum_sum_a_c > nat,Rep2: sum_sum_a_c > $o,Abs2: $o > sum_sum_a_c,Rep3: product_unit > $o,Abs3: $o > product_unit,M: $o > $o,M1: option_Sum_sum_a_c > $o,M2: $o > option_Sum_sum_a_c] :
      ( ( type_d2102347926421078773um_a_c @ Rep @ Abs @ top_to2662137674714748765um_a_c )
     => ( ( type_d6698783210362209795_a_c_o @ Rep2 @ Abs2 @ top_top_set_o )
       => ( ( type_d6188575255521822967unit_o @ Rep3 @ Abs3 @ top_top_set_o )
         => ( ( ( comp_o2269694853038864111t_unit @ ( comp_o_Sum_sum_a_c_o @ Abs2 @ M ) @ Rep3 )
              = ( comp_n6451336636183913521t_unit @ ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ Abs2 @ M1 ) @ Rep ) @ ( comp_o5682674704259303302t_unit @ ( comp_o251945112900462188_nat_o @ Abs @ M2 ) @ Rep3 ) ) )
           => ( ( comp_o2345080802303217242_c_o_o @ M1 @ M2 )
              = M ) ) ) ) ) ).

% type_copy_map_comp0_undo
thf(fact_380_asym__iff,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( asym_on_o @ top_top_set_o @ R2 )
      = ( ! [X: $o,Y: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y @ X ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_381_asym__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
      = ( ! [X: nat,Y: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ R2 ) ) ) ) ).

% asym_iff
thf(fact_382_asymD,axiom,
    ! [R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( asym_on_o @ top_top_set_o @ R2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
       => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R2 ) ) ) ).

% asymD
thf(fact_383_asymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
       => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 ) ) ) ).

% asymD
thf(fact_384_type__copy__set__map0,axiom,
    ! [Rep: sum_sum_a_c > $o,Abs: $o > sum_sum_a_c,S: $o > set_list_Sum_sum_a_c,M: option_Sum_sum_a_c > $o,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,S3: option_Sum_sum_a_c > set_nat_Sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( type_d6698783210362209795_a_c_o @ Rep @ Abs @ top_top_set_o )
     => ( ( ( comp_o2486460743544778971um_a_c @ S @ M )
          = ( comp_s3496158340084682261um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ S3 ) )
       => ( ( comp_S425929331446500553_c_nat @ ( comp_o5808114760531824395um_a_c @ S @ Rep ) @ ( comp_o2608084742246830931_c_nat @ ( comp_o1083264402525676181um_a_c @ Abs @ M ) @ G ) )
          = ( comp_s682445721244654_c_nat @ ( image_7318554134589591124um_a_c @ F ) @ ( comp_o89929621031998200_c_nat @ S3 @ G ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_385_type__copy__set__map0,axiom,
    ! [Rep: sum_sum_a_c > nat,Abs: nat > sum_sum_a_c,S: nat > set_list_Sum_sum_a_c,M: option_Sum_sum_a_c > nat,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,S3: option_Sum_sum_a_c > set_nat_Sum_sum_a_c,G: nat > option_Sum_sum_a_c] :
      ( ( type_d4811220683359377957_c_nat @ Rep @ Abs @ top_top_set_nat )
     => ( ( ( comp_n8881718936295027993um_a_c @ S @ M )
          = ( comp_s3496158340084682261um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ S3 ) )
       => ( ( comp_S425929331446500553_c_nat @ ( comp_n3700659604615366729um_a_c @ S @ Rep ) @ ( comp_o2608084742246830931_c_nat @ ( comp_n3187222931342693843um_a_c @ Abs @ M ) @ G ) )
          = ( comp_s682445721244654_c_nat @ ( image_7318554134589591124um_a_c @ F ) @ ( comp_o89929621031998200_c_nat @ S3 @ G ) ) ) ) ) ).

% type_copy_set_map0
thf(fact_386_type__copy__map__id0,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,M: $o > $o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( M = id_o )
       => ( ( comp_o7724735231917782709t_unit @ ( comp_o3916225744453444672unit_o @ Abs @ M ) @ Rep )
          = id_Product_unit ) ) ) ).

% type_copy_map_id0
thf(fact_387_type__copy__Abs__o__Rep,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ top_top_set_o )
     => ( ( comp_o7724735231917782709t_unit @ Abs @ Rep )
        = id_Product_unit ) ) ).

% type_copy_Abs_o_Rep
thf(fact_388_type__definition_Ouniv,axiom,
    ! [Rep: list_Sum_sum_a_c > nat > sum_sum_a_c,Abs: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( type_d6615932268850986235um_a_c @ Rep @ Abs @ A )
     => ( top_to8424319039679220765um_a_c
        = ( image_7318554134589591124um_a_c @ Abs @ A ) ) ) ).

% type_definition.univ
thf(fact_389_type__definition_Ouniv,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A )
     => ( top_to1996260823553986621t_unit
        = ( image_o_Product_unit @ Abs @ A ) ) ) ).

% type_definition.univ
thf(fact_390_type__definition_OAbs__image,axiom,
    ! [Rep: list_Sum_sum_a_c > nat > sum_sum_a_c,Abs: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( type_d6615932268850986235um_a_c @ Rep @ Abs @ A )
     => ( ( image_7318554134589591124um_a_c @ Abs @ A )
        = top_to8424319039679220765um_a_c ) ) ).

% type_definition.Abs_image
thf(fact_391_type__definition_OAbs__image,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A )
     => ( ( image_o_Product_unit @ Abs @ A )
        = top_to1996260823553986621t_unit ) ) ).

% type_definition.Abs_image
thf(fact_392_type__definition_ORep__range,axiom,
    ! [Rep: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,Abs: list_Sum_sum_a_c > nat > sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( type_d5737584791754892155um_a_c @ Rep @ Abs @ A )
     => ( ( image_7318554134589591124um_a_c @ Rep @ top_to3439567294830504444um_a_c )
        = A ) ) ).

% type_definition.Rep_range
thf(fact_393_type__definition_ORep__range,axiom,
    ! [Rep: product_unit > $o,Abs: $o > product_unit,A: set_o] :
      ( ( type_d6188575255521822967unit_o @ Rep @ Abs @ A )
     => ( ( image_Product_unit_o @ Rep @ top_to1996260823553986621t_unit )
        = A ) ) ).

% type_definition.Rep_range
thf(fact_394_these__insert,axiom,
    ! [X2: option_o,A: set_option_o] :
      ( ( these_o @ ( insert_option_o @ X2 @ A ) )
      = ( case_o2021003419215089958et_o_o @ id_set_o @ insert_o @ X2 @ ( these_o @ A ) ) ) ).

% these_insert
thf(fact_395_asymp__asym__eq,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ( asymp_on_o @ top_top_set_o
        @ ^ [X: $o,Y: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 ) )
      = ( asym_on_o @ top_top_set_o @ R2 ) ) ).

% asymp_asym_eq
thf(fact_396_asymp__asym__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( asymp_on_nat @ top_top_set_nat
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
      = ( asym_on_nat @ top_top_set_nat @ R2 ) ) ).

% asymp_asym_eq
thf(fact_397_range__constant,axiom,
    ! [X2: list_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c
        @ ^ [Uu: nat > sum_sum_a_c] : X2
        @ top_to3439567294830504444um_a_c )
      = ( insert7520310754158225127um_a_c @ X2 @ bot_bo3453284597459734017um_a_c ) ) ).

% range_constant
thf(fact_398_range__constant,axiom,
    ! [X2: $o] :
      ( ( image_o_o2
        @ ^ [Uu: $o] : X2
        @ top_top_set_o )
      = ( insert_o @ X2 @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_399_range__constant,axiom,
    ! [X2: $o] :
      ( ( image_nat_o2
        @ ^ [Uu: nat] : X2
        @ top_top_set_nat )
      = ( insert_o @ X2 @ bot_bot_set_o ) ) ).

% range_constant
thf(fact_400_asym__inv__image,axiom,
    ! [R3: set_Product_prod_o_o,F: $o > $o] :
      ( ( asym_on_o @ top_top_set_o @ R3 )
     => ( asym_on_o @ top_top_set_o @ ( inv_image_o_o @ R3 @ F ) ) ) ).

% asym_inv_image
thf(fact_401_asym__inv__image,axiom,
    ! [R3: set_Product_prod_o_o,F: nat > $o] :
      ( ( asym_on_o @ top_top_set_o @ R3 )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_image_o_nat @ R3 @ F ) ) ) ).

% asym_inv_image
thf(fact_402_asym__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: $o > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R3 )
     => ( asym_on_o @ top_top_set_o @ ( inv_image_nat_o @ R3 @ F ) ) ) ).

% asym_inv_image
thf(fact_403_asym__inv__image,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,F: nat > nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R3 )
     => ( asym_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R3 @ F ) ) ) ).

% asym_inv_image
thf(fact_404_empty__Collect__eq,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o] :
      ( ( bot_bo4834947051006887904um_a_c
        = ( collec5227572641185395563um_a_c @ P ) )
      = ( ! [X: nat > sum_sum_a_c] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_405_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_406_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_407_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_408_Collect__empty__eq,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o] :
      ( ( ( collec5227572641185395563um_a_c @ P )
        = bot_bo4834947051006887904um_a_c )
      = ( ! [X: nat > sum_sum_a_c] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_409_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_410_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_411_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_412_all__not__in__conv,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ( ! [X: list_Sum_sum_a_c] :
            ~ ( member7772695417316360142um_a_c @ X @ A ) )
      = ( A = bot_bo3453284597459734017um_a_c ) ) ).

% all_not_in_conv
thf(fact_413_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_414_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X: $o] :
            ~ ( member_o @ X @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_415_empty__iff,axiom,
    ! [C: list_Sum_sum_a_c] :
      ~ ( member7772695417316360142um_a_c @ C @ bot_bo3453284597459734017um_a_c ) ).

% empty_iff
thf(fact_416_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_417_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_418_asymp__onI,axiom,
    ! [A: set_list_Sum_sum_a_c,R3: list_Sum_sum_a_c > list_Sum_sum_a_c > $o] :
      ( ! [X3: list_Sum_sum_a_c,Y4: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( ( member7772695417316360142um_a_c @ Y4 @ A )
           => ( ( R3 @ X3 @ Y4 )
             => ~ ( R3 @ Y4 @ X3 ) ) ) )
     => ( asymp_5276361849251488665um_a_c @ A @ R3 ) ) ).

% asymp_onI
thf(fact_419_asymp__onI,axiom,
    ! [A: set_o,R3: $o > $o > $o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( member_o @ X3 @ A )
         => ( ( member_o @ Y4 @ A )
           => ( ( R3 @ X3 @ Y4 )
             => ~ ( R3 @ Y4 @ X3 ) ) ) )
     => ( asymp_on_o @ A @ R3 ) ) ).

% asymp_onI
thf(fact_420_asymp__onI,axiom,
    ! [A: set_nat,R3: nat > nat > $o] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ( member_nat @ Y4 @ A )
           => ( ( R3 @ X3 @ Y4 )
             => ~ ( R3 @ Y4 @ X3 ) ) ) )
     => ( asymp_on_nat @ A @ R3 ) ) ).

% asymp_onI
thf(fact_421_image__empty,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ bot_bo4834947051006887904um_a_c )
      = bot_bo3453284597459734017um_a_c ) ).

% image_empty
thf(fact_422_image__empty,axiom,
    ! [F: $o > $o] :
      ( ( image_o_o2 @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% image_empty
thf(fact_423_empty__is__image,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( bot_bo3453284597459734017um_a_c
        = ( image_7318554134589591124um_a_c @ F @ A ) )
      = ( A = bot_bo4834947051006887904um_a_c ) ) ).

% empty_is_image
thf(fact_424_empty__is__image,axiom,
    ! [F: $o > $o,A: set_o] :
      ( ( bot_bot_set_o
        = ( image_o_o2 @ F @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% empty_is_image
thf(fact_425_image__is__empty,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ A )
        = bot_bo3453284597459734017um_a_c )
      = ( A = bot_bo4834947051006887904um_a_c ) ) ).

% image_is_empty
thf(fact_426_image__is__empty,axiom,
    ! [F: $o > $o,A: set_o] :
      ( ( ( image_o_o2 @ F @ A )
        = bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% image_is_empty
thf(fact_427_singletonI,axiom,
    ! [A2: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ A2 @ ( insert7520310754158225127um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) ) ).

% singletonI
thf(fact_428_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_429_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_430_these__empty,axiom,
    ( ( these_o @ bot_bot_set_option_o )
    = bot_bot_set_o ) ).

% these_empty
thf(fact_431_asympI,axiom,
    ! [R3: $o > $o > $o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( R3 @ X3 @ Y4 )
         => ~ ( R3 @ Y4 @ X3 ) )
     => ( asymp_on_o @ top_top_set_o @ R3 ) ) ).

% asympI
thf(fact_432_asympI,axiom,
    ! [R3: nat > nat > $o] :
      ( ! [X3: nat,Y4: nat] :
          ( ( R3 @ X3 @ Y4 )
         => ~ ( R3 @ Y4 @ X3 ) )
     => ( asymp_on_nat @ top_top_set_nat @ R3 ) ) ).

% asympI
thf(fact_433_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec5227572641185395563um_a_c
            @ ^ [S4: nat > sum_sum_a_c] : P )
          = top_to3439567294830504444um_a_c ) )
      & ( ~ P
       => ( ( collec5227572641185395563um_a_c
            @ ^ [S4: nat > sum_sum_a_c] : P )
          = bot_bo4834947051006887904um_a_c ) ) ) ).

% Collect_const
thf(fact_434_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S4: product_prod_nat_nat] : P )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ^ [S4: product_prod_nat_nat] : P )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const
thf(fact_435_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_o
            @ ^ [S4: $o] : P )
          = top_top_set_o ) )
      & ( ~ P
       => ( ( collect_o
            @ ^ [S4: $o] : P )
          = bot_bot_set_o ) ) ) ).

% Collect_const
thf(fact_436_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S4: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_437_singleton__conv,axiom,
    ! [A2: nat > sum_sum_a_c] :
      ( ( collec5227572641185395563um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( X = A2 ) )
      = ( insert7583143589955530566um_a_c @ A2 @ bot_bo4834947051006887904um_a_c ) ) ).

% singleton_conv
thf(fact_438_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_439_singleton__conv,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( X = A2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_440_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X: $o] : ( X = A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_441_singleton__conv2,axiom,
    ! [A2: nat > sum_sum_a_c] :
      ( ( collec5227572641185395563um_a_c
        @ ( ^ [Y5: nat > sum_sum_a_c,Z3: nat > sum_sum_a_c] : ( Y5 = Z3 )
          @ A2 ) )
      = ( insert7583143589955530566um_a_c @ A2 @ bot_bo4834947051006887904um_a_c ) ) ).

% singleton_conv2
thf(fact_442_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 )
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_443_singleton__conv2,axiom,
    ! [A2: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y5: product_prod_nat_nat,Z3: product_prod_nat_nat] : ( Y5 = Z3 )
          @ A2 ) )
      = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_444_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z3: $o] : ( Y5 = Z3 )
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_445_Set_Oempty__def,axiom,
    ( bot_bo4834947051006887904um_a_c
    = ( collec5227572641185395563um_a_c
      @ ^ [X: nat > sum_sum_a_c] : $false ) ) ).

% Set.empty_def
thf(fact_446_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_447_Set_Oempty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $false ) ) ).

% Set.empty_def
thf(fact_448_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X: $o] : $false ) ) ).

% Set.empty_def
thf(fact_449_ex__in__conv,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ( ? [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A ) )
      = ( A != bot_bo3453284597459734017um_a_c ) ) ).

% ex_in_conv
thf(fact_450_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_451_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X: $o] : ( member_o @ X @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_452_equals0I,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ! [Y4: list_Sum_sum_a_c] :
          ~ ( member7772695417316360142um_a_c @ Y4 @ A )
     => ( A = bot_bo3453284597459734017um_a_c ) ) ).

% equals0I
thf(fact_453_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_454_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_455_equals0D,axiom,
    ! [A: set_list_Sum_sum_a_c,A2: list_Sum_sum_a_c] :
      ( ( A = bot_bo3453284597459734017um_a_c )
     => ~ ( member7772695417316360142um_a_c @ A2 @ A ) ) ).

% equals0D
thf(fact_456_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_457_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_458_emptyE,axiom,
    ! [A2: list_Sum_sum_a_c] :
      ~ ( member7772695417316360142um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) ).

% emptyE
thf(fact_459_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_460_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_461_asymp__onD,axiom,
    ! [A: set_list_Sum_sum_a_c,R3: list_Sum_sum_a_c > list_Sum_sum_a_c > $o,X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( asymp_5276361849251488665um_a_c @ A @ R3 )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( ( member7772695417316360142um_a_c @ Y3 @ A )
         => ( ( R3 @ X2 @ Y3 )
           => ~ ( R3 @ Y3 @ X2 ) ) ) ) ) ).

% asymp_onD
thf(fact_462_asymp__onD,axiom,
    ! [A: set_o,R3: $o > $o > $o,X2: $o,Y3: $o] :
      ( ( asymp_on_o @ A @ R3 )
     => ( ( member_o @ X2 @ A )
       => ( ( member_o @ Y3 @ A )
         => ( ( R3 @ X2 @ Y3 )
           => ~ ( R3 @ Y3 @ X2 ) ) ) ) ) ).

% asymp_onD
thf(fact_463_asymp__onD,axiom,
    ! [A: set_nat,R3: nat > nat > $o,X2: nat,Y3: nat] :
      ( ( asymp_on_nat @ A @ R3 )
     => ( ( member_nat @ X2 @ A )
       => ( ( member_nat @ Y3 @ A )
         => ( ( R3 @ X2 @ Y3 )
           => ~ ( R3 @ Y3 @ X2 ) ) ) ) ) ).

% asymp_onD
thf(fact_464_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_465_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_466_singleton__inject,axiom,
    ! [A2: $o,B: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_467_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_468_doubleton__eq__iff,axiom,
    ! [A2: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_469_singleton__iff,axiom,
    ! [B: list_Sum_sum_a_c,A2: list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( insert7520310754158225127um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_470_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_471_singleton__iff,axiom,
    ! [B: $o,A2: $o] :
      ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_472_singletonD,axiom,
    ! [B: list_Sum_sum_a_c,A2: list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( insert7520310754158225127um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_473_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_474_singletonD,axiom,
    ! [B: $o,A2: $o] :
      ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_475_asympD,axiom,
    ! [R3: $o > $o > $o,X2: $o,Y3: $o] :
      ( ( asymp_on_o @ top_top_set_o @ R3 )
     => ( ( R3 @ X2 @ Y3 )
       => ~ ( R3 @ Y3 @ X2 ) ) ) ).

% asympD
thf(fact_476_asympD,axiom,
    ! [R3: nat > nat > $o,X2: nat,Y3: nat] :
      ( ( asymp_on_nat @ top_top_set_nat @ R3 )
     => ( ( R3 @ X2 @ Y3 )
       => ~ ( R3 @ Y3 @ X2 ) ) ) ).

% asympD
thf(fact_477_Collect__conv__if,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,A2: nat > sum_sum_a_c] :
      ( ( ( P @ A2 )
       => ( ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert7583143589955530566um_a_c @ A2 @ bot_bo4834947051006887904um_a_c ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bo4834947051006887904um_a_c ) ) ) ).

% Collect_conv_if
thf(fact_478_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_479_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_480_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( X = A2 )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_481_Collect__conv__if2,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,A2: nat > sum_sum_a_c] :
      ( ( ( P @ A2 )
       => ( ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert7583143589955530566um_a_c @ A2 @ bot_bo4834947051006887904um_a_c ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bo4834947051006887904um_a_c ) ) ) ).

% Collect_conv_if2
thf(fact_482_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_483_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
      ( ( ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_484_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X: $o] :
                ( ( A2 = X )
                & ( P @ X ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_485_these__not__empty__eq,axiom,
    ! [B3: set_op4691527921008582829um_a_c] :
      ( ( ( these_Sum_sum_a_c @ B3 )
       != bot_bo8815070702908403697um_a_c )
      = ( ( B3 != bot_bo4057517430891132225um_a_c )
        & ( B3
         != ( insert8830009938712558887um_a_c @ none_Sum_sum_a_c @ bot_bo4057517430891132225um_a_c ) ) ) ) ).

% these_not_empty_eq
thf(fact_486_these__not__empty__eq,axiom,
    ! [B3: set_option_o] :
      ( ( ( these_o @ B3 )
       != bot_bot_set_o )
      = ( ( B3 != bot_bot_set_option_o )
        & ( B3
         != ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_not_empty_eq
thf(fact_487_these__empty__eq,axiom,
    ! [B3: set_op4691527921008582829um_a_c] :
      ( ( ( these_Sum_sum_a_c @ B3 )
        = bot_bo8815070702908403697um_a_c )
      = ( ( B3 = bot_bo4057517430891132225um_a_c )
        | ( B3
          = ( insert8830009938712558887um_a_c @ none_Sum_sum_a_c @ bot_bo4057517430891132225um_a_c ) ) ) ) ).

% these_empty_eq
thf(fact_488_these__empty__eq,axiom,
    ! [B3: set_option_o] :
      ( ( ( these_o @ B3 )
        = bot_bot_set_o )
      = ( ( B3 = bot_bot_set_option_o )
        | ( B3
          = ( insert_option_o @ none_o @ bot_bot_set_option_o ) ) ) ) ).

% these_empty_eq
thf(fact_489_image__constant,axiom,
    ! [X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c,C: list_Sum_sum_a_c] :
      ( ( member4884986500679352621um_a_c @ X2 @ A )
     => ( ( image_7318554134589591124um_a_c
          @ ^ [X: nat > sum_sum_a_c] : C
          @ A )
        = ( insert7520310754158225127um_a_c @ C @ bot_bo3453284597459734017um_a_c ) ) ) ).

% image_constant
thf(fact_490_image__constant,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,C: $o] :
      ( ( member7772695417316360142um_a_c @ X2 @ A )
     => ( ( image_1429702505460152410_a_c_o
          @ ^ [X: list_Sum_sum_a_c] : C
          @ A )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_491_image__constant,axiom,
    ! [X2: $o,A: set_o,C: $o] :
      ( ( member_o @ X2 @ A )
     => ( ( image_o_o2
          @ ^ [X: $o] : C
          @ A )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_492_image__constant,axiom,
    ! [X2: nat,A: set_nat,C: $o] :
      ( ( member_nat @ X2 @ A )
     => ( ( image_nat_o2
          @ ^ [X: nat] : C
          @ A )
        = ( insert_o @ C @ bot_bot_set_o ) ) ) ).

% image_constant
thf(fact_493_image__constant__conv,axiom,
    ! [A: set_nat_Sum_sum_a_c,C: list_Sum_sum_a_c] :
      ( ( ( A = bot_bo4834947051006887904um_a_c )
       => ( ( image_7318554134589591124um_a_c
            @ ^ [X: nat > sum_sum_a_c] : C
            @ A )
          = bot_bo3453284597459734017um_a_c ) )
      & ( ( A != bot_bo4834947051006887904um_a_c )
       => ( ( image_7318554134589591124um_a_c
            @ ^ [X: nat > sum_sum_a_c] : C
            @ A )
          = ( insert7520310754158225127um_a_c @ C @ bot_bo3453284597459734017um_a_c ) ) ) ) ).

% image_constant_conv
thf(fact_494_image__constant__conv,axiom,
    ! [A: set_o,C: $o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( image_o_o2
            @ ^ [X: $o] : C
            @ A )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_o )
       => ( ( image_o_o2
            @ ^ [X: $o] : C
            @ A )
          = ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).

% image_constant_conv
thf(fact_495_range__eq__singletonD,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A2: list_Sum_sum_a_c,X2: nat > sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c )
        = ( insert7520310754158225127um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) )
     => ( ( F @ X2 )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_496_range__eq__singletonD,axiom,
    ! [F: $o > $o,A2: $o,X2: $o] :
      ( ( ( image_o_o2 @ F @ top_top_set_o )
        = ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( ( F @ X2 )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_497_range__eq__singletonD,axiom,
    ! [F: nat > $o,A2: $o,X2: nat] :
      ( ( ( image_nat_o2 @ F @ top_top_set_nat )
        = ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( ( F @ X2 )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_498_is__singletonI,axiom,
    ! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_499_eval__table__def,axiom,
    ( eval_table_a_c
    = ( ^ [Ts2: list_fo_term_a,X8: set_list_Sum_sum_a_c] :
          ( comple5959856935634906944um_a_c
          @ ( image_557424483135642155um_a_c
            @ ^ [Vs2: list_Sum_sum_a_c] :
                ( case_o3473079687290933728um_a_c @ bot_bo3453284597459734017um_a_c
                @ ^ [Sigma4: nat > option_Sum_sum_a_c] : ( insert7520310754158225127um_a_c @ ( map_nat_Sum_sum_a_c @ ( comp_o2608084742246830931_c_nat @ the_Sum_sum_a_c @ Sigma4 ) @ ( fv_fo_terms_list_a @ Ts2 ) ) @ bot_bo3453284597459734017um_a_c )
                @ ( unify_vals_terms_a_c @ Vs2 @ Ts2
                  @ ^ [X: nat] : none_Sum_sum_a_c ) )
            @ X8 ) ) ) ) ).

