TPTP Problem File: SLH0966^1.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    : Safe_Range_RC/0021_Relational_Calculus/prob_01581_058661__17463690_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2092 ( 534 unt; 804 typ;   0 def)
%            Number of atoms       : 4370 (1766 equ;   0 cnn)
%            Maximal formula atoms :   37 (   3 avg)
%            Number of connectives : 16152 ( 725   ~;  79   |; 417   &;12695   @)
%                                         (   0 <=>;2236  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   7 avg)
%            Number of types       :  174 ( 173 usr)
%            Number of type conns  : 3261 (3261   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  634 ( 631 usr;  15 con; 0-4 aty)
%            Number of variables   : 5333 ( 897   ^;4176   !; 260   ?;5333   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:26:20.229
%------------------------------------------------------------------------------
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thf(ty_n_t__List__Olist_It__Nat__Onat_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_Itf__b_J,type,
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thf(ty_n_t__Set__Oset_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_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (631)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofold_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
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    member_nat_o: ( nat > $o ) > set_nat_o > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
    member_nat_a: ( nat > a ) > set_nat_a > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member8404886500659538246list_a: ( product_prod_b_nat > set_list_a ) > set_Pr5048610625260234749list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    member_list_list_a: list_list_a > set_list_list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J_M_062_I_062_It__Nat__Onat_Mtf__a_J_M_062_It__Nat__Onat_Mtf__a_J_J_J,type,
    member5591254206257022164_nat_a: produc1037807075074811947_nat_a > set_Pr4689821003595768459_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_M_062_It__Nat__Onat_Mtf__a_J_J,type,
    member9198066416134578520_nat_a: produc5835360497134304175_nat_a > set_Pr6389665502131816719_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member7820552544826472890at_nat: produc1540777390238407569at_nat > set_Pr7549493975000385137at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mtf__a_J,type,
    member6908590373215729450_nat_a: produc6459537399369964819_nat_a > set_Pr8540246825521615561_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Nat__Onat_J,type,
    member873448109036076442_a_nat: produc424395135190311811_a_nat > set_Pr6248017591093387577_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member1631564025489475386st_nat: produc4575160907756185873st_nat > set_Pr698032948827214833st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    member8189971380475638336list_a: produc7740918406629873705list_a > set_Pr3870785875473343199list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
    member2321988868818003709_nat_a: produc6033039080626123092_nat_a > set_Pr2174358626933722164_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Relational____Calculus__Oterm_Itf__a_J_J,type,
    member2722330264857718789term_a: produc7526962659421474158term_a > set_Pr3863467996586345508term_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member3782324328723991648et_nat: produc2400336064389900727et_nat > set_Pr400265656397884439et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member2637305593812847696list_a: produc3865747550613471015list_a > set_Pr5391735021705396743list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member8873588116083876704at_nat: produc7491599851749785783at_nat > set_Pr3601174868274201367at_nat > $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__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mtf__a_J,type,
    member5257926396607048452_nat_a: produc2157779926878614893_nat_a > set_Pr2123770715006448547_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member1257341626780437706et_nat: produc7380567193906779955et_nat > set_Pr8069033395972952681et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

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

thf(sy_c_member_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    member4680049679412964150la_a_b: relational_fmla_a_b > set_Re381260168593705685la_a_b > $o ).

thf(sy_c_member_001t__Relational____Calculus__Oterm_Itf__a_J,type,
    member5466445112152945266term_a: relational_term_a > set_Re5178783185447174953term_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    member_set_list_a: set_list_a > set_set_list_a > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member8857465052274545133list_a: set_set_list_a > set_set_set_list_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

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

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

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

thf(sy_v_I,type,
    i: product_prod_b_nat > set_list_a ).

thf(sy_v_Q1,type,
    q1: relational_fmla_a_b ).

thf(sy_v_Q2,type,
    q2: relational_fmla_a_b ).

% Relevant facts (1275)
thf(fact_0_fmla_Oinject_I6_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,Y61: relational_fmla_a_b,Y62: relational_fmla_a_b] :
      ( ( ( relational_Disj_a_b @ X61 @ X62 )
        = ( relational_Disj_a_b @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% fmla.inject(6)
thf(fact_1_sat_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) @ I @ Sigma )
      = ( ( relational_sat_a_b @ Phi @ I @ Sigma )
        | ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).

% sat.simps(6)
thf(fact_2_sat__fun__upd,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a,Z: a] :
      ( ~ ( member_nat @ N @ ( relational_fv_a_b @ Q ) )
     => ( ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N @ Z ) )
        = ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).

% sat_fun_upd
thf(fact_3_sat__fv__cong,axiom,
    ! [Phi: relational_fmla_a_b,Sigma: nat > a,Sigma2: nat > a,I: product_prod_b_nat > set_list_a] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