% eval_table_def
thf(fact_500_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
        ? [X: $o] :
          ( A4
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_501_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_502_UnionI,axiom,
    ! [X7: set_list_Sum_sum_a_c,C2: set_se1773346511508022051um_a_c,A: list_Sum_sum_a_c] :
      ( ( member8087264379667941508um_a_c @ X7 @ C2 )
     => ( ( member7772695417316360142um_a_c @ A @ X7 )
       => ( member7772695417316360142um_a_c @ A @ ( comple5959856935634906944um_a_c @ C2 ) ) ) ) ).

% UnionI
thf(fact_503_UnionI,axiom,
    ! [X7: set_o,C2: set_set_o,A: $o] :
      ( ( member_set_o @ X7 @ C2 )
     => ( ( member_o @ A @ X7 )
       => ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).

% UnionI
thf(fact_504_UnionI,axiom,
    ! [X7: set_nat,C2: set_set_nat,A: nat] :
      ( ( member_set_nat @ X7 @ C2 )
     => ( ( member_nat @ A @ X7 )
       => ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).

% UnionI
thf(fact_505_Union__iff,axiom,
    ! [A: list_Sum_sum_a_c,C2: set_se1773346511508022051um_a_c] :
      ( ( member7772695417316360142um_a_c @ A @ ( comple5959856935634906944um_a_c @ C2 ) )
      = ( ? [X: set_list_Sum_sum_a_c] :
            ( ( member8087264379667941508um_a_c @ X @ C2 )
            & ( member7772695417316360142um_a_c @ A @ X ) ) ) ) ).

% Union_iff
thf(fact_506_Union__iff,axiom,
    ! [A: $o,C2: set_set_o] :
      ( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
      = ( ? [X: set_o] :
            ( ( member_set_o @ X @ C2 )
            & ( member_o @ A @ X ) ) ) ) ).

% Union_iff
thf(fact_507_Union__iff,axiom,
    ! [A: nat,C2: set_set_nat] :
      ( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
      = ( ? [X: set_nat] :
            ( ( member_set_nat @ X @ C2 )
            & ( member_nat @ A @ X ) ) ) ) ).

% Union_iff
thf(fact_508_Sup__bot__conv_I2_J,axiom,
    ! [A: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_509_Sup__bot__conv_I2_J,axiom,
    ! [A: set_o] :
      ( ( bot_bot_o
        = ( complete_Sup_Sup_o @ A ) )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A )
           => ( X = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_510_Sup__bot__conv_I1_J,axiom,
    ! [A: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_511_Sup__bot__conv_I1_J,axiom,
    ! [A: set_o] :
      ( ( ( complete_Sup_Sup_o @ A )
        = bot_bot_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A )
           => ( X = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_512_SUP__identity__eq,axiom,
    ! [A: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [X: $o] : X
          @ A ) )
      = ( complete_Sup_Sup_o @ A ) ) ).

% SUP_identity_eq
thf(fact_513_UN__I,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_557424483135642155um_a_c @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_514_UN__I,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: $o,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_8463123914210768698_set_o @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_515_UN__I,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: nat,B3: list_Sum_sum_a_c > set_nat] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_3797473998943575876et_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_516_UN__I,axiom,
    ! [A2: $o,A: set_o,B: list_Sum_sum_a_c,B3: $o > set_list_Sum_sum_a_c] :
      ( ( member_o @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_5735476843758645056um_a_c @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_517_UN__I,axiom,
    ! [A2: $o,A: set_o,B: $o,B3: $o > set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_518_UN__I,axiom,
    ! [A2: $o,A: set_o,B: nat,B3: $o > set_nat] :
      ( ( member_o @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_519_UN__I,axiom,
    ! [A2: nat,A: set_nat,B: list_Sum_sum_a_c,B3: nat > set_list_Sum_sum_a_c] :
      ( ( member_nat @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_1769227344735649220um_a_c @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_520_UN__I,axiom,
    ! [A2: nat,A: set_nat,B: $o,B3: nat > set_o] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_521_UN__I,axiom,
    ! [A2: nat,A: set_nat,B: nat,B3: nat > set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) ) ) ) ).

% UN_I
thf(fact_522_Sup__UNIV,axiom,
    ( ( comple90263536869209701_set_o @ top_top_set_set_o )
    = top_top_set_o ) ).

% Sup_UNIV
thf(fact_523_Sup__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Sup_UNIV
thf(fact_524_Sup__UNIV,axiom,
    ( ( complete_Sup_Sup_o @ top_top_set_o )
    = top_top_o ) ).

% Sup_UNIV
thf(fact_525_Sup__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Sup_empty
thf(fact_526_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_527_SUP__id__eq,axiom,
    ! [A: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ id_o @ A ) )
      = ( complete_Sup_Sup_o @ A ) ) ).

% SUP_id_eq
thf(fact_528_SUP__const,axiom,
    ! [A: set_o,F: $o] :
      ( ( A != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [I: $o] : F
            @ A ) )
        = F ) ) ).

% SUP_const
thf(fact_529_UN__constant,axiom,
    ! [A: set_o,C: set_o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A ) )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A ) )
          = C ) ) ) ).

% UN_constant
thf(fact_530_UN__singleton,axiom,
    ! [A: set_o] :
      ( ( comple90263536869209701_set_o
        @ ( image_o_set_o
          @ ^ [X: $o] : ( insert_o @ X @ bot_bot_set_o )
          @ A ) )
      = A ) ).

% UN_singleton
thf(fact_531_UN__simps_I1_J,axiom,
    ! [C2: set_o,A2: $o,B3: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A2 @ ( B3 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A2 @ ( B3 @ X ) )
              @ C2 ) )
          = ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ C2 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_532_UnionE,axiom,
    ! [A: list_Sum_sum_a_c,C2: set_se1773346511508022051um_a_c] :
      ( ( member7772695417316360142um_a_c @ A @ ( comple5959856935634906944um_a_c @ C2 ) )
     => ~ ! [X9: set_list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ A @ X9 )
           => ~ ( member8087264379667941508um_a_c @ X9 @ C2 ) ) ) ).

% UnionE
thf(fact_533_UnionE,axiom,
    ! [A: $o,C2: set_set_o] :
      ( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
     => ~ ! [X9: set_o] :
            ( ( member_o @ A @ X9 )
           => ~ ( member_set_o @ X9 @ C2 ) ) ) ).

% UnionE
thf(fact_534_UnionE,axiom,
    ! [A: nat,C2: set_set_nat] :
      ( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
     => ~ ! [X9: set_nat] :
            ( ( member_nat @ A @ X9 )
           => ~ ( member_set_nat @ X9 @ C2 ) ) ) ).

% UnionE
thf(fact_535_UN__E,axiom,
    ! [B: list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_557424483135642155um_a_c @ B3 @ A ) ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_536_UN__E,axiom,
    ! [B: list_Sum_sum_a_c,B3: $o > set_list_Sum_sum_a_c,A: set_o] :
      ( ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_5735476843758645056um_a_c @ B3 @ A ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_537_UN__E,axiom,
    ! [B: list_Sum_sum_a_c,B3: nat > set_list_Sum_sum_a_c,A: set_nat] :
      ( ( member7772695417316360142um_a_c @ B @ ( comple5959856935634906944um_a_c @ ( image_1769227344735649220um_a_c @ B3 @ A ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_538_UN__E,axiom,
    ! [B: $o,B3: list_Sum_sum_a_c > set_o,A: set_list_Sum_sum_a_c] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_8463123914210768698_set_o @ B3 @ A ) ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ~ ( member_o @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_539_UN__E,axiom,
    ! [B: $o,B3: $o > set_o,A: set_o] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ A ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ~ ( member_o @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_540_UN__E,axiom,
    ! [B: $o,B3: nat > set_o,A: set_nat] :
      ( ( member_o @ B @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B3 @ A ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ~ ( member_o @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_541_UN__E,axiom,
    ! [B: nat,B3: list_Sum_sum_a_c > set_nat,A: set_list_Sum_sum_a_c] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_3797473998943575876et_nat @ B3 @ A ) ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ~ ( member_nat @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_542_UN__E,axiom,
    ! [B: nat,B3: $o > set_nat,A: set_o] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B3 @ A ) ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ~ ( member_nat @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_543_UN__E,axiom,
    ! [B: nat,B3: nat > set_nat,A: set_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ~ ( member_nat @ B @ ( B3 @ X3 ) ) ) ) ).

% UN_E
thf(fact_544_bot__empty__eq,axiom,
    ( bot_bo6414102808470806980_a_c_o
    = ( ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ bot_bo3453284597459734017um_a_c ) ) ) ).

% bot_empty_eq
thf(fact_545_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_546_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_547_bot__set__def,axiom,
    ( bot_bo4834947051006887904um_a_c
    = ( collec5227572641185395563um_a_c @ bot_bo8889875489019841189_a_c_o ) ) ).

% bot_set_def
thf(fact_548_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_549_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_550_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_551_SUP__cong,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,C2: list_Sum_sum_a_c > $o,D2: list_Sum_sum_a_c > $o] :
      ( ( A = B3 )
     => ( ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ B3 )
           => ( ( C2 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ C2 @ A ) )
          = ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ D2 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_552_SUP__cong,axiom,
    ! [A: set_o,B3: set_o,C2: $o > $o,D2: $o > $o] :
      ( ( A = B3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ B3 )
           => ( ( C2 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ C2 @ A ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ D2 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_553_SUP__cong,axiom,
    ! [A: set_nat,B3: set_nat,C2: nat > $o,D2: nat > $o] :
      ( ( A = B3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B3 )
           => ( ( C2 @ X3 )
              = ( D2 @ X3 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ C2 @ A ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o2 @ D2 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_554_Union__UNIV,axiom,
    ( ( comple90263536869209701_set_o @ top_top_set_set_o )
    = top_top_set_o ) ).

% Union_UNIV
thf(fact_555_Union__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Union_UNIV
thf(fact_556_Union__empty,axiom,
    ( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
    = bot_bot_set_o ) ).

% Union_empty
thf(fact_557_Union__empty__conv,axiom,
    ! [A: set_set_o] :
      ( ( ( comple90263536869209701_set_o @ A )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A )
           => ( X = bot_bot_set_o ) ) ) ) ).

% Union_empty_conv
thf(fact_558_empty__Union__conv,axiom,
    ! [A: set_set_o] :
      ( ( bot_bot_set_o
        = ( comple90263536869209701_set_o @ A ) )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ A )
           => ( X = bot_bot_set_o ) ) ) ) ).

% empty_Union_conv
thf(fact_559_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_Sum_sum_a_c] :
      ( ( Option = none_Sum_sum_a_c )
      = ( case_o3562416616119909748um_a_c @ $true
        @ ^ [Uu: sum_sum_a_c] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_560_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_Sum_sum_a_c] :
      ( ( Option != none_Sum_sum_a_c )
      = ( case_o3562416616119909748um_a_c @ $false
        @ ^ [Uu: sum_sum_a_c] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_561_image__Union,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,S: set_se703004760546936450um_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ ( comple6350876882455579167um_a_c @ S ) )
      = ( comple5959856935634906944um_a_c @ ( image_8538515752806177344um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ S ) ) ) ).

% image_Union
thf(fact_562_Union__natural,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( comp_s5125373028252909175um_a_c @ comple5959856935634906944um_a_c @ ( image_8538515752806177344um_a_c @ ( image_7318554134589591124um_a_c @ F ) ) )
      = ( comp_s8429145977130442144um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ comple6350876882455579167um_a_c ) ) ).

% Union_natural
thf(fact_563_UN__empty,axiom,
    ! [B3: $o > set_o] :
      ( ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ bot_bot_set_o ) )
      = bot_bot_set_o ) ).

% UN_empty
thf(fact_564_UN__insert__distrib,axiom,
    ! [U2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,A2: $o,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7772695417316360142um_a_c @ U2 @ A )
     => ( ( comple90263536869209701_set_o
          @ ( image_8463123914210768698_set_o
            @ ^ [X: list_Sum_sum_a_c] : ( insert_o @ A2 @ ( B3 @ X ) )
            @ A ) )
        = ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_8463123914210768698_set_o @ B3 @ A ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_565_UN__insert__distrib,axiom,
    ! [U2: $o,A: set_o,A2: $o,B3: $o > set_o] :
      ( ( member_o @ U2 @ A )
     => ( ( comple90263536869209701_set_o
          @ ( image_o_set_o
            @ ^ [X: $o] : ( insert_o @ A2 @ ( B3 @ X ) )
            @ A ) )
        = ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ A ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_566_UN__insert__distrib,axiom,
    ! [U2: nat,A: set_nat,A2: $o,B3: nat > set_o] :
      ( ( member_nat @ U2 @ A )
     => ( ( comple90263536869209701_set_o
          @ ( image_nat_set_o
            @ ^ [X: nat] : ( insert_o @ A2 @ ( B3 @ X ) )
            @ A ) )
        = ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B3 @ A ) ) ) ) ) ).

% UN_insert_distrib
thf(fact_567_SUP__eq__const,axiom,
    ! [I2: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,X2: $o] :
      ( ( I2 != bot_bo3453284597459734017um_a_c )
     => ( ! [I3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ I3 @ I2 )
           => ( ( F @ I3 )
              = X2 ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ I2 ) )
          = X2 ) ) ) ).

% SUP_eq_const
thf(fact_568_SUP__eq__const,axiom,
    ! [I2: set_nat,F: nat > $o,X2: $o] :
      ( ( I2 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I2 )
           => ( ( F @ I3 )
              = X2 ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ I2 ) )
          = X2 ) ) ) ).

% SUP_eq_const
thf(fact_569_SUP__eq__const,axiom,
    ! [I2: set_o,F: $o > $o,X2: $o] :
      ( ( I2 != bot_bot_set_o )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I2 )
           => ( ( F @ I3 )
              = X2 ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ I2 ) )
          = X2 ) ) ) ).

% SUP_eq_const
thf(fact_570_SUP__image,axiom,
    ! [G: list_Sum_sum_a_c > $o,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ ( image_7318554134589591124um_a_c @ F @ A ) ) )
      = ( complete_Sup_Sup_o @ ( image_3393300230255336507_a_c_o @ ( comp_l8081135279519614228um_a_c @ G @ F ) @ A ) ) ) ).

% SUP_image
thf(fact_571_case__optionE,axiom,
    ! [P: $o,Q: sum_sum_a_c > $o,X2: option_Sum_sum_a_c] :
      ( ( case_o3562416616119909748um_a_c @ P @ Q @ X2 )
     => ( ( ( X2 = none_Sum_sum_a_c )
         => ~ P )
       => ~ ! [Y4: sum_sum_a_c] :
              ( ( X2
                = ( some_Sum_sum_a_c @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_572_case__optionE,axiom,
    ! [P: $o,Q: ( nat > option_Sum_sum_a_c ) > $o,X2: option6339742336662979638um_a_c] :
      ( ( case_o1806125575262777907um_a_c @ P @ Q @ X2 )
     => ( ( ( X2 = none_n2797051391425193029um_a_c )
         => ~ P )
       => ~ ! [Y4: nat > option_Sum_sum_a_c] :
              ( ( X2
                = ( some_n5886337604800580545um_a_c @ Y4 ) )
             => ~ ( Q @ Y4 ) ) ) ) ).

% case_optionE
thf(fact_573_UN__extend__simps_I1_J,axiom,
    ! [C2: set_o,A2: $o,B3: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ C2 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( insert_o @ A2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ C2 ) ) )
          = ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( insert_o @ A2 @ ( B3 @ X ) )
              @ C2 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_574_is__singletonI_H,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ( A != bot_bo3453284597459734017um_a_c )
     => ( ! [X3: list_Sum_sum_a_c,Y4: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ( ( member7772695417316360142um_a_c @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin8793101897608424131um_a_c @ A ) ) ) ).

% is_singletonI'
thf(fact_575_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( member_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_576_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y4: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( member_o @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_577_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_578_SUP__empty,axiom,
    ! [F: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ bot_bot_set_o ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_579_SUP__constant,axiom,
    ! [A: set_o,C: set_o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A ) )
          = bot_bot_set_o ) )
      & ( ( A != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [Y: $o] : C
              @ A ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_580_SUP__constant,axiom,
    ! [C: $o,A: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [Y: $o] : C
          @ A ) )
      = ( ( ( A = bot_bot_set_o )
         => bot_bot_o )
        & ( ( A != bot_bot_set_o )
         => C ) ) ) ).

% SUP_constant
thf(fact_581_UNION__singleton__eq__range,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( comple5959856935634906944um_a_c
        @ ( image_9084665225475849994um_a_c
          @ ^ [X: nat > sum_sum_a_c] : ( insert7520310754158225127um_a_c @ ( F @ X ) @ bot_bo3453284597459734017um_a_c )
          @ A ) )
      = ( image_7318554134589591124um_a_c @ F @ A ) ) ).

% UNION_singleton_eq_range
thf(fact_582_cSUP__const,axiom,
    ! [A: set_o,C: $o] :
      ( ( A != bot_bot_set_o )
     => ( ( complete_Sup_Sup_o
          @ ( image_o_o2
            @ ^ [X: $o] : C
            @ A ) )
        = C ) ) ).

% cSUP_const
thf(fact_583_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
          ( A4
          = ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_584_the__elem__eq,axiom,
    ! [X2: $o] :
      ( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
      = X2 ) ).

% the_elem_eq
thf(fact_585_Sup__SUP__eq,axiom,
    ( comple2583835846132560197_a_c_o
    = ( ^ [S5: set_li8768942875727007800_a_c_o,X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ ( comple5959856935634906944um_a_c @ ( image_4575336749219106048um_a_c @ collec8219452656984879116um_a_c @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_586_Sup__SUP__eq,axiom,
    ( complete_Sup_Sup_o_o
    = ( ^ [S5: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_587_Sup__SUP__eq,axiom,
    ( comple3252014395245574438_a_c_o
    = ( ^ [S5: set_na4498582200620068761_a_c_o,X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ ( comple6350876882455579167um_a_c @ ( image_1246991150420438848um_a_c @ collec5227572641185395563um_a_c @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_588_Sup__SUP__eq,axiom,
    ( comple8317665133742190828_nat_o
    = ( ^ [S5: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_589_Sup__SUP__eq,axiom,
    ( comple1901448594430626575_nat_o
    = ( ^ [S5: set_Pr5582243495563764594_nat_o,X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ S5 ) ) ) ) ) ).

% Sup_SUP_eq
thf(fact_590_SUP__Sup__eq,axiom,
    ! [S: set_se1773346511508022051um_a_c] :
      ( ( comple2583835846132560197_a_c_o
        @ ( image_93493330030733284_a_c_o
          @ ^ [I: set_list_Sum_sum_a_c,X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ I )
          @ S ) )
      = ( ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ ( comple5959856935634906944um_a_c @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_591_SUP__Sup__eq,axiom,
    ! [S: set_set_o] :
      ( ( complete_Sup_Sup_o_o
        @ ( image_set_o_o_o
          @ ^ [I: set_o,X: $o] : ( member_o @ X @ I )
          @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_592_SUP__Sup__eq,axiom,
    ! [S: set_set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I: set_nat,X: nat] : ( member_nat @ X @ I )
          @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).

% SUP_Sup_eq
thf(fact_593_Sup__set__def,axiom,
    ( comple5959856935634906944um_a_c
    = ( ^ [A4: set_se1773346511508022051um_a_c] :
          ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] : ( complete_Sup_Sup_o @ ( image_2401522433795426852_a_c_o @ ( member7772695417316360142um_a_c @ X ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_594_Sup__set__def,axiom,
    ( comple90263536869209701_set_o
    = ( ^ [A4: set_set_o] :
          ( collect_o
          @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_set_o_o @ ( member_o @ X ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_595_Sup__set__def,axiom,
    ( comple6350876882455579167um_a_c
    = ( ^ [A4: set_se703004760546936450um_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] : ( complete_Sup_Sup_o @ ( image_5931342554959902853_a_c_o @ ( member4884986500679352621um_a_c @ X ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_596_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A4: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_597_Sup__set__def,axiom,
    ( comple5685304695842803022at_nat
    = ( ^ [A4: set_se7855581050983116737at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] : ( complete_Sup_Sup_o @ ( image_7457375456213313148_nat_o @ ( member8440522571783428010at_nat @ X ) @ A4 ) ) ) ) ) ).

% Sup_set_def
thf(fact_598_the__elem__image__unique,axiom,
    ! [A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,X2: nat > sum_sum_a_c] :
      ( ( A != bot_bo4834947051006887904um_a_c )
     => ( ! [Y4: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ Y4 @ A )
           => ( ( F @ Y4 )
              = ( F @ X2 ) ) )
       => ( ( the_el2610808913227880450um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) )
          = ( F @ X2 ) ) ) ) ).

% the_elem_image_unique
thf(fact_599_cSUP__UNION,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_o,F: $o > $o] :
      ( ( A != bot_bo3453284597459734017um_a_c )
     => ( ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ( ( B3 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_8463123914210768698_set_o
                @ ^ [X: list_Sum_sum_a_c] : ( image_o_o2 @ F @ ( B3 @ X ) )
                @ A ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_8463123914210768698_set_o @ B3 @ A ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_1429702505460152410_a_c_o
                @ ^ [X: list_Sum_sum_a_c] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B3 @ X ) ) )
                @ A ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_600_cSUP__UNION,axiom,
    ! [A: set_nat,B3: nat > set_o,F: $o > $o] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( B3 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_nat_set_o
                @ ^ [X: nat] : ( image_o_o2 @ F @ ( B3 @ X ) )
                @ A ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B3 @ A ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_nat_o2
                @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B3 @ X ) ) )
                @ A ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_601_cSUP__UNION,axiom,
    ! [A: set_o,B3: $o > set_o,F: $o > $o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( B3 @ X3 )
             != bot_bot_set_o ) )
       => ( ( condit5488710616941104124bove_o
            @ ( comple90263536869209701_set_o
              @ ( image_o_set_o
                @ ^ [X: $o] : ( image_o_o2 @ F @ ( B3 @ X ) )
                @ A ) ) )
         => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ A ) ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o2
                @ ^ [X: $o] : ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( B3 @ X ) ) )
                @ A ) ) ) ) ) ) ).

% cSUP_UNION
thf(fact_602_refl__on__singleton,axiom,
    ! [X2: $o] : ( refl_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).

% refl_on_singleton
thf(fact_603_UN__simps_I3_J,axiom,
    ! [C2: set_o,A: set_o,B3: $o > set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B3 @ X ) )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B3 @ X ) )
              @ C2 ) )
          = ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B3 @ C2 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_604_UN__simps_I2_J,axiom,
    ! [C2: set_o,A: $o > set_o,B3: set_o] :
      ( ( ( C2 = bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B3 )
              @ C2 ) )
          = bot_bot_set_o ) )
      & ( ( C2 != bot_bot_set_o )
       => ( ( comple90263536869209701_set_o
            @ ( image_o_set_o
              @ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B3 )
              @ C2 ) )
          = ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ C2 ) ) @ B3 ) ) ) ) ).

% UN_simps(2)
thf(fact_605_UnCI,axiom,
    ! [C: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( ~ ( member7772695417316360142um_a_c @ C @ B3 )
       => ( member7772695417316360142um_a_c @ C @ A ) )
     => ( member7772695417316360142um_a_c @ C @ ( sup_su3984685238261546137um_a_c @ A @ B3 ) ) ) ).

% UnCI
thf(fact_606_UnCI,axiom,
    ! [C: $o,B3: set_o,A: set_o] :
      ( ( ~ ( member_o @ C @ B3 )
       => ( member_o @ C @ A ) )
     => ( member_o @ C @ ( sup_sup_set_o @ A @ B3 ) ) ) ).

% UnCI
thf(fact_607_UnCI,axiom,
    ! [C: nat,B3: set_nat,A: set_nat] :
      ( ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ A ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).

% UnCI
thf(fact_608_Un__iff,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( sup_su3984685238261546137um_a_c @ A @ B3 ) )
      = ( ( member7772695417316360142um_a_c @ C @ A )
        | ( member7772695417316360142um_a_c @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_609_Un__iff,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A @ B3 ) )
      = ( ( member_o @ C @ A )
        | ( member_o @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_610_Un__iff,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) )
      = ( ( member_nat @ C @ A )
        | ( member_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_611_Un__empty,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ( sup_sup_set_o @ A @ B3 )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_612_Un__insert__left,axiom,
    ! [A2: $o,B3: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A2 @ B3 ) @ C2 )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ B3 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_613_Un__insert__right,axiom,
    ! [A: set_o,A2: $o,B3: set_o] :
      ( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B3 ) )
      = ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B3 ) ) ) ).

% Un_insert_right
thf(fact_614_Sup__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( complete_Sup_Sup_o @ ( insert_o @ A2 @ A ) )
      = ( sup_sup_o @ A2 @ ( complete_Sup_Sup_o @ A ) ) ) ).

% Sup_insert
thf(fact_615_cSUP__union,axiom,
    ! [A: set_o,F: $o > $o,B3: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A ) )
       => ( ( B3 != bot_bot_set_o )
         => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ B3 ) )
           => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( sup_sup_set_o @ A @ B3 ) ) )
              = ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ B3 ) ) ) ) ) ) ) ) ).

% cSUP_union
thf(fact_616_Sup__union__distrib,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B3 ) )
      = ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ).

% Sup_union_distrib
thf(fact_617_Sup__bool__def,axiom,
    ( complete_Sup_Sup_o
    = ( member_o @ $true ) ) ).

% Sup_bool_def
thf(fact_618_gt__ex,axiom,
    ! [X2: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).

% gt_ex
thf(fact_619_less__imp__neq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% less_imp_neq
thf(fact_620_order_Oasym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ~ ( ord_less_nat @ B @ A2 ) ) ).

% order.asym
thf(fact_621_ord__eq__less__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_622_ord__less__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_623_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P @ Y6 ) )
         => ( P @ X3 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_624_antisym__conv3,axiom,
    ! [Y3: nat,X2: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X2 )
     => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
        = ( X2 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_625_linorder__cases,axiom,
    ! [X2: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X2 @ Y3 )
     => ( ( X2 != Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_cases
thf(fact_626_dual__order_Oasym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ~ ( ord_less_nat @ A2 @ B ) ) ).

% dual_order.asym
thf(fact_627_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_628_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X5: nat] : ( P2 @ X5 ) )
    = ( ^ [P3: nat > $o] :
        ? [N4: nat] :
          ( ( P3 @ N4 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_629_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
         => ( P @ A3 @ B2 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B2: nat] :
              ( ( P @ B2 @ A3 )
             => ( P @ A3 @ B2 ) )
         => ( P @ A2 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_630_order_Ostrict__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A2 @ C ) ) ) ).

% order.strict_trans
thf(fact_631_not__less__iff__gr__or__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X2 )
        | ( X2 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_632_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_633_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( A2 != B ) ) ).

% order.strict_implies_not_eq
thf(fact_634_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_nat @ B @ A2 )
     => ( A2 != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_635_linorder__neqE,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE
thf(fact_636_order__less__asym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_asym
thf(fact_637_linorder__neq__iff,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
      = ( ( ord_less_nat @ X2 @ Y3 )
        | ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neq_iff
thf(fact_638_order__less__asym_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ~ ( ord_less_nat @ B @ A2 ) ) ).

% order_less_asym'
thf(fact_639_order__less__trans,axiom,
    ! [X2: nat,Y3: nat,Z: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z )
       => ( ord_less_nat @ X2 @ Z ) ) ) ).

% order_less_trans
thf(fact_640_ord__eq__less__subst,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A2
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_641_ord__less__eq__subst,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_642_order__less__irrefl,axiom,
    ! [X2: nat] :
      ~ ( ord_less_nat @ X2 @ X2 ) ).

% order_less_irrefl
thf(fact_643_order__less__subst1,axiom,
    ! [A2: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_644_order__less__subst2,axiom,
    ! [A2: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_nat @ X3 @ Y4 )
             => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_645_order__less__not__sym,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_not_sym
thf(fact_646_order__less__imp__triv,axiom,
    ! [X2: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X2 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_647_linorder__less__linear,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
      | ( X2 = Y3 )
      | ( ord_less_nat @ Y3 @ X2 ) ) ).

% linorder_less_linear
thf(fact_648_order__less__imp__not__eq,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( X2 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_649_order__less__imp__not__eq2,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ( Y3 != X2 ) ) ).

% order_less_imp_not_eq2
thf(fact_650_order__less__imp__not__less,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ord_less_nat @ X2 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).

% order_less_imp_not_less
thf(fact_651_Collect__disj__eq,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( collec5227572641185395563um_a_c
        @ ^ [X: nat > sum_sum_a_c] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su6350494296397389432um_a_c @ ( collec5227572641185395563um_a_c @ P ) @ ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_652_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_653_Collect__disj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_654_Un__def,axiom,
    ( sup_su3984685238261546137um_a_c
    = ( ^ [A4: set_list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X @ A4 )
              | ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_655_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A4 )
              | ( member_o @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_656_Un__def,axiom,
    ( sup_su6350494296397389432um_a_c
    = ( ^ [A4: set_nat_Sum_sum_a_c,B5: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ X @ A4 )
              | ( member4884986500679352621um_a_c @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_657_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A4 )
              | ( member_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_658_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A4 )
              | ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_659_UnE,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( sup_su3984685238261546137um_a_c @ A @ B3 ) )
     => ( ~ ( member7772695417316360142um_a_c @ C @ A )
       => ( member7772695417316360142um_a_c @ C @ B3 ) ) ) ).

% UnE
thf(fact_660_UnE,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A @ B3 ) )
     => ( ~ ( member_o @ C @ A )
       => ( member_o @ C @ B3 ) ) ) ).

% UnE
thf(fact_661_UnE,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) )
     => ( ~ ( member_nat @ C @ A )
       => ( member_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_662_UnI1,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ A )
     => ( member7772695417316360142um_a_c @ C @ ( sup_su3984685238261546137um_a_c @ A @ B3 ) ) ) ).

% UnI1
thf(fact_663_UnI1,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ A )
     => ( member_o @ C @ ( sup_sup_set_o @ A @ B3 ) ) ) ).

% UnI1
thf(fact_664_UnI1,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).

% UnI1
thf(fact_665_UnI2,axiom,
    ! [C: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ B3 )
     => ( member7772695417316360142um_a_c @ C @ ( sup_su3984685238261546137um_a_c @ A @ B3 ) ) ) ).

% UnI2
thf(fact_666_UnI2,axiom,
    ! [C: $o,B3: set_o,A: set_o] :
      ( ( member_o @ C @ B3 )
     => ( member_o @ C @ ( sup_sup_set_o @ A @ B3 ) ) ) ).

% UnI2
thf(fact_667_UnI2,axiom,
    ! [C: nat,B3: set_nat,A: set_nat] :
      ( ( member_nat @ C @ B3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A @ B3 ) ) ) ).

% UnI2
thf(fact_668_Un__UNIV__left,axiom,
    ! [B3: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ B3 )
      = top_top_set_o ) ).

% Un_UNIV_left
thf(fact_669_Un__UNIV__left,axiom,
    ! [B3: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ B3 )
      = top_top_set_nat ) ).

% Un_UNIV_left
thf(fact_670_Un__UNIV__right,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ top_top_set_o )
      = top_top_set_o ) ).

% Un_UNIV_right
thf(fact_671_Un__UNIV__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ top_top_set_nat )
      = top_top_set_nat ) ).

% Un_UNIV_right
thf(fact_672_image__Un,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c @ F @ ( sup_su6350494296397389432um_a_c @ A @ B3 ) )
      = ( sup_su3984685238261546137um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ ( image_7318554134589591124um_a_c @ F @ B3 ) ) ) ).

% image_Un
thf(fact_673_Un__empty__left,axiom,
    ! [B3: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_674_Un__empty__right,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Un_empty_right
thf(fact_675_cSUP__lessD,axiom,
    ! [F: list_Sum_sum_a_c > nat,A: set_list_Sum_sum_a_c,Y3: nat,I4: list_Sum_sum_a_c] :
      ( ( condit2214826472909112428ve_nat @ ( image_7712311756382514062_c_nat @ F @ A ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_7712311756382514062_c_nat @ F @ A ) ) @ Y3 )
       => ( ( member7772695417316360142um_a_c @ I4 @ A )
         => ( ord_less_nat @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_676_cSUP__lessD,axiom,
    ! [F: $o > nat,A: set_o,Y3: nat,I4: $o] :
      ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) @ Y3 )
       => ( ( member_o @ I4 @ A )
         => ( ord_less_nat @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_677_cSUP__lessD,axiom,
    ! [F: nat > nat,A: set_nat,Y3: nat,I4: nat] :
      ( ( condit2214826472909112428ve_nat @ ( image_nat_nat2 @ F @ A ) )
     => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat2 @ F @ A ) ) @ Y3 )
       => ( ( member_nat @ I4 @ A )
         => ( ord_less_nat @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_678_cSUP__lessD,axiom,
    ! [F: list_Sum_sum_a_c > $o,A: set_list_Sum_sum_a_c,Y3: $o,I4: list_Sum_sum_a_c] :
      ( ( condit5488710616941104124bove_o @ ( image_1429702505460152410_a_c_o @ F @ A ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) @ Y3 )
       => ( ( member7772695417316360142um_a_c @ I4 @ A )
         => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_679_cSUP__lessD,axiom,
    ! [F: $o > $o,A: set_o,Y3: $o,I4: $o] :
      ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ Y3 )
       => ( ( member_o @ I4 @ A )
         => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_680_cSUP__lessD,axiom,
    ! [F: nat > $o,A: set_nat,Y3: $o,I4: nat] :
      ( ( condit5488710616941104124bove_o @ ( image_nat_o2 @ F @ A ) )
     => ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) @ Y3 )
       => ( ( member_nat @ I4 @ A )
         => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ) ).

% cSUP_lessD
thf(fact_681_top_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != top_top_set_o )
      = ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).

% top.not_eq_extremum
thf(fact_682_top_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != top_top_set_nat )
      = ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).

% top.not_eq_extremum
thf(fact_683_top_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).

% top.extremum_strict
thf(fact_684_top_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).

% top.extremum_strict
thf(fact_685_refl__onD2,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( refl_o5556642055249044088um_a_c @ A @ R2 )
     => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X2 @ Y3 ) @ R2 )
       => ( member7772695417316360142um_a_c @ Y3 @ A ) ) ) ).

% refl_onD2
thf(fact_686_refl__onD2,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( refl_on_o @ A @ R2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
       => ( member_o @ Y3 @ A ) ) ) ).

% refl_onD2
thf(fact_687_refl__onD2,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( refl_on_nat @ A @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
       => ( member_nat @ Y3 @ A ) ) ) ).

% refl_onD2
thf(fact_688_refl__onD1,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( refl_o5556642055249044088um_a_c @ A @ R2 )
     => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X2 @ Y3 ) @ R2 )
       => ( member7772695417316360142um_a_c @ X2 @ A ) ) ) ).

% refl_onD1
thf(fact_689_refl__onD1,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( refl_on_o @ A @ R2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
       => ( member_o @ X2 @ A ) ) ) ).

% refl_onD1
thf(fact_690_refl__onD1,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( refl_on_nat @ A @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
       => ( member_nat @ X2 @ A ) ) ) ).

% refl_onD1
thf(fact_691_refl__onD,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,A2: list_Sum_sum_a_c] :
      ( ( refl_o5556642055249044088um_a_c @ A @ R2 )
     => ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ A2 ) @ R2 ) ) ) ).

% refl_onD
thf(fact_692_refl__onD,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o] :
      ( ( refl_on_o @ A @ R2 )
     => ( ( member_o @ A2 @ A )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R2 ) ) ) ).

% refl_onD
thf(fact_693_refl__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( ( refl_on_nat @ A @ R2 )
     => ( ( member_nat @ A2 @ A )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ) ).

% refl_onD
thf(fact_694_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_695_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_696_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_697_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_698_not__psubset__empty,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_699_SUP__UNIV__bool__expand,axiom,
    ! [A: $o > $o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ A @ top_top_set_o ) )
      = ( sup_sup_o @ ( A @ $true ) @ ( A @ $false ) ) ) ).

% SUP_UNIV_bool_expand
thf(fact_700_asymp__on__less,axiom,
    ! [A: set_nat] : ( asymp_on_nat @ A @ ord_less_nat ) ).

% asymp_on_less
thf(fact_701_Set_Oinsert__def,axiom,
    ( insert_o
    = ( ^ [A5: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X: $o] : ( X = A5 ) ) ) ) ) ).

% Set.insert_def
thf(fact_702_Set_Oinsert__def,axiom,
    ( insert7583143589955530566um_a_c
    = ( ^ [A5: nat > sum_sum_a_c] :
          ( sup_su6350494296397389432um_a_c
          @ ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] : ( X = A5 ) ) ) ) ) ).

% Set.insert_def
thf(fact_703_Set_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [A5: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A5 ) ) ) ) ) ).

% Set.insert_def
thf(fact_704_Set_Oinsert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A5: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] : ( X = A5 ) ) ) ) ) ).

% Set.insert_def
thf(fact_705_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: set_o,F: $o > $o,G: $o > $o] :
      ( ( A != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A ) )
       => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ G @ A ) )
         => ( ( sup_sup_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ A ) ) )
            = ( complete_Sup_Sup_o
              @ ( image_o_o2
                @ ^ [A5: $o] : ( sup_sup_o @ ( F @ A5 ) @ ( G @ A5 ) )
                @ A ) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_706_less__cSUP__iff,axiom,
    ! [A: set_o,F: $o > nat,A2: nat] :
      ( ( A != bot_bot_set_o )
     => ( ( condit2214826472909112428ve_nat @ ( image_o_nat2 @ F @ A ) )
       => ( ( ord_less_nat @ A2 @ ( complete_Sup_Sup_nat @ ( image_o_nat2 @ F @ A ) ) )
          = ( ? [X: $o] :
                ( ( member_o @ X @ A )
                & ( ord_less_nat @ A2 @ ( F @ X ) ) ) ) ) ) ) ).

% less_cSUP_iff
thf(fact_707_cSUP__insert,axiom,
    ! [A: set_o,F: $o > $o,A2: $o] :
      ( ( A != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ F @ A ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( insert_o @ A2 @ A ) ) )
          = ( sup_sup_o @ ( F @ A2 ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) ) ) ) ) ).

% cSUP_insert
thf(fact_708_asymp__on__greater,axiom,
    ! [A: set_nat] :
      ( asymp_on_nat @ A
      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% asymp_on_greater
thf(fact_709_singleton__Un__iff,axiom,
    ! [X2: $o,A: set_o,B3: set_o] :
      ( ( ( insert_o @ X2 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A @ B3 ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B3
            = ( insert_o @ X2 @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B3 = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B3
            = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_710_Un__singleton__iff,axiom,
    ! [A: set_o,B3: set_o,X2: $o] :
      ( ( ( sup_sup_set_o @ A @ B3 )
        = ( insert_o @ X2 @ bot_bot_set_o ) )
      = ( ( ( A = bot_bot_set_o )
          & ( B3
            = ( insert_o @ X2 @ bot_bot_set_o ) ) )
        | ( ( A
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B3 = bot_bot_set_o ) )
        | ( ( A
            = ( insert_o @ X2 @ bot_bot_set_o ) )
          & ( B3
            = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_711_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_712_reflI,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ! [X3: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R2 )
     => ( refl_on_o @ top_top_set_o @ R2 ) ) ).

% reflI
thf(fact_713_reflI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R2 )
     => ( refl_on_nat @ top_top_set_nat @ R2 ) ) ).

% reflI
thf(fact_714_reflD,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o] :
      ( ( refl_on_o @ top_top_set_o @ R2 )
     => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ R2 ) ) ).

% reflD
thf(fact_715_reflD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat] :
      ( ( refl_on_nat @ top_top_set_nat @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ R2 ) ) ).

% reflD
thf(fact_716_refl__on__empty,axiom,
    refl_on_o @ bot_bot_set_o @ bot_bo7073875226086086771od_o_o ).

% refl_on_empty
thf(fact_717_SUP__absorb,axiom,
    ! [K: list_Sum_sum_a_c,I2: set_list_Sum_sum_a_c,A: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ K @ I2 )
     => ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ A @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ A @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_718_SUP__absorb,axiom,
    ! [K: $o,I2: set_o,A: $o > $o] :
      ( ( member_o @ K @ I2 )
     => ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ A @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_o_o2 @ A @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_719_SUP__absorb,axiom,
    ! [K: nat,I2: set_nat,A: nat > $o] :
      ( ( member_nat @ K @ I2 )
     => ( ( sup_sup_o @ ( A @ K ) @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ A @ I2 ) ) )
        = ( complete_Sup_Sup_o @ ( image_nat_o2 @ A @ I2 ) ) ) ) ).

% SUP_absorb
thf(fact_720_SUP__lessD,axiom,
    ! [F: list_Sum_sum_a_c > $o,A: set_list_Sum_sum_a_c,Y3: $o,I4: list_Sum_sum_a_c] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) @ Y3 )
     => ( ( member7772695417316360142um_a_c @ I4 @ A )
       => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ).

% SUP_lessD
thf(fact_721_SUP__lessD,axiom,
    ! [F: $o > $o,A: set_o,Y3: $o,I4: $o] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ Y3 )
     => ( ( member_o @ I4 @ A )
       => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ).

% SUP_lessD
thf(fact_722_SUP__lessD,axiom,
    ! [F: nat > $o,A: set_nat,Y3: $o,I4: nat] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) @ Y3 )
     => ( ( member_nat @ I4 @ A )
       => ( ord_less_o @ ( F @ I4 ) @ Y3 ) ) ) ).

% SUP_lessD
thf(fact_723_SUP__insert,axiom,
    ! [F: $o > $o,A2: $o,A: set_o] :
      ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ ( insert_o @ A2 @ A ) ) )
      = ( sup_sup_o @ ( F @ A2 ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) ) ) ).

% SUP_insert
thf(fact_724_boolean__algebra_Odisj__one__right,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ X2 @ top_top_set_o )
      = top_top_set_o ) ).

% boolean_algebra.disj_one_right
thf(fact_725_boolean__algebra_Odisj__one__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_right
thf(fact_726_boolean__algebra_Odisj__one__left,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ X2 )
      = top_top_set_o ) ).

% boolean_algebra.disj_one_left
thf(fact_727_boolean__algebra_Odisj__one__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
      = top_top_set_nat ) ).

% boolean_algebra.disj_one_left
thf(fact_728_sup__top__right,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ X2 @ top_top_set_o )
      = top_top_set_o ) ).

% sup_top_right
thf(fact_729_sup__top__right,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% sup_top_right
thf(fact_730_sup__top__left,axiom,
    ! [X2: set_o] :
      ( ( sup_sup_set_o @ top_top_set_o @ X2 )
      = top_top_set_o ) ).

% sup_top_left
thf(fact_731_sup__top__left,axiom,
    ! [X2: set_nat] :
      ( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
      = top_top_set_nat ) ).

% sup_top_left
thf(fact_732_Id__onI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ A2 ) @ ( id_on_8540856796995947866um_a_c @ A ) ) ) ).

% Id_onI
thf(fact_733_Id__onI,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ A2 ) @ ( id_on_o @ A ) ) ) ).

% Id_onI
thf(fact_734_Id__onI,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( id_on_nat @ A ) ) ) ).

% Id_onI
thf(fact_735_Id__on__empty,axiom,
    ( ( id_on_o @ bot_bot_set_o )
    = bot_bo7073875226086086771od_o_o ) ).

% Id_on_empty
thf(fact_736_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_737_psubsetD,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,C: list_Sum_sum_a_c] :
      ( ( ord_le7452102681940104641um_a_c @ A @ B3 )
     => ( ( member7772695417316360142um_a_c @ C @ A )
       => ( member7772695417316360142um_a_c @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_738_psubsetD,axiom,
    ! [A: set_o,B3: set_o,C: $o] :
      ( ( ord_less_set_o @ A @ B3 )
     => ( ( member_o @ C @ A )
       => ( member_o @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_739_psubsetD,axiom,
    ! [A: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A @ B3 )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_740_less__set__def,axiom,
    ( ord_le7452102681940104641um_a_c
    = ( ^ [A4: set_list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( ord_le137615947321231876_a_c_o
          @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A4 )
          @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_741_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( ord_less_o_o
          @ ^ [X: $o] : ( member_o @ X @ A4 )
          @ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_742_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A4 )
          @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_743_Id__onE,axiom,
    ! [C: produc7878403009594063951um_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ C @ ( id_on_8540856796995947866um_a_c @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ( C
             != ( produc7367857235639141383um_a_c @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_744_Id__onE,axiom,
    ! [C: product_prod_o_o,A: set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( id_on_o @ A ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ( C
             != ( product_Pair_o_o @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_745_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( C
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_746_Id__on__eqI,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( A2 = B )
     => ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ ( id_on_8540856796995947866um_a_c @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_747_Id__on__eqI,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( A2 = B )
     => ( ( member_o @ A2 @ A )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( id_on_o @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_748_Id__on__eqI,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( A2 = B )
     => ( ( member_nat @ A2 @ A )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( id_on_nat @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_749_Id__on__iff,axiom,
    ! [X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X2 @ Y3 ) @ ( id_on_8540856796995947866um_a_c @ A ) )
      = ( ( X2 = Y3 )
        & ( member7772695417316360142um_a_c @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_750_Id__on__iff,axiom,
    ! [X2: $o,Y3: $o,A: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ ( id_on_o @ A ) )
      = ( ( X2 = Y3 )
        & ( member_o @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_751_Id__on__iff,axiom,
    ! [X2: nat,Y3: nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( id_on_nat @ A ) )
      = ( ( X2 = Y3 )
        & ( member_nat @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_752_sup__set__def,axiom,
    ( sup_su3984685238261546137um_a_c
    = ( ^ [A4: set_list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ( sup_su8851308358271025452_a_c_o
            @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A4 )
            @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_753_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X: $o] : ( member_o @ X @ A4 )
            @ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_754_sup__set__def,axiom,
    ( sup_su6350494296397389432um_a_c
    = ( ^ [A4: set_nat_Sum_sum_a_c,B5: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ( sup_su3894212136040546829_a_c_o
            @ ^ [X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ A4 )
            @ ^ [X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_755_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A4 )
            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_756_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A4 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_757_sup__Un__eq,axiom,
    ! [R3: set_list_Sum_sum_a_c,S: set_list_Sum_sum_a_c] :
      ( ( sup_su8851308358271025452_a_c_o
        @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ R3 )
        @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ S ) )
      = ( ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ ( sup_su3984685238261546137um_a_c @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_758_sup__Un__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X: $o] : ( member_o @ X @ R3 )
        @ ^ [X: $o] : ( member_o @ X @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( sup_sup_set_o @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_759_sup__Un__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_760_refl__on__domain,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c] :
      ( ( refl_o5556642055249044088um_a_c @ A @ R2 )
     => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ R2 )
       => ( ( member7772695417316360142um_a_c @ A2 @ A )
          & ( member7772695417316360142um_a_c @ B @ A ) ) ) ) ).

% refl_on_domain
thf(fact_761_refl__on__domain,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( ( refl_on_o @ A @ R2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
       => ( ( member_o @ A2 @ A )
          & ( member_o @ B @ A ) ) ) ) ).

% refl_on_domain
thf(fact_762_refl__on__domain,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( ( refl_on_nat @ A @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
       => ( ( member_nat @ A2 @ A )
          & ( member_nat @ B @ A ) ) ) ) ).

% refl_on_domain
thf(fact_763_Field__insert,axiom,
    ! [A2: $o,B: $o,R2: set_Product_prod_o_o] :
      ( ( field_o @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) )
      = ( sup_sup_set_o @ ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) @ ( field_o @ R2 ) ) ) ).

% Field_insert
thf(fact_764_SigmaI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ ( produc7224206333055223096um_a_c @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_765_SigmaI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: $o,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ A2 @ B ) @ ( produc7856722269759015511_a_c_o @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_766_SigmaI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: nat,B3: list_Sum_sum_a_c > set_nat] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ A2 @ B ) @ ( produc3632111297172148945_c_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_767_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: list_Sum_sum_a_c,B3: $o > set_list_Sum_sum_a_c] :
      ( ( member_o @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ ( produc7916830319660886151um_a_c @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_768_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: $o,B3: $o > set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_769_SigmaI,axiom,
    ! [A2: $o,A: set_o,B: nat,B3: $o > set_nat] :
      ( ( member_o @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_770_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: list_Sum_sum_a_c,B3: nat > set_list_Sum_sum_a_c] :
      ( ( member_nat @ A2 @ A )
     => ( ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) )
       => ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ A2 @ B ) @ ( produc6191802370236767313um_a_c @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_771_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: $o,B3: nat > set_o] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_o @ B @ ( B3 @ A2 ) )
       => ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_772_SigmaI,axiom,
    ! [A2: nat,A: set_nat,B: nat,B3: nat > set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( member_nat @ B @ ( B3 @ A2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B3 ) ) ) ) ).

% SigmaI
thf(fact_773_mem__Sigma__iff,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ ( produc7224206333055223096um_a_c @ A @ B3 ) )
      = ( ( member7772695417316360142um_a_c @ A2 @ A )
        & ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_774_mem__Sigma__iff,axiom,
    ! [A2: list_Sum_sum_a_c,B: $o,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ A2 @ B ) @ ( produc7856722269759015511_a_c_o @ A @ B3 ) )
      = ( ( member7772695417316360142um_a_c @ A2 @ A )
        & ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_775_mem__Sigma__iff,axiom,
    ! [A2: list_Sum_sum_a_c,B: nat,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ A2 @ B ) @ ( produc3632111297172148945_c_nat @ A @ B3 ) )
      = ( ( member7772695417316360142um_a_c @ A2 @ A )
        & ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_776_mem__Sigma__iff,axiom,
    ! [A2: $o,B: list_Sum_sum_a_c,A: set_o,B3: $o > set_list_Sum_sum_a_c] :
      ( ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ ( produc7916830319660886151um_a_c @ A @ B3 ) )
      = ( ( member_o @ A2 @ A )
        & ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_777_mem__Sigma__iff,axiom,
    ! [A2: $o,B: $o,A: set_o,B3: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B3 ) )
      = ( ( member_o @ A2 @ A )
        & ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_778_mem__Sigma__iff,axiom,
    ! [A2: $o,B: nat,A: set_o,B3: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B3 ) )
      = ( ( member_o @ A2 @ A )
        & ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_779_mem__Sigma__iff,axiom,
    ! [A2: nat,B: list_Sum_sum_a_c,A: set_nat,B3: nat > set_list_Sum_sum_a_c] :
      ( ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ A2 @ B ) @ ( produc6191802370236767313um_a_c @ A @ B3 ) )
      = ( ( member_nat @ A2 @ A )
        & ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_780_mem__Sigma__iff,axiom,
    ! [A2: nat,B: $o,A: set_nat,B3: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B3 ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_781_mem__Sigma__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B3 ) )
      = ( ( member_nat @ A2 @ A )
        & ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% mem_Sigma_iff
thf(fact_782_Field__empty,axiom,
    ( ( field_o @ bot_bo7073875226086086771od_o_o )
    = bot_bot_set_o ) ).

% Field_empty
thf(fact_783_Times__empty,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ( product_Sigma_o_o @ A
          @ ^ [Uu: $o] : B3 )
        = bot_bo7073875226086086771od_o_o )
      = ( ( A = bot_bot_set_o )
        | ( B3 = bot_bot_set_o ) ) ) ).

% Times_empty
thf(fact_784_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_o_o @ top_top_set_o
      @ ^ [Uu: $o] : top_top_set_o )
    = top_to7721136755696657239od_o_o ) ).

% UNIV_Times_UNIV
thf(fact_785_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_o_nat @ top_top_set_o
      @ ^ [Uu: $o] : top_top_set_nat )
    = top_to7022684507342537725_o_nat ) ).

% UNIV_Times_UNIV
thf(fact_786_UNIV__Times__UNIV,axiom,
    ( ( product_Sigma_nat_o @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_o )
    = top_to8070287629520841379_nat_o ) ).

% UNIV_Times_UNIV
thf(fact_787_UNIV__Times__UNIV,axiom,
    ( ( produc457027306803732586at_nat @ top_top_set_nat
      @ ^ [Uu: nat] : top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% UNIV_Times_UNIV
thf(fact_788_insert__Times__insert,axiom,
    ! [A2: $o,A: set_o,B: $o,B3: set_o] :
      ( ( product_Sigma_o_o @ ( insert_o @ A2 @ A )
        @ ^ [Uu: $o] : ( insert_o @ B @ B3 ) )
      = ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ A2 @ B )
        @ ( sup_su5769328420594410459od_o_o
          @ ( product_Sigma_o_o @ A
            @ ^ [Uu: $o] : ( insert_o @ B @ B3 ) )
          @ ( product_Sigma_o_o @ ( insert_o @ A2 @ A )
            @ ^ [Uu: $o] : B3 ) ) ) ) ).

% insert_Times_insert
thf(fact_789_FieldI1,axiom,
    ! [I4: list_Sum_sum_a_c,J: list_Sum_sum_a_c,R3: set_Pr8580000064110529967um_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ I4 @ J ) @ R3 )
     => ( member7772695417316360142um_a_c @ I4 @ ( field_4321413778972592298um_a_c @ R3 ) ) ) ).

% FieldI1
thf(fact_790_FieldI1,axiom,
    ! [I4: $o,J: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I4 @ J ) @ R3 )
     => ( member_o @ I4 @ ( field_o @ R3 ) ) ) ).

% FieldI1
thf(fact_791_FieldI1,axiom,
    ! [I4: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J ) @ R3 )
     => ( member_nat @ I4 @ ( field_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_792_FieldI2,axiom,
    ! [I4: list_Sum_sum_a_c,J: list_Sum_sum_a_c,R3: set_Pr8580000064110529967um_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ I4 @ J ) @ R3 )
     => ( member7772695417316360142um_a_c @ J @ ( field_4321413778972592298um_a_c @ R3 ) ) ) ).

% FieldI2
thf(fact_793_FieldI2,axiom,
    ! [I4: $o,J: $o,R3: set_Product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ I4 @ J ) @ R3 )
     => ( member_o @ J @ ( field_o @ R3 ) ) ) ).

% FieldI2
thf(fact_794_FieldI2,axiom,
    ! [I4: nat,J: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J ) @ R3 )
     => ( member_nat @ J @ ( field_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_795_times__eq__iff,axiom,
    ! [A: set_o,B3: set_o,C2: set_o,D2: set_o] :
      ( ( ( product_Sigma_o_o @ A
          @ ^ [Uu: $o] : B3 )
        = ( product_Sigma_o_o @ C2
          @ ^ [Uu: $o] : D2 ) )
      = ( ( ( A = C2 )
          & ( B3 = D2 ) )
        | ( ( ( A = bot_bot_set_o )
            | ( B3 = bot_bot_set_o ) )
          & ( ( C2 = bot_bot_set_o )
            | ( D2 = bot_bot_set_o ) ) ) ) ) ).

% times_eq_iff
thf(fact_796_SigmaE2,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ ( produc7224206333055223096um_a_c @ A @ B3 ) )
     => ~ ( ( member7772695417316360142um_a_c @ A2 @ A )
         => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_797_SigmaE2,axiom,
    ! [A2: list_Sum_sum_a_c,B: $o,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ A2 @ B ) @ ( produc7856722269759015511_a_c_o @ A @ B3 ) )
     => ~ ( ( member7772695417316360142um_a_c @ A2 @ A )
         => ~ ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_798_SigmaE2,axiom,
    ! [A2: list_Sum_sum_a_c,B: nat,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ A2 @ B ) @ ( produc3632111297172148945_c_nat @ A @ B3 ) )
     => ~ ( ( member7772695417316360142um_a_c @ A2 @ A )
         => ~ ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_799_SigmaE2,axiom,
    ! [A2: $o,B: list_Sum_sum_a_c,A: set_o,B3: $o > set_list_Sum_sum_a_c] :
      ( ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ ( produc7916830319660886151um_a_c @ A @ B3 ) )
     => ~ ( ( member_o @ A2 @ A )
         => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_800_SigmaE2,axiom,
    ! [A2: $o,B: $o,A: set_o,B3: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ ( product_Sigma_o_o @ A @ B3 ) )
     => ~ ( ( member_o @ A2 @ A )
         => ~ ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_801_SigmaE2,axiom,
    ! [A2: $o,B: nat,A: set_o,B3: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ ( product_Sigma_o_nat @ A @ B3 ) )
     => ~ ( ( member_o @ A2 @ A )
         => ~ ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_802_SigmaE2,axiom,
    ! [A2: nat,B: list_Sum_sum_a_c,A: set_nat,B3: nat > set_list_Sum_sum_a_c] :
      ( ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ A2 @ B ) @ ( produc6191802370236767313um_a_c @ A @ B3 ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member7772695417316360142um_a_c @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_803_SigmaE2,axiom,
    ! [A2: nat,B: $o,A: set_nat,B3: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ ( product_Sigma_nat_o @ A @ B3 ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_o @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_804_SigmaE2,axiom,
    ! [A2: nat,B: nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( produc457027306803732586at_nat @ A @ B3 ) )
     => ~ ( ( member_nat @ A2 @ A )
         => ~ ( member_nat @ B @ ( B3 @ A2 ) ) ) ) ).

% SigmaE2
thf(fact_805_SigmaE,axiom,
    ! [C: produc7878403009594063951um_a_c,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ C @ ( produc7224206333055223096um_a_c @ A @ B3 ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ! [Y4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc7367857235639141383um_a_c @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_806_SigmaE,axiom,
    ! [C: produc5829762120286046742_a_c_o,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_o] :
      ( ( member7213031410929038765_a_c_o @ C @ ( produc7856722269759015511_a_c_o @ A @ B3 ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc8438544629413731400_a_c_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_807_SigmaE,axiom,
    ! [C: produc1537306021406423912_c_nat,A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( member6914143599624037265_c_nat @ C @ ( produc3632111297172148945_c_nat @ A @ B3 ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc8518022002054455072_c_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_808_SigmaE,axiom,
    ! [C: produc6487107448139085310um_a_c,A: set_o,B3: $o > set_list_Sum_sum_a_c] :
      ( ( member7870376738782077333um_a_c @ C @ ( produc7916830319660886151um_a_c @ A @ B3 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ! [Y4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc8498652679315602040um_a_c @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_809_SigmaE,axiom,
    ! [C: product_prod_o_o,A: set_o,B3: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ C @ ( product_Sigma_o_o @ A @ B3 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( product_Pair_o_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_810_SigmaE,axiom,
    ! [C: product_prod_o_nat,A: set_o,B3: $o > set_nat] :
      ( ( member2802428098988154798_o_nat @ C @ ( product_Sigma_o_nat @ A @ B3 ) )
     => ~ ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( product_Pair_o_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_811_SigmaE,axiom,
    ! [C: produc5100097535369517928um_a_c,A: set_nat,B3: nat > set_list_Sum_sum_a_c] :
      ( ( member1253563076732355473um_a_c @ C @ ( produc6191802370236767313um_a_c @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( produc1854341038264297632um_a_c @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_812_SigmaE,axiom,
    ! [C: product_prod_nat_o,A: set_nat,B3: nat > set_o] :
      ( ( member6310962623043647828_nat_o @ C @ ( product_Sigma_nat_o @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y4: $o] :
                ( ( member_o @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_o @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_813_SigmaE,axiom,
    ! [C: product_prod_nat_nat,A: set_nat,B3: nat > set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( produc457027306803732586at_nat @ A @ B3 ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ! [Y4: nat] :
                ( ( member_nat @ Y4 @ ( B3 @ X3 ) )
               => ( C
                 != ( product_Pair_nat_nat @ X3 @ Y4 ) ) ) ) ) ).

% SigmaE
thf(fact_814_linear__order__on__singleton,axiom,
    ! [X2: $o] : ( order_6238756238976269133r_on_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( insert6201435330877294327od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ bot_bo7073875226086086771od_o_o ) ) ).

% linear_order_on_singleton
thf(fact_815_Rep__unit,axiom,
    ! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_816_Abs__unit__cases,axiom,
    ! [X2: product_unit] :
      ~ ! [Y4: $o] :
          ( ( X2
            = ( product_Abs_unit @ Y4 ) )
         => ~ ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_817_Rep__unit__inverse,axiom,
    ! [X2: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X2 ) )
      = X2 ) ).

% Rep_unit_inverse
thf(fact_818_Rep__unit__inject,axiom,
    ! [X2: product_unit,Y3: product_unit] :
      ( ( ( product_Rep_unit @ X2 )
        = ( product_Rep_unit @ Y3 ) )
      = ( X2 = Y3 ) ) ).

% Rep_unit_inject
thf(fact_819_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_820_Abs__unit__inverse,axiom,
    ! [Y3: $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
        = Y3 ) ) ).

% Abs_unit_inverse
thf(fact_821_Rep__unit__induct,axiom,
    ! [Y3: $o,P: $o > $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X3: product_unit] : ( P @ ( product_Rep_unit @ X3 ) )
       => ( P @ Y3 ) ) ) ).

% Rep_unit_induct
thf(fact_822_Abs__unit__inject,axiom,
    ! [X2: $o,Y3: $o] :
      ( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X2 )
            = ( product_Abs_unit @ Y3 ) )
          = ( X2 = Y3 ) ) ) ) ).

% Abs_unit_inject
thf(fact_823_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X2: product_unit] :
      ( ! [Y4: $o] :
          ( ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y4 ) ) )
     => ( P @ X2 ) ) ).

% Abs_unit_induct
thf(fact_824_Rep__unit__cases,axiom,
    ! [Y3: $o] :
      ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X3: product_unit] :
            ( Y3
            = ( ~ ( product_Rep_unit @ X3 ) ) ) ) ).

% Rep_unit_cases
thf(fact_825_aboveS__def,axiom,
    ( order_727316718645315563um_a_c
    = ( ^ [R4: set_Pr1218573636708433773um_a_c,A5: nat > sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [B4: nat > sum_sum_a_c] :
              ( ( B4 != A5 )
              & ( member1490001167679721142um_a_c @ ( produc7129626048164379717um_a_c @ A5 @ B4 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_826_aboveS__def,axiom,
    ( order_aboveS_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,A5: nat] :
          ( collect_nat
          @ ^ [B4: nat] :
              ( ( B4 != A5 )
              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_827_aboveS__def,axiom,
    ( order_4940263126897443436at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,A5: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] :
              ( ( B4 != A5 )
              & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ R4 ) ) ) ) ) ).

% aboveS_def
thf(fact_828_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A2: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
      = ( F @ A2 @ B ) ) ).

% case_prod_conv
thf(fact_829_pair__imageI,axiom,
    ! [A2: nat,B: nat,A: set_Pr1261947904930325089at_nat,F: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ A )
     => ( member_o @ ( F @ A2 @ B ) @ ( image_3693632289388996572_nat_o @ ( produc6081775807080527818_nat_o @ F ) @ A ) ) ) ).

% pair_imageI
thf(fact_830_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y4: nat] :
          ( ( F @ X3 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y4 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_831_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_832_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( Q @ ( P @ X3 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_833_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_834_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > $o,G: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y4 )
            = Q2 )
         => ( ( F @ X3 @ Y4 )
            = ( G @ X3 @ Y4 ) ) )
     => ( ( P4 = Q2 )
       => ( ( produc6081775807080527818_nat_o @ F @ P4 )
          = ( produc6081775807080527818_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_835_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X24: nat] : ( H @ ( F @ X12 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_836_internal__case__prod__def,axiom,
    produc9139807813499881076_nat_o = produc6081775807080527818_nat_o ).

% internal_case_prod_def
thf(fact_837_case__prodI2,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B2: nat] :
          ( ( P4
            = ( product_Pair_nat_nat @ A3 @ B2 ) )
         => ( C @ A3 @ B2 ) )
     => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_838_case__prodI,axiom,
    ! [F: nat > nat > $o,A2: nat,B: nat] :
      ( ( F @ A2 @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A2 @ B ) ) ) ).

% case_prodI
thf(fact_839_Collect__case__prod,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( collec3026100613349326840um_a_c
        @ ( produc2106327337280714430_a_c_o
          @ ^ [A5: nat > sum_sum_a_c,B4: nat > sum_sum_a_c] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc5938743151663733622um_a_c @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [Uu: nat > sum_sum_a_c] : ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_case_prod
thf(fact_840_Collect__case__prod,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: nat > $o] :
      ( ( collec8523534512140322354_c_nat
        @ ( produc8035988671407038212_nat_o
          @ ^ [A5: nat > sum_sum_a_c,B4: nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc7547139578838631984_c_nat @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [Uu: nat > sum_sum_a_c] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_841_Collect__case__prod,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec2968159817709868213at_nat
        @ ( produc1918003690408168217_nat_o
          @ ^ [A5: nat > sum_sum_a_c,B4: product_prod_nat_nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc6510659468026324641at_nat @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [Uu: nat > sum_sum_a_c] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_842_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( collec2681008756847244722um_a_c
        @ ( produc1315642468178905732_a_c_o
          @ ^ [A5: nat,B4: nat > sum_sum_a_c] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc4613874750863342256um_a_c @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_case_prod
thf(fact_843_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [A5: nat,B4: product_prod_nat_nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_844_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( collec7765197881830713499um_a_c
        @ ( produc5409643230324462187_a_c_o
          @ ^ [A5: product_prod_nat_nat,B4: nat > sum_sum_a_c] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc508052045600165839um_a_c @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_case_prod
thf(fact_845_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [A5: product_prod_nat_nat,B4: nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_846_Collect__case__prod,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [Uu: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_847_Collect__case__prod,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [A5: nat,B4: nat] :
              ( ( P @ A5 )
              & ( Q @ B4 ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [Uu: nat] : ( collect_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_848_Collect__const__case__prod,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A5: nat,B4: nat] : P ) )
          = top_to4669805908274784177at_nat ) )
      & ( ~ P
       => ( ( collec3392354462482085612at_nat
            @ ( produc6081775807080527818_nat_o
              @ ^ [A5: nat,B4: nat] : P ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_const_case_prod
thf(fact_849_finite__option__UNIV,axiom,
    ( ( finite4093902646404507527tion_o @ top_top_set_option_o )
    = ( finite_finite_o @ top_top_set_o ) ) ).

% finite_option_UNIV
thf(fact_850_finite__option__UNIV,axiom,
    ( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_option_UNIV
thf(fact_851_case__prodE,axiom,
    ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P4 )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X3 @ Y4 ) )
           => ~ ( C @ X3 @ Y4 ) ) ) ).

% case_prodE
thf(fact_852_case__prodD,axiom,
    ! [F: nat > nat > $o,A2: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A2 @ B ) )
     => ( F @ A2 @ B ) ) ).

% case_prodD
thf(fact_853_same__fst__def,axiom,
    ( same_fst_nat_nat
    = ( ^ [P3: nat > $o,R: nat > set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [X10: nat,Y7: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [X: nat,Y: nat] :
                      ( ( X10 = X )
                      & ( P3 @ X )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y ) @ ( R @ X ) ) ) ) ) ) ) ) ) ).

% same_fst_def
thf(fact_854_Collect__case__prod__Sigma,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > ( nat > sum_sum_a_c ) > $o] :
      ( ( collec3026100613349326840um_a_c
        @ ( produc2106327337280714430_a_c_o
          @ ^ [X: nat > sum_sum_a_c,Y: nat > sum_sum_a_c] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc5938743151663733622um_a_c @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [X: nat > sum_sum_a_c] : ( collec5227572641185395563um_a_c @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_855_Collect__case__prod__Sigma,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > nat > $o] :
      ( ( collec8523534512140322354_c_nat
        @ ( produc8035988671407038212_nat_o
          @ ^ [X: nat > sum_sum_a_c,Y: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc7547139578838631984_c_nat @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [X: nat > sum_sum_a_c] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_856_Collect__case__prod__Sigma,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > product_prod_nat_nat > $o] :
      ( ( collec2968159817709868213at_nat
        @ ( produc1918003690408168217_nat_o
          @ ^ [X: nat > sum_sum_a_c,Y: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc6510659468026324641at_nat @ ( collec5227572641185395563um_a_c @ P )
        @ ^ [X: nat > sum_sum_a_c] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_857_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > ( nat > sum_sum_a_c ) > $o] :
      ( ( collec2681008756847244722um_a_c
        @ ( produc1315642468178905732_a_c_o
          @ ^ [X: nat,Y: nat > sum_sum_a_c] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc4613874750863342256um_a_c @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collec5227572641185395563um_a_c @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_858_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > product_prod_nat_nat > $o] :
      ( ( collec5903703980526211963at_nat
        @ ( produc5864757623865647827_nat_o
          @ ^ [X: nat,Y: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc1809337555817847783at_nat @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_859_Collect__case__prod__Sigma,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > ( nat > sum_sum_a_c ) > $o] :
      ( ( collec7765197881830713499um_a_c
        @ ( produc5409643230324462187_a_c_o
          @ ^ [X: product_prod_nat_nat,Y: nat > sum_sum_a_c] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc508052045600165839um_a_c @ ( collec3392354462482085612at_nat @ P )
        @ ^ [X: product_prod_nat_nat] : ( collec5227572641185395563um_a_c @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_860_Collect__case__prod__Sigma,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > nat > $o] :
      ( ( collec7029190964493513045at_nat
        @ ( produc8758269395934548017_nat_o
          @ ^ [X: product_prod_nat_nat,Y: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc7672662199629908489at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [X: product_prod_nat_nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_861_Collect__case__prod__Sigma,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc2761391749766926216at_nat @ ( collec3392354462482085612at_nat @ P )
        @ ^ [X: product_prod_nat_nat] : ( collec3392354462482085612at_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_862_Collect__case__prod__Sigma,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( P @ X )
              & ( Q @ X @ Y ) ) ) )
      = ( produc457027306803732586at_nat @ ( collect_nat @ P )
        @ ^ [X: nat] : ( collect_nat @ ( Q @ X ) ) ) ) ).

% Collect_case_prod_Sigma
thf(fact_863_finite__range__Some,axiom,
    ( ( finite1377308360061600845um_a_c @ ( image_1096817385502557859um_a_c @ some_n5886337604800580545um_a_c @ top_to5366858714144749388um_a_c ) )
    = ( finite2642776695614460285um_a_c @ top_to5366858714144749388um_a_c ) ) ).

% finite_range_Some
thf(fact_864_finite__range__Some,axiom,
    ( ( finite4093902646404507527tion_o @ ( image_o_option_o @ some_o @ top_top_set_o ) )
    = ( finite_finite_o @ top_top_set_o ) ) ).

% finite_range_Some
thf(fact_865_finite__range__Some,axiom,
    ( ( finite5523153139673422903on_nat @ ( image_nat_option_nat @ some_nat @ top_top_set_nat ) )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% finite_range_Some
thf(fact_866_Id__on__def_H,axiom,
    ! [A: ( nat > sum_sum_a_c ) > $o] :
      ( ( id_on_1666287213725549241um_a_c @ ( collec5227572641185395563um_a_c @ A ) )
      = ( collec3026100613349326840um_a_c
        @ ( produc2106327337280714430_a_c_o
          @ ^ [X: nat > sum_sum_a_c,Y: nat > sum_sum_a_c] :
              ( ( X = Y )
              & ( A @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_867_Id__on__def_H,axiom,
    ! [A: product_prod_nat_nat > $o] :
      ( ( id_on_2554058798563519774at_nat @ ( collec3392354462482085612at_nat @ A ) )
      = ( collec7088162979684241874at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
              ( ( X = Y )
              & ( A @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_868_Id__on__def_H,axiom,
    ! [A: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( X = Y )
              & ( A @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_869_finite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( finite_finite_nat @ ( field_nat @ R2 ) ) ) ).

% finite_Field
thf(fact_870_finite__fv__fo__terms__set,axiom,
    ! [Ts: list_fo_term_a] : ( finite_finite_nat @ ( fv_fo_terms_set_a @ Ts ) ) ).

% finite_fv_fo_terms_set
thf(fact_871_image__paired__Times,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c] :
      ( ( image_744226690754329991um_a_c
        @ ( produc4480726505017772917um_a_c
          @ ^ [X: nat > sum_sum_a_c,Y: nat > sum_sum_a_c] : ( produc7367857235639141383um_a_c @ ( F @ X ) @ ( G @ Y ) ) )
        @ ( produc5938743151663733622um_a_c @ A
          @ ^ [Uu: nat > sum_sum_a_c] : B3 ) )
      = ( produc7224206333055223096um_a_c @ ( image_7318554134589591124um_a_c @ F @ A )
        @ ^ [Uu: list_Sum_sum_a_c] : ( image_7318554134589591124um_a_c @ G @ B3 ) ) ) ).

% image_paired_Times
thf(fact_872_Sup__SUP__eq2,axiom,
    ( comple3592611370556534995_nat_o
    = ( ^ [S5: set_nat_nat_o,X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( comple5685304695842803022at_nat @ ( image_7124889717316225246at_nat @ collec3392354462482085612at_nat @ ( image_7429393840292777309_nat_o @ produc6081775807080527818_nat_o @ S5 ) ) ) ) ) ) ).

% Sup_SUP_eq2
thf(fact_873_finite__UN__I,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( finite3830171671145977038um_a_c @ A )
     => ( ! [A3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_3797473998943575876et_nat @ B3 @ A ) ) ) ) ) ).

% finite_UN_I
thf(fact_874_finite__UN__I,axiom,
    ! [A: set_o,B3: $o > set_nat] :
      ( ( finite_finite_o @ A )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B3 @ A ) ) ) ) ) ).

% finite_UN_I
thf(fact_875_finite__UN__I,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A ) ) ) ) ) ).

% finite_UN_I
thf(fact_876_finite__SigmaI,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( finite3830171671145977038um_a_c @ A )
     => ( ! [A3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite6100139010601684625_c_nat @ ( produc3632111297172148945_c_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_877_finite__SigmaI,axiom,
    ! [A: set_o,B3: $o > set_nat] :
      ( ( finite_finite_o @ A )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite1846473907987936430_o_nat @ ( product_Sigma_o_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_878_finite__SigmaI,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI
thf(fact_879_finite__Collect__conjI,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ P ) )
        | ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ Q ) ) )
     => ( finite541027974130308653um_a_c
        @ ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_880_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_881_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_882_finite__Collect__disjI,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( finite541027974130308653um_a_c
        @ ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ P ) )
        & ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_883_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_884_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_885_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A5: nat,B4: nat] :
            ( P
            & ( Q @ A5 @ B4 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_886_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_887_finite__Plus__UNIV__iff,axiom,
    ( ( finite6699802884135759036um_o_o @ top_to1686961084667892491um_o_o )
    = ( ( finite_finite_o @ top_top_set_o )
      & ( finite_finite_o @ top_top_set_o ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_888_finite__Plus__UNIV__iff,axiom,
    ( ( finite5809725721784815170_o_nat @ top_to6072511757011528009_o_nat )
    = ( ( finite_finite_o @ top_top_set_o )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_889_finite__Plus__UNIV__iff,axiom,
    ( ( finite94888208985532392_nat_o @ top_to7120114879189831663_nat_o )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_o @ top_top_set_o ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_890_finite__Plus__UNIV__iff,axiom,
    ( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Plus_UNIV_iff
thf(fact_891_finite__imageI,axiom,
    ! [F4: set_nat_Sum_sum_a_c,H: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( finite541027974130308653um_a_c @ F4 )
     => ( finite3830171671145977038um_a_c @ ( image_7318554134589591124um_a_c @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_892_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_nat @ ( image_nat_nat2 @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_893_finite__Collect__not,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o] :
      ( ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ P ) )
     => ( ( finite541027974130308653um_a_c
          @ ( collec5227572641185395563um_a_c
            @ ^ [X: nat > sum_sum_a_c] :
                ~ ( P @ X ) ) )
        = ( finite541027974130308653um_a_c @ top_to3439567294830504444um_a_c ) ) ) ).

% finite_Collect_not
thf(fact_894_finite__Collect__not,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ( ( finite6177210948735845034at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ~ ( P @ X ) ) )
        = ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Collect_not
thf(fact_895_finite__Collect__not,axiom,
    ! [P: $o > $o] :
      ( ( finite_finite_o @ ( collect_o @ P ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [X: $o] :
                ~ ( P @ X ) ) )
        = ( finite_finite_o @ top_top_set_o ) ) ) ).

% finite_Collect_not
thf(fact_896_finite__Collect__not,axiom,
    ! [P: nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_Collect_not
thf(fact_897_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_898_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_899_not__finite__existsD,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o] :
      ( ~ ( finite541027974130308653um_a_c @ ( collec5227572641185395563um_a_c @ P ) )
     => ? [X_1: nat > sum_sum_a_c] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_900_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_901_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_902_pigeonhole__infinite__rel,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_nat,R3: list_Sum_sum_a_c > nat > $o] :
      ( ~ ( finite3830171671145977038um_a_c @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite3830171671145977038um_a_c
                  @ ( collec8219452656984879116um_a_c
                    @ ^ [A5: list_Sum_sum_a_c] :
                        ( ( member7772695417316360142um_a_c @ A5 @ A )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_903_pigeonhole__infinite__rel,axiom,
    ! [A: set_o,B3: set_nat,R3: $o > nat > $o] :
      ( ~ ( finite_finite_o @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: $o] :
              ( ( member_o @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_o
                  @ ( collect_o
                    @ ^ [A5: $o] :
                        ( ( member_o @ A5 @ A )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_904_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_nat,R3: ( nat > sum_sum_a_c ) > nat > $o] :
      ( ~ ( finite541027974130308653um_a_c @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite541027974130308653um_a_c
                  @ ( collec5227572641185395563um_a_c
                    @ ^ [A5: nat > sum_sum_a_c] :
                        ( ( member4884986500679352621um_a_c @ A5 @ A )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_905_pigeonhole__infinite__rel,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: set_nat,R3: product_prod_nat_nat > nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A5: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A5 @ A )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_906_pigeonhole__infinite__rel,axiom,
    ! [A: set_nat,B3: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B3 )
                  & ( R3 @ X3 @ Xa2 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A5: nat] :
                        ( ( member_nat @ A5 @ A )
                        & ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_907_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A4: set_nat] :
        ? [N4: nat,F2: nat > nat] :
          ( A4
          = ( image_nat_nat2 @ F2
            @ ( collect_nat
              @ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_908_nat__seg__image__imp__finite,axiom,
    ! [A: set_nat,F: nat > nat,N2: nat] :
      ( ( A
        = ( image_nat_nat2 @ F
          @ ( collect_nat
            @ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
     => ( finite_finite_nat @ A ) ) ).

% nat_seg_image_imp_finite
thf(fact_909_infinite__UNIV__char__0,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_char_0
thf(fact_910_ex__new__if__finite,axiom,
    ! [A: set_list_Sum_sum_a_c] :
      ( ~ ( finite3830171671145977038um_a_c @ top_to8424319039679220765um_a_c )
     => ( ( finite3830171671145977038um_a_c @ A )
       => ? [A3: list_Sum_sum_a_c] :
            ~ ( member7772695417316360142um_a_c @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_911_ex__new__if__finite,axiom,
    ! [A: set_o] :
      ( ~ ( finite_finite_o @ top_top_set_o )
     => ( ( finite_finite_o @ A )
       => ? [A3: $o] :
            ~ ( member_o @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_912_ex__new__if__finite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ top_top_set_nat )
     => ( ( finite_finite_nat @ A )
       => ? [A3: nat] :
            ~ ( member_nat @ A3 @ A ) ) ) ).

% ex_new_if_finite
thf(fact_913_finite__class_Ofinite__UNIV,axiom,
    finite_finite_o @ top_top_set_o ).

% finite_class.finite_UNIV
thf(fact_914_pigeonhole__infinite,axiom,
    ! [A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ~ ( finite541027974130308653um_a_c @ A )
     => ( ( finite3830171671145977038um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) )
       => ? [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ A )
            & ~ ( finite541027974130308653um_a_c
                @ ( collec5227572641185395563um_a_c
                  @ ^ [A5: nat > sum_sum_a_c] :
                      ( ( member4884986500679352621um_a_c @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_915_pigeonhole__infinite,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > nat] :
      ( ~ ( finite3830171671145977038um_a_c @ A )
     => ( ( finite_finite_nat @ ( image_7712311756382514062_c_nat @ F @ A ) )
       => ? [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
            & ~ ( finite3830171671145977038um_a_c
                @ ( collec8219452656984879116um_a_c
                  @ ^ [A5: list_Sum_sum_a_c] :
                      ( ( member7772695417316360142um_a_c @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_916_pigeonhole__infinite,axiom,
    ! [A: set_o,F: $o > nat] :
      ( ~ ( finite_finite_o @ A )
     => ( ( finite_finite_nat @ ( image_o_nat2 @ F @ A ) )
       => ? [X3: $o] :
            ( ( member_o @ X3 @ A )
            & ~ ( finite_finite_o
                @ ( collect_o
                  @ ^ [A5: $o] :
                      ( ( member_o @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_917_pigeonhole__infinite,axiom,
    ! [A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > nat] :
      ( ~ ( finite541027974130308653um_a_c @ A )
     => ( ( finite_finite_nat @ ( image_930363360967426541_c_nat @ F @ A ) )
       => ? [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ A )
            & ~ ( finite541027974130308653um_a_c
                @ ( collec5227572641185395563um_a_c
                  @ ^ [A5: nat > sum_sum_a_c] :
                      ( ( member4884986500679352621um_a_c @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_918_pigeonhole__infinite,axiom,
    ! [A: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ~ ( finite6177210948735845034at_nat @ A )
     => ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F @ A ) )
       => ? [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A5: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_919_pigeonhole__infinite,axiom,
    ! [A: set_nat,F: nat > nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ ( image_nat_nat2 @ F @ A ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A5: nat] :
                      ( ( member_nat @ A5 @ A )
                      & ( ( F @ A5 )
                        = ( F @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_920_finite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( finite6177210948735845034at_nat
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B3 ) ) ) ) ).

% finite_cartesian_product
thf(fact_921_finite__prod,axiom,
    ( ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o )
    = ( ( finite_finite_o @ top_top_set_o )
      & ( finite_finite_o @ top_top_set_o ) ) ) ).

% finite_prod
thf(fact_922_finite__prod,axiom,
    ( ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat )
    = ( ( finite_finite_o @ top_top_set_o )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_923_finite__prod,axiom,
    ( ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_o @ top_top_set_o ) ) ) ).

% finite_prod
thf(fact_924_finite__prod,axiom,
    ( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
    = ( ( finite_finite_nat @ top_top_set_nat )
      & ( finite_finite_nat @ top_top_set_nat ) ) ) ).

% finite_prod
thf(fact_925_finite__Prod__UNIV,axiom,
    ( ( finite_finite_o @ top_top_set_o )
   => ( ( finite_finite_o @ top_top_set_o )
     => ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o ) ) ) ).

% finite_Prod_UNIV
thf(fact_926_finite__Prod__UNIV,axiom,
    ( ( finite_finite_o @ top_top_set_o )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_927_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_o @ top_top_set_o )
     => ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o ) ) ) ).

% finite_Prod_UNIV
thf(fact_928_finite__Prod__UNIV,axiom,
    ( ( finite_finite_nat @ top_top_set_nat )
   => ( ( finite_finite_nat @ top_top_set_nat )
     => ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).

% finite_Prod_UNIV
thf(fact_929_Finite__Set_Ofinite__set,axiom,
    ( ( finite_finite_set_o @ top_top_set_set_o )
    = ( finite_finite_o @ top_top_set_o ) ) ).

% Finite_Set.finite_set
thf(fact_930_Finite__Set_Ofinite__set,axiom,
    ( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
    = ( finite_finite_nat @ top_top_set_nat ) ) ).

% Finite_Set.finite_set
thf(fact_931_finite__range__imageI,axiom,
    ! [G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c] :
      ( ( finite3830171671145977038um_a_c @ ( image_7318554134589591124um_a_c @ G @ top_to3439567294830504444um_a_c ) )
     => ( finite3830171671145977038um_a_c
        @ ( image_7318554134589591124um_a_c
          @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
          @ top_to3439567294830504444um_a_c ) ) ) ).

% finite_range_imageI
thf(fact_932_finite__range__imageI,axiom,
    ! [G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,F: list_Sum_sum_a_c > nat] :
      ( ( finite3830171671145977038um_a_c @ ( image_7318554134589591124um_a_c @ G @ top_to3439567294830504444um_a_c ) )
     => ( finite_finite_nat
        @ ( image_930363360967426541_c_nat
          @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
          @ top_to3439567294830504444um_a_c ) ) ) ).

% finite_range_imageI
thf(fact_933_finite__range__imageI,axiom,
    ! [G: ( nat > sum_sum_a_c ) > nat,F: nat > list_Sum_sum_a_c] :
      ( ( finite_finite_nat @ ( image_930363360967426541_c_nat @ G @ top_to3439567294830504444um_a_c ) )
     => ( finite3830171671145977038um_a_c
        @ ( image_7318554134589591124um_a_c
          @ ^ [X: nat > sum_sum_a_c] : ( F @ ( G @ X ) )
          @ top_to3439567294830504444um_a_c ) ) ) ).

% finite_range_imageI
thf(fact_934_finite__range__imageI,axiom,
    ! [G: $o > nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( image_o_nat2 @ G @ top_top_set_o ) )
     => ( finite_finite_nat
        @ ( image_o_nat2
          @ ^ [X: $o] : ( F @ ( G @ X ) )
          @ top_top_set_o ) ) ) ).

% finite_range_imageI
thf(fact_935_finite__range__imageI,axiom,
    ! [G: nat > nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( image_nat_nat2 @ G @ top_top_set_nat ) )
     => ( finite_finite_nat
        @ ( image_nat_nat2
          @ ^ [X: nat] : ( F @ ( G @ X ) )
          @ top_top_set_nat ) ) ) ).

% finite_range_imageI
thf(fact_936_finite__SigmaI2,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_nat] :
      ( ( finite3830171671145977038um_a_c
        @ ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite6100139010601684625_c_nat @ ( produc3632111297172148945_c_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_937_finite__SigmaI2,axiom,
    ! [A: set_o,B3: $o > set_nat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite1846473907987936430_o_nat @ ( product_Sigma_o_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_938_finite__SigmaI2,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: ( nat > sum_sum_a_c ) > set_nat] :
      ( ( finite541027974130308653um_a_c
        @ ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite161735112584112368_c_nat @ ( produc7547139578838631984_c_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_939_finite__SigmaI2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite8785817246233100311at_nat @ ( produc7672662199629908489at_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_940_finite__SigmaI2,axiom,
    ! [A: set_nat,B3: nat > set_nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_nat ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_nat @ ( B3 @ A3 ) ) )
       => ( finite6177210948735845034at_nat @ ( produc457027306803732586at_nat @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_941_finite__SigmaI2,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: list_Sum_sum_a_c > set_o] :
      ( ( finite3830171671145977038um_a_c
        @ ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_o ) ) ) )
     => ( ! [A3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ A3 @ A )
           => ( finite_finite_o @ ( B3 @ A3 ) ) )
       => ( finite3557940929864092333_a_c_o @ ( produc7856722269759015511_a_c_o @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_942_finite__SigmaI2,axiom,
    ! [A: set_o,B3: $o > set_o] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_o ) ) ) )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A )
           => ( finite_finite_o @ ( B3 @ A3 ) ) )
       => ( finite6120865539452801872od_o_o @ ( product_Sigma_o_o @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_943_finite__SigmaI2,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: ( nat > sum_sum_a_c ) > set_o] :
      ( ( finite541027974130308653um_a_c
        @ ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_o ) ) ) )
     => ( ! [A3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ A3 @ A )
           => ( finite_finite_o @ ( B3 @ A3 ) ) )
       => ( finite8387064983604144782_a_c_o @ ( produc5959874039126855992_a_c_o @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_944_finite__SigmaI2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat > set_o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_o ) ) ) )
     => ( ! [A3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A3 @ A )
           => ( finite_finite_o @ ( B3 @ A3 ) ) )
       => ( finite2699364969320418215_nat_o @ ( produc203973861852599583_nat_o @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_945_finite__SigmaI2,axiom,
    ! [A: set_nat,B3: nat > set_o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( ( B3 @ X )
               != bot_bot_set_o ) ) ) )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A )
           => ( finite_finite_o @ ( B3 @ A3 ) ) )
       => ( finite5355008432043429460_nat_o @ ( product_Sigma_nat_o @ A @ B3 ) ) ) ) ).

% finite_SigmaI2
thf(fact_946_finite__cartesian__productD1,axiom,
    ! [A: set_nat,B3: set_o] :
      ( ( finite5355008432043429460_nat_o
        @ ( product_Sigma_nat_o @ A
          @ ^ [Uu: nat] : B3 ) )
     => ( ( B3 != bot_bot_set_o )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_cartesian_productD1
thf(fact_947_finite__cartesian__productD2,axiom,
    ! [A: set_o,B3: set_nat] :
      ( ( finite1846473907987936430_o_nat
        @ ( product_Sigma_o_nat @ A
          @ ^ [Uu: $o] : B3 ) )
     => ( ( A != bot_bot_set_o )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_cartesian_productD2
thf(fact_948_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite6177210948735845034at_nat
        @ ( produc457027306803732586at_nat @ A
          @ ^ [Uu: nat] : B3 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B3 = bot_bot_set_nat )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_nat @ B3 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_949_finite__cartesian__product__iff,axiom,
    ! [A: set_nat,B3: set_o] :
      ( ( finite5355008432043429460_nat_o
        @ ( product_Sigma_nat_o @ A
          @ ^ [Uu: nat] : B3 ) )
      = ( ( A = bot_bot_set_nat )
        | ( B3 = bot_bot_set_o )
        | ( ( finite_finite_nat @ A )
          & ( finite_finite_o @ B3 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_950_finite__cartesian__product__iff,axiom,
    ! [A: set_o,B3: set_nat] :
      ( ( finite1846473907987936430_o_nat
        @ ( product_Sigma_o_nat @ A
          @ ^ [Uu: $o] : B3 ) )
      = ( ( A = bot_bot_set_o )
        | ( B3 = bot_bot_set_nat )
        | ( ( finite_finite_o @ A )
          & ( finite_finite_nat @ B3 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_951_finite__cartesian__product__iff,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( finite6120865539452801872od_o_o
        @ ( product_Sigma_o_o @ A
          @ ^ [Uu: $o] : B3 ) )
      = ( ( A = bot_bot_set_o )
        | ( B3 = bot_bot_set_o )
        | ( ( finite_finite_o @ A )
          & ( finite_finite_o @ B3 ) ) ) ) ).

% finite_cartesian_product_iff
thf(fact_952_infinite__cartesian__product,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ~ ( finite_finite_nat @ B3 )
       => ~ ( finite6177210948735845034at_nat
            @ ( produc457027306803732586at_nat @ A
              @ ^ [Uu: nat] : B3 ) ) ) ) ).

% infinite_cartesian_product
thf(fact_953_sup__SUP__fold__sup,axiom,
    ! [A: set_nat,B3: $o,F: nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( sup_sup_o @ B3 @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) )
        = ( finite_fold_nat_o @ ( comp_o_o_o_nat @ sup_sup_o @ F ) @ B3 @ A ) ) ) ).

% sup_SUP_fold_sup
thf(fact_954_image__fold__insert,axiom,
    ! [A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( finite541027974130308653um_a_c @ A )
     => ( ( image_7318554134589591124um_a_c @ F @ A )
        = ( finite1019145557901317965um_a_c
          @ ^ [K2: nat > sum_sum_a_c] : ( insert7520310754158225127um_a_c @ ( F @ K2 ) )
          @ bot_bo3453284597459734017um_a_c
          @ A ) ) ) ).

% image_fold_insert
thf(fact_955_image__fold__insert,axiom,
    ! [A: set_nat,F: nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( image_nat_o2 @ F @ A )
        = ( finite3217087857726763998_set_o
          @ ^ [K2: nat] : ( insert_o @ ( F @ K2 ) )
          @ bot_bot_set_o
          @ A ) ) ) ).

% image_fold_insert
thf(fact_956_SUP__fold__sup,axiom,
    ! [A: set_nat,F: nat > set_o] :
      ( ( finite_finite_nat @ A )
     => ( ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) )
        = ( finite3217087857726763998_set_o @ ( comp_s4424444925706540684_o_nat @ sup_sup_set_o @ F ) @ bot_bot_set_o @ A ) ) ) ).

% SUP_fold_sup
thf(fact_957_SUP__fold__sup,axiom,
    ! [A: set_nat,F: nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) )
        = ( finite_fold_nat_o @ ( comp_o_o_o_nat @ sup_sup_o @ F ) @ bot_bot_o @ A ) ) ) ).

% SUP_fold_sup
thf(fact_958_product__fold,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( finite_finite_nat @ B3 )
       => ( ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : B3 )
          = ( finite3745491028973389255at_nat
            @ ^ [X: nat,Z4: set_Pr1261947904930325089at_nat] :
                ( finite3745491028973389255at_nat
                @ ^ [Y: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) )
                @ Z4
                @ B3 )
            @ bot_bo2099793752762293965at_nat
            @ A ) ) ) ) ).

% product_fold
thf(fact_959_Id__on__fold,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( id_on_nat @ A )
        = ( finite3745491028973389255at_nat
          @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) )
          @ bot_bo2099793752762293965at_nat
          @ A ) ) ) ).

% Id_on_fold
thf(fact_960_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I4: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I4 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_961_Set__filter__fold,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( filter_nat @ P @ A )
        = ( finite5529483035118572448et_nat
          @ ^ [X: nat,A6: set_nat] : ( if_set_nat @ ( P @ X ) @ ( insert_nat @ X @ A6 ) @ A6 )
          @ bot_bot_set_nat
          @ A ) ) ) ).

% Set_filter_fold
thf(fact_962_Set__filter__fold,axiom,
    ! [A: set_o,P: $o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( filter_o @ P @ A )
        = ( finite_fold_o_set_o
          @ ^ [X: $o,A6: set_o] : ( if_set_o @ ( P @ X ) @ ( insert_o @ X @ A6 ) @ A6 )
          @ bot_bot_set_o
          @ A ) ) ) ).

% Set_filter_fold
thf(fact_963_Set_Omember__filter,axiom,
    ! [X2: list_Sum_sum_a_c,P: list_Sum_sum_a_c > $o,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ X2 @ ( filter8929528674883905222um_a_c @ P @ A ) )
      = ( ( member7772695417316360142um_a_c @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Set.member_filter
thf(fact_964_Set_Omember__filter,axiom,
    ! [X2: $o,P: $o > $o,A: set_o] :
      ( ( member_o @ X2 @ ( filter_o @ P @ A ) )
      = ( ( member_o @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Set.member_filter
thf(fact_965_Set_Omember__filter,axiom,
    ! [X2: nat,P: nat > $o,A: set_nat] :
      ( ( member_nat @ X2 @ ( filter_nat @ P @ A ) )
      = ( ( member_nat @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Set.member_filter
thf(fact_966_Set_Ofilter__def,axiom,
    ( filter8929528674883905222um_a_c
    = ( ^ [P3: list_Sum_sum_a_c > $o,A4: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ^ [A5: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ A5 @ A4 )
              & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_967_Set_Ofilter__def,axiom,
    ( filter_o
    = ( ^ [P3: $o > $o,A4: set_o] :
          ( collect_o
          @ ^ [A5: $o] :
              ( ( member_o @ A5 @ A4 )
              & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_968_Set_Ofilter__def,axiom,
    ( filter6709415106449780773um_a_c
    = ( ^ [P3: ( nat > sum_sum_a_c ) > $o,A4: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [A5: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ A5 @ A4 )
              & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_969_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P3: nat > $o,A4: set_nat] :
          ( collect_nat
          @ ^ [A5: nat] :
              ( ( member_nat @ A5 @ A4 )
              & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_970_Set_Ofilter__def,axiom,
    ( filter5640266504077782706at_nat
    = ( ^ [P3: product_prod_nat_nat > $o,A4: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [A5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A5 @ A4 )
              & ( P3 @ A5 ) ) ) ) ) ).

% Set.filter_def
thf(fact_971_Pow__fold,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( pow_nat @ A )
        = ( finite4178521680790401110et_nat
          @ ^ [X: nat,A4: set_set_nat] : ( sup_sup_set_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ A4 ) )
          @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
          @ A ) ) ) ).

% Pow_fold
thf(fact_972_Pow__fold,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( pow_o @ A )
        = ( finite2775045464208429320_set_o
          @ ^ [X: $o,A4: set_set_o] : ( sup_sup_set_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X ) @ A4 ) )
          @ ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o )
          @ A ) ) ) ).

% Pow_fold
thf(fact_973_infinite__UNIV,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV
thf(fact_974_Pow__UNIV,axiom,
    ( ( pow_o @ top_top_set_o )
    = top_top_set_set_o ) ).

% Pow_UNIV
thf(fact_975_Pow__UNIV,axiom,
    ( ( pow_nat @ top_top_set_nat )
    = top_top_set_set_nat ) ).

% Pow_UNIV
thf(fact_976_Pow__empty,axiom,
    ( ( pow_o @ bot_bot_set_o )
    = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).

% Pow_empty
thf(fact_977_Pow__singleton__iff,axiom,
    ! [X7: set_o,Y8: set_o] :
      ( ( ( pow_o @ X7 )
        = ( insert_set_o @ Y8 @ bot_bot_set_set_o ) )
      = ( ( X7 = bot_bot_set_o )
        & ( Y8 = bot_bot_set_o ) ) ) ).

% Pow_singleton_iff
thf(fact_978_comp__fun__commute__filter__fold,axiom,
    ! [P: $o > $o] :
      ( finite7927022897302369457_set_o
      @ ^ [X: $o,A6: set_o] : ( if_set_o @ ( P @ X ) @ ( insert_o @ X @ A6 ) @ A6 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_979_Pow__bottom,axiom,
    ! [B3: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B3 ) ) ).

% Pow_bottom
thf(fact_980_image__Pow__surj,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ( image_7318554134589591124um_a_c @ F @ A )
        = B3 )
     => ( ( image_8538515752806177344um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ ( pow_nat_Sum_sum_a_c @ A ) )
        = ( pow_list_Sum_sum_a_c @ B3 ) ) ) ).

% image_Pow_surj
thf(fact_981_comp__fun__commute__Image__fold,axiom,
    ! [S: set_list_Sum_sum_a_c] :
      ( finite1433128912581391311_set_o
      @ ( produc3823239332121889754_set_o
        @ ^ [X: list_Sum_sum_a_c,Y: $o,A4: set_o] : ( if_set_o @ ( member7772695417316360142um_a_c @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) ) ) ).

% comp_fun_commute_Image_fold
thf(fact_982_comp__fun__commute__Image__fold,axiom,
    ! [S: set_o] :
      ( finite4312298596831040610_set_o
      @ ( produc391208049229117841_set_o
        @ ^ [X: $o,Y: $o,A4: set_o] : ( if_set_o @ ( member_o @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) ) ) ).

% comp_fun_commute_Image_fold
thf(fact_983_comp__fun__commute__Image__fold,axiom,
    ! [S: set_nat] :
      ( finite6084612287169574120_set_o
      @ ( produc4840876362065759105_set_o
        @ ^ [X: nat,Y: $o,A4: set_o] : ( if_set_o @ ( member_nat @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) ) ) ).

% comp_fun_commute_Image_fold
thf(fact_984_Pow__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( pow_o @ ( insert_o @ A2 @ A ) )
      = ( sup_sup_set_set_o @ ( pow_o @ A ) @ ( image_set_o_set_o @ ( insert_o @ A2 ) @ ( pow_o @ A ) ) ) ) ).

% Pow_insert
thf(fact_985_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_986_in__finite__psubset,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B3 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A @ B3 )
        & ( finite_finite_nat @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_987_relation__of__def,axiom,
    ( order_1962693441708344834of_nat
    = ( ^ [P3: nat > nat > $o,A4: set_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A5: nat,B4: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 )
                  @ ( produc457027306803732586at_nat @ A4
                    @ ^ [Uu: nat] : A4 ) )
                & ( P3 @ A5 @ B4 ) ) ) ) ) ) ).

% relation_of_def
thf(fact_988_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A4: set_nat,B5: set_nat] :
            ( ( ord_less_set_nat @ A4 @ B5 )
            & ( finite_finite_nat @ B5 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_989_comp__fun__commute__Pow__fold,axiom,
    ( finite6772287081031898001_set_o
    @ ^ [X: $o,A4: set_set_o] : ( sup_sup_set_set_o @ A4 @ ( image_set_o_set_o @ ( insert_o @ X ) @ A4 ) ) ) ).

% comp_fun_commute_Pow_fold
thf(fact_990_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A5: nat,B4: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A7: nat,B7: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A7 ) @ Ra )
                      | ( ( A5 = A7 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_991_max__ext_Omax__extI,axiom,
    ! [X7: set_list_Sum_sum_a_c,Y8: set_list_Sum_sum_a_c,R3: set_Pr8580000064110529967um_a_c] :
      ( ( finite3830171671145977038um_a_c @ X7 )
     => ( ( finite3830171671145977038um_a_c @ Y8 )
       => ( ( Y8 != bot_bo3453284597459734017um_a_c )
         => ( ! [X3: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ X3 @ X7 )
               => ? [Xa2: list_Sum_sum_a_c] :
                    ( ( member7772695417316360142um_a_c @ Xa2 @ Y8 )
                    & ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X3 @ Xa2 ) @ R3 ) ) )
           => ( member1684655781224426852um_a_c @ ( produc9098613746998175347um_a_c @ X7 @ Y8 ) @ ( max_ex1156648277730443196um_a_c @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_992_max__ext_Omax__extI,axiom,
    ! [X7: set_nat,Y8: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X7 )
     => ( ( finite_finite_nat @ Y8 )
       => ( ( Y8 != bot_bot_set_nat )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ X7 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y8 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Xa2 ) @ R3 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X7 @ Y8 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_993_max__ext_Omax__extI,axiom,
    ! [X7: set_o,Y8: set_o,R3: set_Product_prod_o_o] :
      ( ( finite_finite_o @ X7 )
     => ( ( finite_finite_o @ Y8 )
       => ( ( Y8 != bot_bot_set_o )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ X7 )
               => ? [Xa2: $o] :
                    ( ( member_o @ Xa2 @ Y8 )
                    & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Xa2 ) @ R3 ) ) )
           => ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ X7 @ Y8 ) @ ( max_ext_o @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_994_asym__lex__prod,axiom,
    ! [R_A: set_Product_prod_o_o,R_B: set_Product_prod_o_o] :
      ( ( asym_on_o @ top_top_set_o @ R_A )
     => ( ( asym_on_o @ top_top_set_o @ R_B )
       => ( asym_o744349338269926489od_o_o @ top_to7721136755696657239od_o_o @ ( lex_prod_o_o @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_995_asym__lex__prod,axiom,
    ! [R_A: set_Product_prod_o_o,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_o @ top_top_set_o @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o1567361650835871717_o_nat @ top_to7022684507342537725_o_nat @ ( lex_prod_o_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_996_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Product_prod_o_o] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_on_o @ top_top_set_o @ R_B )
       => ( asym_o5075896174891364747_nat_o @ top_to8070287629520841379_nat_o @ ( lex_prod_nat_o @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_997_asym__lex__prod,axiom,
    ! [R_A: set_Pr1261947904930325089at_nat,R_B: set_Pr1261947904930325089at_nat] :
      ( ( asym_on_nat @ top_top_set_nat @ R_A )
     => ( ( asym_on_nat @ top_top_set_nat @ R_B )
       => ( asym_o8546584678760569651at_nat @ top_to4669805908274784177at_nat @ ( lex_prod_nat_nat @ R_A @ R_B ) ) ) ) ).

% asym_lex_prod
thf(fact_998_max__ext_Ocases,axiom,
    ! [A1: set_list_Sum_sum_a_c,A22: set_list_Sum_sum_a_c,R3: set_Pr8580000064110529967um_a_c] :
      ( ( member1684655781224426852um_a_c @ ( produc9098613746998175347um_a_c @ A1 @ A22 ) @ ( max_ex1156648277730443196um_a_c @ R3 ) )
     => ~ ( ( finite3830171671145977038um_a_c @ A1 )
         => ( ( finite3830171671145977038um_a_c @ A22 )
           => ( ( A22 != bot_bo3453284597459734017um_a_c )
             => ~ ! [X4: list_Sum_sum_a_c] :
                    ( ( member7772695417316360142um_a_c @ X4 @ A1 )
                   => ? [Xa3: list_Sum_sum_a_c] :
                        ( ( member7772695417316360142um_a_c @ Xa3 @ A22 )
                        & ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X4 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_999_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X4: nat] :
                    ( ( member_nat @ X4 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1000_max__ext_Ocases,axiom,
    ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
      ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
     => ~ ( ( finite_finite_o @ A1 )
         => ( ( finite_finite_o @ A22 )
           => ( ( A22 != bot_bot_set_o )
             => ~ ! [X4: $o] :
                    ( ( member_o @ X4 @ A1 )
                   => ? [Xa3: $o] :
                        ( ( member_o @ Xa3 @ A22 )
                        & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1001_max__ext_Osimps,axiom,
    ! [A1: set_list_Sum_sum_a_c,A22: set_list_Sum_sum_a_c,R3: set_Pr8580000064110529967um_a_c] :
      ( ( member1684655781224426852um_a_c @ ( produc9098613746998175347um_a_c @ A1 @ A22 ) @ ( max_ex1156648277730443196um_a_c @ R3 ) )
      = ( ( finite3830171671145977038um_a_c @ A1 )
        & ( finite3830171671145977038um_a_c @ A22 )
        & ( A22 != bot_bo3453284597459734017um_a_c )
        & ! [X: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X @ A1 )
           => ? [Y: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ Y @ A22 )
                & ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X @ Y ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1002_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X: nat] :
            ( ( member_nat @ X @ A1 )
           => ? [Y: nat] :
                ( ( member_nat @ Y @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1003_max__ext_Osimps,axiom,
    ! [A1: set_o,A22: set_o,R3: set_Product_prod_o_o] :
      ( ( member9116954335612470352_set_o @ ( produc5838405689764958487_set_o @ A1 @ A22 ) @ ( max_ext_o @ R3 ) )
      = ( ( finite_finite_o @ A1 )
        & ( finite_finite_o @ A22 )
        & ( A22 != bot_bot_set_o )
        & ! [X: $o] :
            ( ( member_o @ X @ A1 )
           => ? [Y: $o] :
                ( ( member_o @ Y @ A22 )
                & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1004_max__extp__eq,axiom,
    ( max_extp_nat
    = ( ^ [R4: nat > nat > $o,X: set_nat,Y: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y ) @ ( max_ext_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ R4 ) ) ) ) ) ) ).

% max_extp_eq
thf(fact_1005_Image__fold,axiom,
    ! [R3: set_Pr2109328554291888588_a_c_o,S: set_list_Sum_sum_a_c] :
      ( ( finite3557940929864092333_a_c_o @ R3 )
     => ( ( image_4064336973628159244_a_c_o @ R3 @ S )
        = ( finite5687714443756313368_set_o
          @ ( produc3823239332121889754_set_o
            @ ^ [X: list_Sum_sum_a_c,Y: $o,A4: set_o] : ( if_set_o @ ( member7772695417316360142um_a_c @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) )
          @ bot_bot_set_o
          @ R3 ) ) ) ).

% Image_fold
thf(fact_1006_Image__fold,axiom,
    ! [R3: set_Product_prod_o_o,S: set_o] :
      ( ( finite6120865539452801872od_o_o @ R3 )
     => ( ( image_o_o @ R3 @ S )
        = ( finite631754462958660185_set_o
          @ ( produc391208049229117841_set_o
            @ ^ [X: $o,Y: $o,A4: set_o] : ( if_set_o @ ( member_o @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) )
          @ bot_bot_set_o
          @ R3 ) ) ) ).

% Image_fold
thf(fact_1007_Image__fold,axiom,
    ! [R3: set_Pr3149072824959771635_nat_o,S: set_nat] :
      ( ( finite5355008432043429460_nat_o @ R3 )
     => ( ( image_nat_o @ R3 @ S )
        = ( finite8586300639982209713_set_o
          @ ( produc4840876362065759105_set_o
            @ ^ [X: nat,Y: $o,A4: set_o] : ( if_set_o @ ( member_nat @ X @ S ) @ ( insert_o @ Y @ A4 ) @ A4 ) )
          @ bot_bot_set_o
          @ R3 ) ) ) ).

% Image_fold
thf(fact_1008_ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ R2 )
     => ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_8392244845493202499um_a_c @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1009_ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,B: $o,R2: set_Pr2109328554291888588_a_c_o,A: set_list_Sum_sum_a_c] :
      ( ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ A2 @ B ) @ R2 )
     => ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( member_o @ B @ ( image_4064336973628159244_a_c_o @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1010_ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,B: nat,R2: set_Pr6185592159103593672_c_nat,A: set_list_Sum_sum_a_c] :
      ( ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ A2 @ B ) @ R2 )
     => ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( member_nat @ B @ ( image_785291739673018204_c_nat @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1011_ImageI,axiom,
    ! [A2: $o,B: list_Sum_sum_a_c,R2: set_Pr2999174882133146036um_a_c,A: set_o] :
      ( ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ R2 )
     => ( ( member_o @ A2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_4124445023530029884um_a_c @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1012_ImageI,axiom,
    ! [A2: $o,B: $o,R2: set_Product_prod_o_o,A: set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
     => ( ( member_o @ A2 @ A )
       => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1013_ImageI,axiom,
    ! [A2: $o,B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
      ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
     => ( ( member_o @ A2 @ A )
       => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1014_ImageI,axiom,
    ! [A2: nat,B: list_Sum_sum_a_c,R2: set_Pr2809021152199675080um_a_c,A: set_nat] :
      ( ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ A2 @ B ) @ R2 )
     => ( ( member_nat @ A2 @ A )
       => ( member7772695417316360142um_a_c @ B @ ( image_3344982812737636572um_a_c @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1015_ImageI,axiom,
    ! [A2: nat,B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
      ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
     => ( ( member_nat @ A2 @ A )
       => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1016_ImageI,axiom,
    ! [A2: nat,B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
     => ( ( member_nat @ A2 @ A )
       => ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1017_Image__empty2,axiom,
    ! [R3: set_Product_prod_o_o] :
      ( ( image_o_o @ R3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Image_empty2
thf(fact_1018_case__swap,axiom,
    ! [F: nat > nat > $o,P4: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [Y: nat,X: nat] : ( F @ X @ Y )
        @ ( product_swap_nat_nat @ P4 ) )
      = ( produc6081775807080527818_nat_o @ F @ P4 ) ) ).

% case_swap
thf(fact_1019_Image__singleton__iff,axiom,
    ! [B: list_Sum_sum_a_c,R2: set_Pr2999174882133146036um_a_c,A2: $o] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_4124445023530029884um_a_c @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1020_Image__singleton__iff,axiom,
    ! [B: $o,R2: set_Product_prod_o_o,A2: $o] :
      ( ( member_o @ B @ ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1021_Image__singleton__iff,axiom,
    ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A2: $o] :
      ( ( member_nat @ B @ ( image_o_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1022_ImageE,axiom,
    ! [B: list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,A: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_8392244845493202499um_a_c @ R2 @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X3 @ B ) @ R2 )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1023_ImageE,axiom,
    ! [B: list_Sum_sum_a_c,R2: set_Pr2999174882133146036um_a_c,A: set_o] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_4124445023530029884um_a_c @ R2 @ A ) )
     => ~ ! [X3: $o] :
            ( ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ X3 @ B ) @ R2 )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1024_ImageE,axiom,
    ! [B: list_Sum_sum_a_c,R2: set_Pr2809021152199675080um_a_c,A: set_nat] :
      ( ( member7772695417316360142um_a_c @ B @ ( image_3344982812737636572um_a_c @ R2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ X3 @ B ) @ R2 )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1025_ImageE,axiom,
    ! [B: $o,R2: set_Pr2109328554291888588_a_c_o,A: set_list_Sum_sum_a_c] :
      ( ( member_o @ B @ ( image_4064336973628159244_a_c_o @ R2 @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ X3 @ B ) @ R2 )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1026_ImageE,axiom,
    ! [B: $o,R2: set_Product_prod_o_o,A: set_o] :
      ( ( member_o @ B @ ( image_o_o @ R2 @ A ) )
     => ~ ! [X3: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ B ) @ R2 )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1027_ImageE,axiom,
    ! [B: $o,R2: set_Pr3149072824959771635_nat_o,A: set_nat] :
      ( ( member_o @ B @ ( image_nat_o @ R2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ X3 @ B ) @ R2 )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1028_ImageE,axiom,
    ! [B: nat,R2: set_Pr6185592159103593672_c_nat,A: set_list_Sum_sum_a_c] :
      ( ( member_nat @ B @ ( image_785291739673018204_c_nat @ R2 @ A ) )
     => ~ ! [X3: list_Sum_sum_a_c] :
            ( ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ X3 @ B ) @ R2 )
           => ~ ( member7772695417316360142um_a_c @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1029_ImageE,axiom,
    ! [B: nat,R2: set_Pr2101469702781467981_o_nat,A: set_o] :
      ( ( member_nat @ B @ ( image_o_nat @ R2 @ A ) )
     => ~ ! [X3: $o] :
            ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X3 @ B ) @ R2 )
           => ~ ( member_o @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1030_ImageE,axiom,
    ! [B: nat,R2: set_Pr1261947904930325089at_nat,A: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) )
     => ~ ! [X3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ R2 )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1031_rev__ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ A2 @ B ) @ R2 )
       => ( member7772695417316360142um_a_c @ B @ ( image_8392244845493202499um_a_c @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1032_rev__ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: $o,R2: set_Pr2109328554291888588_a_c_o] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member7213031410929038765_a_c_o @ ( produc8438544629413731400_a_c_o @ A2 @ B ) @ R2 )
       => ( member_o @ B @ ( image_4064336973628159244_a_c_o @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1033_rev__ImageI,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B: nat,R2: set_Pr6185592159103593672_c_nat] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( member6914143599624037265_c_nat @ ( produc8518022002054455072_c_nat @ A2 @ B ) @ R2 )
       => ( member_nat @ B @ ( image_785291739673018204_c_nat @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1034_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: list_Sum_sum_a_c,R2: set_Pr2999174882133146036um_a_c] :
      ( ( member_o @ A2 @ A )
     => ( ( member7870376738782077333um_a_c @ ( produc8498652679315602040um_a_c @ A2 @ B ) @ R2 )
       => ( member7772695417316360142um_a_c @ B @ ( image_4124445023530029884um_a_c @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1035_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: $o,R2: set_Product_prod_o_o] :
      ( ( member_o @ A2 @ A )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B ) @ R2 )
       => ( member_o @ B @ ( image_o_o @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1036_rev__ImageI,axiom,
    ! [A2: $o,A: set_o,B: nat,R2: set_Pr2101469702781467981_o_nat] :
      ( ( member_o @ A2 @ A )
     => ( ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B ) @ R2 )
       => ( member_nat @ B @ ( image_o_nat @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1037_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: list_Sum_sum_a_c,R2: set_Pr2809021152199675080um_a_c] :
      ( ( member_nat @ A2 @ A )
     => ( ( member1253563076732355473um_a_c @ ( produc1854341038264297632um_a_c @ A2 @ B ) @ R2 )
       => ( member7772695417316360142um_a_c @ B @ ( image_3344982812737636572um_a_c @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1038_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: $o,R2: set_Pr3149072824959771635_nat_o] :
      ( ( member_nat @ A2 @ A )
     => ( ( member6310962623043647828_nat_o @ ( product_Pair_nat_o @ A2 @ B ) @ R2 )
       => ( member_o @ B @ ( image_nat_o @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1039_rev__ImageI,axiom,
    ! [A2: nat,A: set_nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ R2 )
       => ( member_nat @ B @ ( image_nat_nat @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1040_Image__singleton,axiom,
    ! [R2: set_Pr8041419848822008851um_a_c,A2: $o] :
      ( ( image_4747166508673656475um_a_c @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( collec5227572641185395563um_a_c
        @ ^ [B4: nat > sum_sum_a_c] : ( member5403367102549231476um_a_c @ ( produc1925496998193254871um_a_c @ A2 @ B4 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1041_Image__singleton,axiom,
    ! [R2: set_Pr2101469702781467981_o_nat,A2: $o] :
      ( ( image_o_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( collect_nat
        @ ^ [B4: nat] : ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ A2 @ B4 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1042_Image__singleton,axiom,
    ! [R2: set_Pr2390076351701138800at_nat,A2: $o] :
      ( ( image_5697180853172723644at_nat @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( collec3392354462482085612at_nat
        @ ^ [B4: product_prod_nat_nat] : ( member6694634564379471929at_nat @ ( produc1017754141990422400at_nat @ A2 @ B4 ) @ R2 ) ) ) ).

% Image_singleton
thf(fact_1043_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr8580000064110529967um_a_c,A2: list_Sum_sum_a_c,B: list_Sum_sum_a_c] :
      ( ( refl_o5556642055249044088um_a_c @ ( field_4321413778972592298um_a_c @ R2 ) @ R2 )
     => ( ( antisy5664394440168933524um_a_c @ top_to8424319039679220765um_a_c @ R2 )
       => ( ( member7772695417316360142um_a_c @ A2 @ ( field_4321413778972592298um_a_c @ R2 ) )
         => ( ( member7772695417316360142um_a_c @ B @ ( field_4321413778972592298um_a_c @ R2 ) )
           => ( ( ( image_8392244845493202499um_a_c @ R2 @ ( insert7520310754158225127um_a_c @ A2 @ bot_bo3453284597459734017um_a_c ) )
                = ( image_8392244845493202499um_a_c @ R2 @ ( insert7520310754158225127um_a_c @ B @ bot_bo3453284597459734017um_a_c ) ) )
              = ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1044_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_o_o,A2: $o,B: $o] :
      ( ( refl_on_o @ ( field_o @ R2 ) @ R2 )
     => ( ( antisym_on_o @ top_top_set_o @ R2 )
       => ( ( member_o @ A2 @ ( field_o @ R2 ) )
         => ( ( member_o @ B @ ( field_o @ R2 ) )
           => ( ( ( image_o_o @ R2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
                = ( image_o_o @ R2 @ ( insert_o @ B @ bot_bot_set_o ) ) )
              = ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1045_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A2: nat,B: nat] :
      ( ( refl_on_nat @ ( field_nat @ R2 ) @ R2 )
     => ( ( antisym_on_nat @ top_top_set_nat @ R2 )
       => ( ( member_nat @ A2 @ ( field_nat @ R2 ) )
         => ( ( member_nat @ B @ ( field_nat @ R2 ) )
           => ( ( ( image_nat_nat @ R2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
                = ( image_nat_nat @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
              = ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1046_quotient__def,axiom,
    ( equiv_quotient_o
    = ( ^ [A4: set_o,R4: set_Product_prod_o_o] :
          ( comple4436988014476444997_set_o
          @ ( image_o_set_set_o
            @ ^ [X: $o] : ( insert_set_o @ ( image_o_o @ R4 @ ( insert_o @ X @ bot_bot_set_o ) ) @ bot_bot_set_set_o )
            @ A4 ) ) ) ) ).

% quotient_def
thf(fact_1047_IntI,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ A )
     => ( ( member7772695417316360142um_a_c @ C @ B3 )
       => ( member7772695417316360142um_a_c @ C @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_1048_IntI,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ A )
     => ( ( member_o @ C @ B3 )
       => ( member_o @ C @ ( inf_inf_set_o @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_1049_IntI,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A )
     => ( ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% IntI
thf(fact_1050_Int__iff,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) )
      = ( ( member7772695417316360142um_a_c @ C @ A )
        & ( member7772695417316360142um_a_c @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_1051_Int__iff,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A @ B3 ) )
      = ( ( member_o @ C @ A )
        & ( member_o @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_1052_Int__iff,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
      = ( ( member_nat @ C @ A )
        & ( member_nat @ C @ B3 ) ) ) ).

% Int_iff
thf(fact_1053_inf__top_Oright__neutral,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ top_top_set_o )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_1054_inf__top_Oright__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
      = A2 ) ).

% inf_top.right_neutral
thf(fact_1055_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( top_top_set_o
        = ( inf_inf_set_o @ A2 @ B ) )
      = ( ( A2 = top_top_set_o )
        & ( B = top_top_set_o ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_1056_inf__top_Oneutr__eq__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ A2 @ B ) )
      = ( ( A2 = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
thf(fact_1057_inf__top_Oleft__neutral,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_1058_inf__top_Oleft__neutral,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
      = A2 ) ).

% inf_top.left_neutral
thf(fact_1059_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_o,B: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B )
        = top_top_set_o )
      = ( ( A2 = top_top_set_o )
        & ( B = top_top_set_o ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_1060_inf__top_Oeq__neutr__iff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B )
        = top_top_set_nat )
      = ( ( A2 = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
thf(fact_1061_top__eq__inf__iff,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( top_top_set_o
        = ( inf_inf_set_o @ X2 @ Y3 ) )
      = ( ( X2 = top_top_set_o )
        & ( Y3 = top_top_set_o ) ) ) ).

% top_eq_inf_iff
thf(fact_1062_top__eq__inf__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( top_top_set_nat
        = ( inf_inf_set_nat @ X2 @ Y3 ) )
      = ( ( X2 = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
thf(fact_1063_inf__eq__top__iff,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( ( inf_inf_set_o @ X2 @ Y3 )
        = top_top_set_o )
      = ( ( X2 = top_top_set_o )
        & ( Y3 = top_top_set_o ) ) ) ).

% inf_eq_top_iff
thf(fact_1064_inf__eq__top__iff,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ X2 @ Y3 )
        = top_top_set_nat )
      = ( ( X2 = top_top_set_nat )
        & ( Y3 = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
thf(fact_1065_inf__top__right,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ X2 @ top_top_set_o )
      = X2 ) ).

% inf_top_right
thf(fact_1066_inf__top__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
      = X2 ) ).

% inf_top_right
thf(fact_1067_inf__top__left,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ X2 )
      = X2 ) ).

% inf_top_left
thf(fact_1068_inf__top__left,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ X2 )
      = X2 ) ).

% inf_top_left
thf(fact_1069_Int__UNIV,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A @ B3 )
        = top_top_set_o )
      = ( ( A = top_top_set_o )
        & ( B3 = top_top_set_o ) ) ) ).

% Int_UNIV
thf(fact_1070_Int__UNIV,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B3 )
        = top_top_set_nat )
      = ( ( A = top_top_set_nat )
        & ( B3 = top_top_set_nat ) ) ) ).

% Int_UNIV
thf(fact_1071_Int__insert__right__if1,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( inf_in1719665111139814143um_a_c @ A @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) )
        = ( insert7520310754158225127um_a_c @ A2 @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1072_Int__insert__right__if1,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B3 ) )
        = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1073_Int__insert__right__if1,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1074_Int__insert__right__if0,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ~ ( member7772695417316360142um_a_c @ A2 @ A )
     => ( ( inf_in1719665111139814143um_a_c @ A @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) )
        = ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1075_Int__insert__right__if0,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B3 ) )
        = ( inf_inf_set_o @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1076_Int__insert__right__if0,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
        = ( inf_inf_set_nat @ A @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1077_insert__inter__insert,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B3 ) )
      = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_1078_Int__insert__left__if1,axiom,
    ! [A2: list_Sum_sum_a_c,C2: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ A2 @ C2 )
     => ( ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) @ C2 )
        = ( insert7520310754158225127um_a_c @ A2 @ ( inf_in1719665111139814143um_a_c @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1079_Int__insert__left__if1,axiom,
    ! [A2: $o,C2: set_o,B3: set_o] :
      ( ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B3 ) @ C2 )
        = ( insert_o @ A2 @ ( inf_inf_set_o @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1080_Int__insert__left__if1,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
        = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1081_Int__insert__left__if0,axiom,
    ! [A2: list_Sum_sum_a_c,C2: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ~ ( member7772695417316360142um_a_c @ A2 @ C2 )
     => ( ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) @ C2 )
        = ( inf_in1719665111139814143um_a_c @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1082_Int__insert__left__if0,axiom,
    ! [A2: $o,C2: set_o,B3: set_o] :
      ( ~ ( member_o @ A2 @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B3 ) @ C2 )
        = ( inf_inf_set_o @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1083_Int__insert__left__if0,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ A2 @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
        = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_1084_insert__disjoint_I1_J,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ A ) @ B3 )
        = bot_bo3453284597459734017um_a_c )
      = ( ~ ( member7772695417316360142um_a_c @ A2 @ B3 )
        & ( ( inf_in1719665111139814143um_a_c @ A @ B3 )
          = bot_bo3453284597459734017um_a_c ) ) ) ).

% insert_disjoint(1)
thf(fact_1085_insert__disjoint_I1_J,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( ( inf_inf_set_nat @ A @ B3 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1086_insert__disjoint_I1_J,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B3 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B3 )
        & ( ( inf_inf_set_o @ A @ B3 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1087_insert__disjoint_I2_J,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( bot_bo3453284597459734017um_a_c
        = ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ A ) @ B3 ) )
      = ( ~ ( member7772695417316360142um_a_c @ A2 @ B3 )
        & ( bot_bo3453284597459734017um_a_c
          = ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1088_insert__disjoint_I2_J,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1089_insert__disjoint_I2_J,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B3 ) )
      = ( ~ ( member_o @ A2 @ B3 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1090_disjoint__insert_I1_J,axiom,
    ! [B3: set_list_Sum_sum_a_c,A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( ( inf_in1719665111139814143um_a_c @ B3 @ ( insert7520310754158225127um_a_c @ A2 @ A ) )
        = bot_bo3453284597459734017um_a_c )
      = ( ~ ( member7772695417316360142um_a_c @ A2 @ B3 )
        & ( ( inf_in1719665111139814143um_a_c @ B3 @ A )
          = bot_bo3453284597459734017um_a_c ) ) ) ).

% disjoint_insert(1)
thf(fact_1091_disjoint__insert_I1_J,axiom,
    ! [B3: set_nat,A2: nat,A: set_nat] :
      ( ( ( inf_inf_set_nat @ B3 @ ( insert_nat @ A2 @ A ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A2 @ B3 )
        & ( ( inf_inf_set_nat @ B3 @ A )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1092_disjoint__insert_I1_J,axiom,
    ! [B3: set_o,A2: $o,A: set_o] :
      ( ( ( inf_inf_set_o @ B3 @ ( insert_o @ A2 @ A ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A2 @ B3 )
        & ( ( inf_inf_set_o @ B3 @ A )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1093_disjoint__insert_I2_J,axiom,
    ! [A: set_list_Sum_sum_a_c,B: list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( bot_bo3453284597459734017um_a_c
        = ( inf_in1719665111139814143um_a_c @ A @ ( insert7520310754158225127um_a_c @ B @ B3 ) ) )
      = ( ~ ( member7772695417316360142um_a_c @ B @ A )
        & ( bot_bo3453284597459734017um_a_c
          = ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1094_disjoint__insert_I2_J,axiom,
    ! [A: set_nat,B: nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B3 ) ) )
      = ( ~ ( member_nat @ B @ A )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1095_disjoint__insert_I2_J,axiom,
    ! [A: set_o,B: $o,B3: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A @ ( insert_o @ B @ B3 ) ) )
      = ( ~ ( member_o @ B @ A )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1096_if__image__distrib,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,G: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,S: set_nat_Sum_sum_a_c] :
      ( ( image_7318554134589591124um_a_c
        @ ^ [X: nat > sum_sum_a_c] : ( if_list_Sum_sum_a_c @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
        @ S )
      = ( sup_su3984685238261546137um_a_c @ ( image_7318554134589591124um_a_c @ F @ ( inf_in5970604416696682206um_a_c @ S @ ( collec5227572641185395563um_a_c @ P ) ) )
        @ ( image_7318554134589591124um_a_c @ G
          @ ( inf_in5970604416696682206um_a_c @ S
            @ ( collec5227572641185395563um_a_c
              @ ^ [X: nat > sum_sum_a_c] :
                  ~ ( P @ X ) ) ) ) ) ) ).

% if_image_distrib
thf(fact_1097_Int__UNIV__left,axiom,
    ! [B3: set_o] :
      ( ( inf_inf_set_o @ top_top_set_o @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_1098_Int__UNIV__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ top_top_set_nat @ B3 )
      = B3 ) ).

% Int_UNIV_left
thf(fact_1099_Int__UNIV__right,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ A @ top_top_set_o )
      = A ) ).

% Int_UNIV_right
thf(fact_1100_Int__UNIV__right,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ top_top_set_nat )
      = A ) ).

% Int_UNIV_right
thf(fact_1101_boolean__algebra_Oconj__one__right,axiom,
    ! [X2: set_o] :
      ( ( inf_inf_set_o @ X2 @ top_top_set_o )
      = X2 ) ).

% boolean_algebra.conj_one_right
thf(fact_1102_boolean__algebra_Oconj__one__right,axiom,
    ! [X2: set_nat] :
      ( ( inf_inf_set_nat @ X2 @ top_top_set_nat )
      = X2 ) ).

% boolean_algebra.conj_one_right
thf(fact_1103_Collect__conj__eq,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( collec5227572641185395563um_a_c
        @ ^ [X: nat > sum_sum_a_c] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in5970604416696682206um_a_c @ ( collec5227572641185395563um_a_c @ P ) @ ( collec5227572641185395563um_a_c @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1104_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1105_Collect__conj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( P @ X )
            & ( Q @ X ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_1106_Int__Collect,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,P: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ X2 @ ( inf_in1719665111139814143um_a_c @ A @ ( collec8219452656984879116um_a_c @ P ) ) )
      = ( ( member7772695417316360142um_a_c @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1107_Int__Collect,axiom,
    ! [X2: $o,A: set_o,P: $o > $o] :
      ( ( member_o @ X2 @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) )
      = ( ( member_o @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1108_Int__Collect,axiom,
    ! [X2: nat > sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: ( nat > sum_sum_a_c ) > $o] :
      ( ( member4884986500679352621um_a_c @ X2 @ ( inf_in5970604416696682206um_a_c @ A @ ( collec5227572641185395563um_a_c @ P ) ) )
      = ( ( member4884986500679352621um_a_c @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1109_Int__Collect,axiom,
    ! [X2: nat,A: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1110_Int__Collect,axiom,
    ! [X2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A @ ( collec3392354462482085612at_nat @ P ) ) )
      = ( ( member8440522571783428010at_nat @ X2 @ A )
        & ( P @ X2 ) ) ) ).

% Int_Collect
thf(fact_1111_Int__def,axiom,
    ( inf_in1719665111139814143um_a_c
    = ( ^ [A4: set_list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ^ [X: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X @ A4 )
              & ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_1112_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( collect_o
          @ ^ [X: $o] :
              ( ( member_o @ X @ A4 )
              & ( member_o @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_1113_Int__def,axiom,
    ( inf_in5970604416696682206um_a_c
    = ( ^ [A4: set_nat_Sum_sum_a_c,B5: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [X: nat > sum_sum_a_c] :
              ( ( member4884986500679352621um_a_c @ X @ A4 )
              & ( member4884986500679352621um_a_c @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_1114_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( member_nat @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_1115_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A4 )
              & ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_1116_IntE,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) )
     => ~ ( ( member7772695417316360142um_a_c @ C @ A )
         => ~ ( member7772695417316360142um_a_c @ C @ B3 ) ) ) ).

% IntE
thf(fact_1117_IntE,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A @ B3 ) )
     => ~ ( ( member_o @ C @ A )
         => ~ ( member_o @ C @ B3 ) ) ) ).

% IntE
thf(fact_1118_IntE,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ~ ( ( member_nat @ C @ A )
         => ~ ( member_nat @ C @ B3 ) ) ) ).

% IntE
thf(fact_1119_IntD1,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) )
     => ( member7772695417316360142um_a_c @ C @ A ) ) ).

% IntD1
thf(fact_1120_IntD1,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A @ B3 ) )
     => ( member_o @ C @ A ) ) ).

% IntD1
thf(fact_1121_IntD1,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ( member_nat @ C @ A ) ) ).

% IntD1
thf(fact_1122_IntD2,axiom,
    ! [C: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( member7772695417316360142um_a_c @ C @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) )
     => ( member7772695417316360142um_a_c @ C @ B3 ) ) ).

% IntD2
thf(fact_1123_IntD2,axiom,
    ! [C: $o,A: set_o,B3: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A @ B3 ) )
     => ( member_o @ C @ B3 ) ) ).

% IntD2
thf(fact_1124_IntD2,axiom,
    ! [C: nat,A: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B3 ) )
     => ( member_nat @ C @ B3 ) ) ).

% IntD2
thf(fact_1125_Int__insert__left,axiom,
    ! [A2: list_Sum_sum_a_c,C2: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ( member7772695417316360142um_a_c @ A2 @ C2 )
       => ( ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) @ C2 )
          = ( insert7520310754158225127um_a_c @ A2 @ ( inf_in1719665111139814143um_a_c @ B3 @ C2 ) ) ) )
      & ( ~ ( member7772695417316360142um_a_c @ A2 @ C2 )
       => ( ( inf_in1719665111139814143um_a_c @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) @ C2 )
          = ( inf_in1719665111139814143um_a_c @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1126_Int__insert__left,axiom,
    ! [A2: $o,C2: set_o,B3: set_o] :
      ( ( ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B3 ) @ C2 )
          = ( insert_o @ A2 @ ( inf_inf_set_o @ B3 @ C2 ) ) ) )
      & ( ~ ( member_o @ A2 @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B3 ) @ C2 )
          = ( inf_inf_set_o @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1127_Int__insert__left,axiom,
    ! [A2: nat,C2: set_nat,B3: set_nat] :
      ( ( ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C2 ) ) ) )
      & ( ~ ( member_nat @ A2 @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B3 ) @ C2 )
          = ( inf_inf_set_nat @ B3 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_1128_Int__insert__right,axiom,
    ! [A2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ( member7772695417316360142um_a_c @ A2 @ A )
       => ( ( inf_in1719665111139814143um_a_c @ A @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) )
          = ( insert7520310754158225127um_a_c @ A2 @ ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) )
      & ( ~ ( member7772695417316360142um_a_c @ A2 @ A )
       => ( ( inf_in1719665111139814143um_a_c @ A @ ( insert7520310754158225127um_a_c @ A2 @ B3 ) )
          = ( inf_in1719665111139814143um_a_c @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_1129_Int__insert__right,axiom,
    ! [A2: $o,A: set_o,B3: set_o] :
      ( ( ( member_o @ A2 @ A )
       => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B3 ) )
          = ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B3 ) ) ) )
      & ( ~ ( member_o @ A2 @ A )
       => ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B3 ) )
          = ( inf_inf_set_o @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_1130_Int__insert__right,axiom,
    ! [A2: nat,A: set_nat,B3: set_nat] :
      ( ( ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
          = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B3 ) ) ) )
      & ( ~ ( member_nat @ A2 @ A )
       => ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
          = ( inf_inf_set_nat @ A @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_1131_Int__emptyI,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ~ ( member7772695417316360142um_a_c @ X3 @ B3 ) )
     => ( ( inf_in1719665111139814143um_a_c @ A @ B3 )
        = bot_bo3453284597459734017um_a_c ) ) ).

% Int_emptyI
thf(fact_1132_Int__emptyI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ~ ( member_nat @ X3 @ B3 ) )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_1133_Int__emptyI,axiom,
    ! [A: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ~ ( member_o @ X3 @ B3 ) )
     => ( ( inf_inf_set_o @ A @ B3 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_1134_disjoint__iff,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ( inf_in1719665111139814143um_a_c @ A @ B3 )
        = bot_bo3453284597459734017um_a_c )
      = ( ! [X: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X @ A )
           => ~ ( member7772695417316360142um_a_c @ X @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1135_disjoint__iff,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A @ B3 )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ A )
           => ~ ( member_nat @ X @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1136_disjoint__iff,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A @ B3 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A )
           => ~ ( member_o @ X @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_1137_Int__empty__left,axiom,
    ! [B3: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B3 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_1138_Int__empty__right,axiom,
    ! [A: set_o] :
      ( ( inf_inf_set_o @ A @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_1139_disjoint__iff__not__equal,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A @ B3 )
        = bot_bot_set_o )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A )
           => ! [Y: $o] :
                ( ( member_o @ Y @ B3 )
               => ( X = (~ Y) ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1140_antisym__onD,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c,X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( antisy5664394440168933524um_a_c @ A @ R2 )
     => ( ( member7772695417316360142um_a_c @ X2 @ A )
       => ( ( member7772695417316360142um_a_c @ Y3 @ A )
         => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X2 @ Y3 ) @ R2 )
           => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ Y3 @ X2 ) @ R2 )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1141_antisym__onD,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( antisym_on_o @ A @ R2 )
     => ( ( member_o @ X2 @ A )
       => ( ( member_o @ Y3 @ A )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R2 )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1142_antisym__onD,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( antisym_on_nat @ A @ R2 )
     => ( ( member_nat @ X2 @ A )
       => ( ( member_nat @ Y3 @ A )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 )
             => ( X2 = Y3 ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1143_antisym__onI,axiom,
    ! [A: set_list_Sum_sum_a_c,R2: set_Pr8580000064110529967um_a_c] :
      ( ! [X3: list_Sum_sum_a_c,Y4: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( ( member7772695417316360142um_a_c @ Y4 @ A )
           => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ X3 @ Y4 ) @ R2 )
             => ( ( member4235642376205993464um_a_c @ ( produc7367857235639141383um_a_c @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisy5664394440168933524um_a_c @ A @ R2 ) ) ).

% antisym_onI
thf(fact_1144_antisym__onI,axiom,
    ! [A: set_o,R2: set_Product_prod_o_o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( member_o @ X3 @ A )
         => ( ( member_o @ Y4 @ A )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R2 )
             => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisym_on_o @ A @ R2 ) ) ).

% antisym_onI
thf(fact_1145_antisym__onI,axiom,
    ! [A: set_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member_nat @ X3 @ A )
         => ( ( member_nat @ Y4 @ A )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
             => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisym_on_nat @ A @ R2 ) ) ).

% antisym_onI
thf(fact_1146_Union__disjoint,axiom,
    ! [C2: set_set_o,A: set_o] :
      ( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C2 ) @ A )
        = bot_bot_set_o )
      = ( ! [X: set_o] :
            ( ( member_set_o @ X @ C2 )
           => ( ( inf_inf_set_o @ X @ A )
              = bot_bot_set_o ) ) ) ) ).

% Union_disjoint
thf(fact_1147_antisymI,axiom,
    ! [R2: set_Product_prod_o_o] :
      ( ! [X3: $o,Y4: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y4 ) @ R2 )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ X3 ) @ R2 )
           => ( X3 = Y4 ) ) )
     => ( antisym_on_o @ top_top_set_o @ R2 ) ) ).

% antisymI
thf(fact_1148_antisymI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ X3 ) @ R2 )
           => ( X3 = Y4 ) ) )
     => ( antisym_on_nat @ top_top_set_nat @ R2 ) ) ).

% antisymI
thf(fact_1149_antisymD,axiom,
    ! [R2: set_Product_prod_o_o,X2: $o,Y3: $o] :
      ( ( antisym_on_o @ top_top_set_o @ R2 )
     => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R2 )
       => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ X2 ) @ R2 )
         => ( X2 = Y3 ) ) ) ) ).

% antisymD
thf(fact_1150_antisymD,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,X2: nat,Y3: nat] :
      ( ( antisym_on_nat @ top_top_set_nat @ R2 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X2 ) @ R2 )
         => ( X2 = Y3 ) ) ) ) ).

% antisymD
thf(fact_1151_antisym__empty,axiom,
    antisym_on_o @ top_top_set_o @ bot_bo7073875226086086771od_o_o ).

% antisym_empty
thf(fact_1152_antisym__empty,axiom,
    antisym_on_nat @ top_top_set_nat @ bot_bo2099793752762293965at_nat ).

% antisym_empty
thf(fact_1153_antisym__Id__on,axiom,
    ! [A: set_o] : ( antisym_on_o @ top_top_set_o @ ( id_on_o @ A ) ) ).

% antisym_Id_on
thf(fact_1154_antisym__Id__on,axiom,
    ! [A: set_nat] : ( antisym_on_nat @ top_top_set_nat @ ( id_on_nat @ A ) ) ).

% antisym_Id_on
thf(fact_1155_inter__Set__filter,axiom,
    ! [B3: set_list_Sum_sum_a_c,A: set_list_Sum_sum_a_c] :
      ( ( finite3830171671145977038um_a_c @ B3 )
     => ( ( inf_in1719665111139814143um_a_c @ A @ B3 )
        = ( filter8929528674883905222um_a_c
          @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1156_inter__Set__filter,axiom,
    ! [B3: set_o,A: set_o] :
      ( ( finite_finite_o @ B3 )
     => ( ( inf_inf_set_o @ A @ B3 )
        = ( filter_o
          @ ^ [X: $o] : ( member_o @ X @ A )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1157_inter__Set__filter,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = ( filter_nat
          @ ^ [X: nat] : ( member_nat @ X @ A )
          @ B3 ) ) ) ).

% inter_Set_filter
thf(fact_1158_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_o,X2: set_o,Y3: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ X2 )
        = bot_bot_set_o )
     => ( ( ( sup_sup_set_o @ A2 @ X2 )
          = top_top_set_o )
       => ( ( ( inf_inf_set_o @ A2 @ Y3 )
            = bot_bot_set_o )
         => ( ( ( sup_sup_set_o @ A2 @ Y3 )
              = top_top_set_o )
           => ( X2 = Y3 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1159_boolean__algebra_Ocomplement__unique,axiom,
    ! [A2: set_nat,X2: set_nat,Y3: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ X2 )
        = bot_bot_set_nat )
     => ( ( ( sup_sup_set_nat @ A2 @ X2 )
          = top_top_set_nat )
       => ( ( ( inf_inf_set_nat @ A2 @ Y3 )
            = bot_bot_set_nat )
         => ( ( ( sup_sup_set_nat @ A2 @ Y3 )
              = top_top_set_nat )
           => ( X2 = Y3 ) ) ) ) ) ).

% boolean_algebra.complement_unique
thf(fact_1160_antisym__singleton,axiom,
    ! [X2: product_prod_o_o] : ( antisym_on_o @ top_top_set_o @ ( insert6201435330877294327od_o_o @ X2 @ bot_bo7073875226086086771od_o_o ) ) ).

% antisym_singleton
thf(fact_1161_antisym__singleton,axiom,
    ! [X2: product_prod_nat_nat] : ( antisym_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).

% antisym_singleton
thf(fact_1162_antisym__Restr,axiom,
    ! [R2: set_Product_prod_o_o,A: set_o] :
      ( ( antisym_on_o @ top_top_set_o @ R2 )
     => ( antisym_on_o @ top_top_set_o
        @ ( inf_in4898592226082374645od_o_o @ R2
          @ ( product_Sigma_o_o @ A
            @ ^ [Uu: $o] : A ) ) ) ) ).

% antisym_Restr
thf(fact_1163_antisym__Restr,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,A: set_nat] :
      ( ( antisym_on_nat @ top_top_set_nat @ R2 )
     => ( antisym_on_nat @ top_top_set_nat
        @ ( inf_in2572325071724192079at_nat @ R2
          @ ( produc457027306803732586at_nat @ A
            @ ^ [Uu: nat] : A ) ) ) ) ).

% antisym_Restr
thf(fact_1164_inf__Int__eq,axiom,
    ! [R3: set_list_Sum_sum_a_c,S: set_list_Sum_sum_a_c] :
      ( ( inf_in6430343424421996102_a_c_o
        @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ R3 )
        @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ S ) )
      = ( ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ ( inf_in1719665111139814143um_a_c @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1165_inf__Int__eq,axiom,
    ! [R3: set_o,S: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X: $o] : ( member_o @ X @ R3 )
        @ ^ [X: $o] : ( member_o @ X @ S ) )
      = ( ^ [X: $o] : ( member_o @ X @ ( inf_inf_set_o @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1166_inf__Int__eq,axiom,
    ! [R3: set_nat,S: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R3 )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq
thf(fact_1167_inf__set__def,axiom,
    ( inf_in1719665111139814143um_a_c
    = ( ^ [A4: set_list_Sum_sum_a_c,B5: set_list_Sum_sum_a_c] :
          ( collec8219452656984879116um_a_c
          @ ( inf_in6430343424421996102_a_c_o
            @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ A4 )
            @ ^ [X: list_Sum_sum_a_c] : ( member7772695417316360142um_a_c @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_1168_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A4: set_o,B5: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X: $o] : ( member_o @ X @ A4 )
            @ ^ [X: $o] : ( member_o @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_1169_inf__set__def,axiom,
    ( inf_in5970604416696682206um_a_c
    = ( ^ [A4: set_nat_Sum_sum_a_c,B5: set_nat_Sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ( inf_in139034771126963495_a_c_o
            @ ^ [X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ A4 )
            @ ^ [X: nat > sum_sum_a_c] : ( member4884986500679352621um_a_c @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_1170_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A4: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A4 )
            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_1171_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A4 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_1172_above__def,axiom,
    ( order_192891853552807446um_a_c
    = ( ^ [R4: set_Pr1218573636708433773um_a_c,A5: nat > sum_sum_a_c] :
          ( collec5227572641185395563um_a_c
          @ ^ [B4: nat > sum_sum_a_c] : ( member1490001167679721142um_a_c @ ( produc7129626048164379717um_a_c @ A5 @ B4 ) @ R4 ) ) ) ) ).

% above_def
thf(fact_1173_above__def,axiom,
    ( order_above_nat
    = ( ^ [R4: set_Pr1261947904930325089at_nat,A5: nat] :
          ( collect_nat
          @ ^ [B4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R4 ) ) ) ) ).

% above_def
thf(fact_1174_above__def,axiom,
    ( order_793026671032835073at_nat
    = ( ^ [R4: set_Pr8693737435421807431at_nat,A5: product_prod_nat_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [B4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ R4 ) ) ) ) ).

% above_def
thf(fact_1175_inf__Sup,axiom,
    ! [A2: $o,B3: set_o] :
      ( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ B3 ) )
      = ( complete_Sup_Sup_o @ ( image_o_o2 @ ( inf_inf_o @ A2 ) @ B3 ) ) ) ).

% inf_Sup
thf(fact_1176_Sup__inf,axiom,
    ! [B3: set_o,A2: $o] :
      ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B3 ) @ A2 )
      = ( complete_Sup_Sup_o
        @ ( image_o_o2
          @ ^ [B4: $o] : ( inf_inf_o @ B4 @ A2 )
          @ B3 ) ) ) ).

% Sup_inf
thf(fact_1177_Fpow__Pow__finite,axiom,
    ( finite_Fpow_nat
    = ( ^ [A4: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A4 ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).

% Fpow_Pow_finite
thf(fact_1178_subsetI,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( member7772695417316360142um_a_c @ X3 @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ A @ B3 ) ) ).

% subsetI
thf(fact_1179_subsetI,axiom,
    ! [A: set_o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ( member_o @ X3 @ B3 ) )
     => ( ord_less_eq_set_o @ A @ B3 ) ) ).

% subsetI
thf(fact_1180_subsetI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B3 ) )
     => ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% subsetI
thf(fact_1181_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1182_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_1183_insert__subset,axiom,
    ! [X2: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ ( insert7520310754158225127um_a_c @ X2 @ A ) @ B3 )
      = ( ( member7772695417316360142um_a_c @ X2 @ B3 )
        & ( ord_le9174549834299225549um_a_c @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_1184_insert__subset,axiom,
    ! [X2: $o,A: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A ) @ B3 )
      = ( ( member_o @ X2 @ B3 )
        & ( ord_less_eq_set_o @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_1185_insert__subset,axiom,
    ! [X2: nat,A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B3 )
      = ( ( member_nat @ X2 @ B3 )
        & ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_1186_singleton__insert__inj__eq,axiom,
    ! [B: $o,A2: $o,A: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1187_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B: $o] :
      ( ( ( insert_o @ A2 @ A )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1188_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_1189_Int__Collect__mono,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,P: list_Sum_sum_a_c > $o,Q: list_Sum_sum_a_c > $o] :
      ( ( ord_le9174549834299225549um_a_c @ A @ B3 )
     => ( ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le9174549834299225549um_a_c @ ( inf_in1719665111139814143um_a_c @ A @ ( collec8219452656984879116um_a_c @ P ) ) @ ( inf_in1719665111139814143um_a_c @ B3 @ ( collec8219452656984879116um_a_c @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1190_Int__Collect__mono,axiom,
    ! [A: set_o,B3: set_o,P: $o > $o,Q: $o > $o] :
      ( ( ord_less_eq_set_o @ A @ B3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B3 @ ( collect_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1191_Int__Collect__mono,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c,P: ( nat > sum_sum_a_c ) > $o,Q: ( nat > sum_sum_a_c ) > $o] :
      ( ( ord_le4530483805337101228um_a_c @ A @ B3 )
     => ( ! [X3: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le4530483805337101228um_a_c @ ( inf_in5970604416696682206um_a_c @ A @ ( collec5227572641185395563um_a_c @ P ) ) @ ( inf_in5970604416696682206um_a_c @ B3 @ ( collec5227572641185395563um_a_c @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1192_Int__Collect__mono,axiom,
    ! [A: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1193_Int__Collect__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A @ B3 )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A )
           => ( ( P @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_1194_image__Int__subset,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c] : ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ ( inf_in5970604416696682206um_a_c @ A @ B3 ) ) @ ( inf_in1719665111139814143um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ ( image_7318554134589591124um_a_c @ F @ B3 ) ) ) ).

% image_Int_subset
thf(fact_1195_SUP__eqI,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,X2: $o] :
      ( ! [I3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X2 ) )
     => ( ! [Y4: $o] :
            ( ! [I5: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ I5 @ A )
               => ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
           => ( ord_less_eq_o @ X2 @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_1196_SUP__eqI,axiom,
    ! [A: set_o,F: $o > $o,X2: $o] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X2 ) )
     => ( ! [Y4: $o] :
            ( ! [I5: $o] :
                ( ( member_o @ I5 @ A )
               => ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
           => ( ord_less_eq_o @ X2 @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_1197_SUP__eqI,axiom,
    ! [A: set_nat,F: nat > $o,X2: $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X2 ) )
     => ( ! [Y4: $o] :
            ( ! [I5: nat] :
                ( ( member_nat @ I5 @ A )
               => ( ord_less_eq_o @ ( F @ I5 ) @ Y4 ) )
           => ( ord_less_eq_o @ X2 @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) )
          = X2 ) ) ) ).

% SUP_eqI
thf(fact_1198_SUP__least,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,U2: $o] :
      ( ! [I3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) @ U2 ) ) ).

% SUP_least
thf(fact_1199_SUP__least,axiom,
    ! [A: set_o,F: $o > $o,U2: $o] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ U2 ) ) ).

% SUP_least
thf(fact_1200_SUP__least,axiom,
    ! [A: set_nat,F: nat > $o,U2: $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U2 ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) @ U2 ) ) ).

% SUP_least
thf(fact_1201_SUP__upper,axiom,
    ! [I4: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ I4 @ A )
     => ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ) ).

% SUP_upper
thf(fact_1202_SUP__upper,axiom,
    ! [I4: $o,A: set_o,F: $o > $o] :
      ( ( member_o @ I4 @ A )
     => ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) ) ) ).

% SUP_upper
thf(fact_1203_SUP__upper,axiom,
    ! [I4: nat,A: set_nat,F: nat > $o] :
      ( ( member_nat @ I4 @ A )
     => ( ord_less_eq_o @ ( F @ I4 ) @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) ) ) ).

% SUP_upper
thf(fact_1204_SUP__upper2,axiom,
    ! [I4: list_Sum_sum_a_c,A: set_list_Sum_sum_a_c,U2: $o,F: list_Sum_sum_a_c > $o] :
      ( ( member7772695417316360142um_a_c @ I4 @ A )
     => ( ( ord_less_eq_o @ U2 @ ( F @ I4 ) )
       => ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) ) ) ) ).

% SUP_upper2
thf(fact_1205_SUP__upper2,axiom,
    ! [I4: $o,A: set_o,U2: $o,F: $o > $o] :
      ( ( member_o @ I4 @ A )
     => ( ( ord_less_eq_o @ U2 @ ( F @ I4 ) )
       => ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) ) ) ) ).

% SUP_upper2
thf(fact_1206_SUP__upper2,axiom,
    ! [I4: nat,A: set_nat,U2: $o,F: nat > $o] :
      ( ( member_nat @ I4 @ A )
     => ( ( ord_less_eq_o @ U2 @ ( F @ I4 ) )
       => ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) ) ) ) ).

% SUP_upper2
thf(fact_1207_cSUP__subset__mono,axiom,
    ! [A: set_list_Sum_sum_a_c,G: list_Sum_sum_a_c > $o,B3: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o] :
      ( ( A != bot_bo3453284597459734017um_a_c )
     => ( ( condit5488710616941104124bove_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) )
       => ( ( ord_le9174549834299225549um_a_c @ A @ B3 )
         => ( ! [X3: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ X3 @ A )
               => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
           => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1208_cSUP__subset__mono,axiom,
    ! [A: set_nat,G: nat > $o,B3: set_nat,F: nat > $o] :
      ( ( A != bot_bot_set_nat )
     => ( ( condit5488710616941104124bove_o @ ( image_nat_o2 @ G @ B3 ) )
       => ( ( ord_less_eq_set_nat @ A @ B3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A )
               => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
           => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ G @ B3 ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1209_cSUP__subset__mono,axiom,
    ! [A: set_o,G: $o > $o,B3: set_o,F: $o > $o] :
      ( ( A != bot_bot_set_o )
     => ( ( condit5488710616941104124bove_o @ ( image_o_o2 @ G @ B3 ) )
       => ( ( ord_less_eq_set_o @ A @ B3 )
         => ( ! [X3: $o] :
                ( ( member_o @ X3 @ A )
               => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
           => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B3 ) ) ) ) ) ) ) ).

% cSUP_subset_mono
thf(fact_1210_image__Pow__mono,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ B3 )
     => ( ord_le6505510949241631619um_a_c @ ( image_8538515752806177344um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ ( pow_nat_Sum_sum_a_c @ A ) ) @ ( pow_list_Sum_sum_a_c @ B3 ) ) ) ).

% image_Pow_mono
thf(fact_1211_image__Fpow__mono,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ B3 )
     => ( ord_le6505510949241631619um_a_c @ ( image_8538515752806177344um_a_c @ ( image_7318554134589591124um_a_c @ F ) @ ( finite6640210490720379440um_a_c @ A ) ) @ ( finite1080387919766963153um_a_c @ B3 ) ) ) ).

% image_Fpow_mono
thf(fact_1212_all__subset__image,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: set_list_Sum_sum_a_c > $o] :
      ( ( ! [B5: set_list_Sum_sum_a_c] :
            ( ( ord_le9174549834299225549um_a_c @ B5 @ ( image_7318554134589591124um_a_c @ F @ A ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat_Sum_sum_a_c] :
            ( ( ord_le4530483805337101228um_a_c @ B5 @ A )
           => ( P @ ( image_7318554134589591124um_a_c @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_1213_image__mono,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( ord_le4530483805337101228um_a_c @ A @ B3 )
     => ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ ( image_7318554134589591124um_a_c @ F @ B3 ) ) ) ).

% image_mono
thf(fact_1214_image__subsetI,axiom,
    ! [A: set_nat_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: nat > sum_sum_a_c] :
          ( ( member4884986500679352621um_a_c @ X3 @ A )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1215_image__subsetI,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_8132764781264612725um_a_c @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1216_image__subsetI,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,B3: set_o] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_1429702505460152410_a_c_o @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1217_image__subsetI,axiom,
    ! [A: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > nat,B3: set_nat] :
      ( ! [X3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_7712311756382514062_c_nat @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1218_image__subsetI,axiom,
    ! [A: set_o,F: $o > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_1489810555362023050um_a_c @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1219_image__subsetI,axiom,
    ! [A: set_o,F: $o > $o,B3: set_o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_o_o2 @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1220_image__subsetI,axiom,
    ! [A: set_o,F: $o > nat,B3: set_nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_o_nat2 @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1221_image__subsetI,axiom,
    ! [A: set_nat,F: nat > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_1048630792592356622um_a_c @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1222_image__subsetI,axiom,
    ! [A: set_nat,F: nat > $o,B3: set_o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1223_image__subsetI,axiom,
    ! [A: set_nat,F: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ A ) @ B3 ) ) ).

% image_subsetI
thf(fact_1224_subset__imageE,axiom,
    ! [B3: set_list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ B3 @ ( image_7318554134589591124um_a_c @ F @ A ) )
     => ~ ! [C4: set_nat_Sum_sum_a_c] :
            ( ( ord_le4530483805337101228um_a_c @ C4 @ A )
           => ( B3
             != ( image_7318554134589591124um_a_c @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_1225_image__subset__iff,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ A ) @ B3 )
      = ( ! [X: nat > sum_sum_a_c] :
            ( ( member4884986500679352621um_a_c @ X @ A )
           => ( member7772695417316360142um_a_c @ ( F @ X ) @ B3 ) ) ) ) ).

% image_subset_iff
thf(fact_1226_subset__image__iff,axiom,
    ! [B3: set_list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ B3 @ ( image_7318554134589591124um_a_c @ F @ A ) )
      = ( ? [AA: set_nat_Sum_sum_a_c] :
            ( ( ord_le4530483805337101228um_a_c @ AA @ A )
            & ( B3
              = ( image_7318554134589591124um_a_c @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_1227_SUP__subset__mono,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,G: list_Sum_sum_a_c > $o] :
      ( ( ord_le9174549834299225549um_a_c @ A @ B3 )
     => ( ! [X3: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ X3 @ A )
           => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1228_SUP__subset__mono,axiom,
    ! [A: set_o,B3: set_o,F: $o > $o,G: $o > $o] :
      ( ( ord_less_eq_set_o @ A @ B3 )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ A )
           => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1229_SUP__subset__mono,axiom,
    ! [A: set_nat,B3: set_nat,F: nat > $o,G: nat > $o] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ord_less_eq_o @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_1230_image__Collect__subsetI,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: nat > sum_sum_a_c] :
          ( ( P @ X3 )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ ( collec5227572641185395563um_a_c @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1231_image__Collect__subsetI,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,F: ( nat > sum_sum_a_c ) > $o,B3: set_o] :
      ( ! [X3: nat > sum_sum_a_c] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_3393300230255336507_a_c_o @ F @ ( collec5227572641185395563um_a_c @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1232_image__Collect__subsetI,axiom,
    ! [P: ( nat > sum_sum_a_c ) > $o,F: ( nat > sum_sum_a_c ) > nat,B3: set_nat] :
      ( ! [X3: nat > sum_sum_a_c] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_930363360967426541_c_nat @ F @ ( collec5227572641185395563um_a_c @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1233_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_1048630792592356622um_a_c @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1234_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > $o,B3: set_o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1235_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F: nat > nat,B3: set_nat] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ ( collect_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1236_image__Collect__subsetI,axiom,
    ! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( member7772695417316360142um_a_c @ ( F @ X3 ) @ B3 ) )
     => ( ord_le9174549834299225549um_a_c @ ( image_6296180510578743155um_a_c @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1237_image__Collect__subsetI,axiom,
    ! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > $o,B3: set_o] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( member_o @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_o @ ( image_3693632289388996572_nat_o @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1238_image__Collect__subsetI,axiom,
    ! [P: product_prod_nat_nat > $o,F: product_prod_nat_nat > nat,B3: set_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ( ( P @ X3 )
         => ( member_nat @ ( F @ X3 ) @ B3 ) )
     => ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ ( collec3392354462482085612at_nat @ P ) ) @ B3 ) ) ).

% image_Collect_subsetI
thf(fact_1239_Sup__upper2,axiom,
    ! [U2: $o,A: set_o,V: $o] :
      ( ( member_o @ U2 @ A )
     => ( ( ord_less_eq_o @ V @ U2 )
       => ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).

% Sup_upper2
thf(fact_1240_Sup__le__iff,axiom,
    ! [A: set_o,B: $o] :
      ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B )
      = ( ! [X: $o] :
            ( ( member_o @ X @ A )
           => ( ord_less_eq_o @ X @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_1241_Sup__upper,axiom,
    ! [X2: $o,A: set_o] :
      ( ( member_o @ X2 @ A )
     => ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A ) ) ) ).

% Sup_upper
thf(fact_1242_Sup__least,axiom,
    ! [A: set_o,Z: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ A )
         => ( ord_less_eq_o @ X3 @ Z ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z ) ) ).

% Sup_least
thf(fact_1243_Sup__mono,axiom,
    ! [A: set_o,B3: set_o] :
      ( ! [A3: $o] :
          ( ( member_o @ A3 @ A )
         => ? [X4: $o] :
              ( ( member_o @ X4 @ B3 )
              & ( ord_less_eq_o @ A3 @ X4 ) ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ).

% Sup_mono
thf(fact_1244_Sup__eqI,axiom,
    ! [A: set_o,X2: $o] :
      ( ! [Y4: $o] :
          ( ( member_o @ Y4 @ A )
         => ( ord_less_eq_o @ Y4 @ X2 ) )
     => ( ! [Y4: $o] :
            ( ! [Z5: $o] :
                ( ( member_o @ Z5 @ A )
               => ( ord_less_eq_o @ Z5 @ Y4 ) )
           => ( ord_less_eq_o @ X2 @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ A )
          = X2 ) ) ) ).

% Sup_eqI
thf(fact_1245_Sup__subset__mono,axiom,
    ! [A: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A @ B3 )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ).

% Sup_subset_mono
thf(fact_1246_range__subsetD,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,I4: nat > sum_sum_a_c] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_7318554134589591124um_a_c @ F @ top_to3439567294830504444um_a_c ) @ B3 )
     => ( member7772695417316360142um_a_c @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1247_range__subsetD,axiom,
    ! [F: $o > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,I4: $o] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_1489810555362023050um_a_c @ F @ top_top_set_o ) @ B3 )
     => ( member7772695417316360142um_a_c @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1248_range__subsetD,axiom,
    ! [F: $o > $o,B3: set_o,I4: $o] :
      ( ( ord_less_eq_set_o @ ( image_o_o2 @ F @ top_top_set_o ) @ B3 )
     => ( member_o @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1249_range__subsetD,axiom,
    ! [F: $o > nat,B3: set_nat,I4: $o] :
      ( ( ord_less_eq_set_nat @ ( image_o_nat2 @ F @ top_top_set_o ) @ B3 )
     => ( member_nat @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1250_range__subsetD,axiom,
    ! [F: nat > list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,I4: nat] :
      ( ( ord_le9174549834299225549um_a_c @ ( image_1048630792592356622um_a_c @ F @ top_top_set_nat ) @ B3 )
     => ( member7772695417316360142um_a_c @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1251_range__subsetD,axiom,
    ! [F: nat > $o,B3: set_o,I4: nat] :
      ( ( ord_less_eq_set_o @ ( image_nat_o2 @ F @ top_top_set_nat ) @ B3 )
     => ( member_o @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1252_range__subsetD,axiom,
    ! [F: nat > nat,B3: set_nat,I4: nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat2 @ F @ top_top_set_nat ) @ B3 )
     => ( member_nat @ ( F @ I4 ) @ B3 ) ) ).

% range_subsetD
thf(fact_1253_SUP__eq,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,F: list_Sum_sum_a_c > $o,G: list_Sum_sum_a_c > $o] :
      ( ! [I3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ I3 @ A )
         => ? [X4: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ J2 @ B3 )
           => ? [X4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1254_SUP__eq,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_o,F: list_Sum_sum_a_c > $o,G: $o > $o] :
      ( ! [I3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ I3 @ A )
         => ? [X4: $o] :
              ( ( member_o @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: $o] :
            ( ( member_o @ J2 @ B3 )
           => ? [X4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1255_SUP__eq,axiom,
    ! [A: set_list_Sum_sum_a_c,B3: set_nat,F: list_Sum_sum_a_c > $o,G: nat > $o] :
      ( ! [I3: list_Sum_sum_a_c] :
          ( ( member7772695417316360142um_a_c @ I3 @ A )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B3 )
           => ? [X4: list_Sum_sum_a_c] :
                ( ( member7772695417316360142um_a_c @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1256_SUP__eq,axiom,
    ! [A: set_o,B3: set_list_Sum_sum_a_c,F: $o > $o,G: list_Sum_sum_a_c > $o] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A )
         => ? [X4: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ J2 @ B3 )
           => ? [X4: $o] :
                ( ( member_o @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1257_SUP__eq,axiom,
    ! [A: set_o,B3: set_o,F: $o > $o,G: $o > $o] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A )
         => ? [X4: $o] :
              ( ( member_o @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: $o] :
            ( ( member_o @ J2 @ B3 )
           => ? [X4: $o] :
                ( ( member_o @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1258_SUP__eq,axiom,
    ! [A: set_o,B3: set_nat,F: $o > $o,G: nat > $o] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B3 )
           => ? [X4: $o] :
                ( ( member_o @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_o_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1259_SUP__eq,axiom,
    ! [A: set_nat,B3: set_list_Sum_sum_a_c,F: nat > $o,G: list_Sum_sum_a_c > $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A )
         => ? [X4: list_Sum_sum_a_c] :
              ( ( member7772695417316360142um_a_c @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: list_Sum_sum_a_c] :
            ( ( member7772695417316360142um_a_c @ J2 @ B3 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_1429702505460152410_a_c_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1260_SUP__eq,axiom,
    ! [A: set_nat,B3: set_o,F: nat > $o,G: $o > $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A )
         => ? [X4: $o] :
              ( ( member_o @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: $o] :
            ( ( member_o @ J2 @ B3 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_o_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1261_SUP__eq,axiom,
    ! [A: set_nat,B3: set_nat,F: nat > $o,G: nat > $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
     => ( ! [J2: nat] :
            ( ( member_nat @ J2 @ B3 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A )
                & ( ord_less_eq_o @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o2 @ F @ A ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o2 @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_1262_finite__surj,axiom,
    ! [A: set_nat_Sum_sum_a_c,B3: set_list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c] :
      ( ( finite541027974130308653um_a_c @ A )
     => ( ( ord_le9174549834299225549um_a_c @ B3 @ ( image_7318554134589591124um_a_c @ F @ A ) )
       => ( finite3830171671145977038um_a_c @ B3 ) ) ) ).

% finite_surj
thf(fact_1263_finite__surj,axiom,
    ! [A: set_nat,B3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat2 @ F @ A ) )
       => ( finite_finite_nat @ B3 ) ) ) ).

% finite_surj
thf(fact_1264_finite__subset__image,axiom,
    ! [B3: set_list_Sum_sum_a_c,F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c] :
      ( ( finite3830171671145977038um_a_c @ B3 )
     => ( ( ord_le9174549834299225549um_a_c @ B3 @ ( image_7318554134589591124um_a_c @ F @ A ) )
       => ? [C4: set_nat_Sum_sum_a_c] :
            ( ( ord_le4530483805337101228um_a_c @ C4 @ A )
            & ( finite541027974130308653um_a_c @ C4 )
            & ( B3
              = ( image_7318554134589591124um_a_c @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1265_finite__subset__image,axiom,
    ! [B3: set_nat,F: nat > nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat2 @ F @ A ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A )
            & ( finite_finite_nat @ C4 )
            & ( B3
              = ( image_nat_nat2 @ F @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1266_ex__finite__subset__image,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: set_list_Sum_sum_a_c > $o] :
      ( ( ? [B5: set_list_Sum_sum_a_c] :
            ( ( finite3830171671145977038um_a_c @ B5 )
            & ( ord_le9174549834299225549um_a_c @ B5 @ ( image_7318554134589591124um_a_c @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat_Sum_sum_a_c] :
            ( ( finite541027974130308653um_a_c @ B5 )
            & ( ord_le4530483805337101228um_a_c @ B5 @ A )
            & ( P @ ( image_7318554134589591124um_a_c @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1267_ex__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A ) )
            & ( P @ B5 ) ) )
      = ( ? [B5: set_nat] :
            ( ( finite_finite_nat @ B5 )
            & ( ord_less_eq_set_nat @ B5 @ A )
            & ( P @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1268_all__finite__subset__image,axiom,
    ! [F: ( nat > sum_sum_a_c ) > list_Sum_sum_a_c,A: set_nat_Sum_sum_a_c,P: set_list_Sum_sum_a_c > $o] :
      ( ( ! [B5: set_list_Sum_sum_a_c] :
            ( ( ( finite3830171671145977038um_a_c @ B5 )
              & ( ord_le9174549834299225549um_a_c @ B5 @ ( image_7318554134589591124um_a_c @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat_Sum_sum_a_c] :
            ( ( ( finite541027974130308653um_a_c @ B5 )
              & ( ord_le4530483805337101228um_a_c @ B5 @ A ) )
           => ( P @ ( image_7318554134589591124um_a_c @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1269_all__finite__subset__image,axiom,
    ! [F: nat > nat,A: set_nat,P: set_nat > $o] :
      ( ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat2 @ F @ A ) ) )
           => ( P @ B5 ) ) )
      = ( ! [B5: set_nat] :
            ( ( ( finite_finite_nat @ B5 )
              & ( ord_less_eq_set_nat @ B5 @ A ) )
           => ( P @ ( image_nat_nat2 @ F @ B5 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1270_less__eq__Sup,axiom,
    ! [A: set_o,U2: $o] :
      ( ! [V3: $o] :
          ( ( member_o @ V3 @ A )
         => ( ord_less_eq_o @ U2 @ V3 ) )
     => ( ( A != bot_bot_set_o )
       => ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A ) ) ) ) ).

% less_eq_Sup
thf(fact_1271_subset__singleton__iff,axiom,
    ! [X7: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X7 = bot_bot_set_o )
        | ( X7
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_1272_subset__singletonD,axiom,
    ! [A: set_o,X2: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_1273_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_1274_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_1275_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum

% Helper facts (7)
thf(help_If_2_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( if_set_o @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_I_Eo_J_T,axiom,
    ! [X2: set_o,Y3: set_o] :
      ( ( if_set_o @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( if_set_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y3: set_nat] :
      ( ( if_set_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_T,axiom,
    ! [X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( if_list_Sum_sum_a_c @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__c_J_J_T,axiom,
    ! [X2: list_Sum_sum_a_c,Y3: list_Sum_sum_a_c] :
      ( ( if_list_Sum_sum_a_c @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( member7772695417316360142um_a_c @ vs
    @ ( proj_v4774688853359684992um_a_c
      @ ( collec5227572641185395563um_a_c
        @ ^ [Sigma4: nat > sum_sum_a_c] : ( member7772695417316360142um_a_c @ ( eval_eterms_a_c @ Sigma4 @ ts ) @ x ) )
      @ ( fv_fo_terms_list_a @ ts ) ) ) ).

%------------------------------------------------------------------------------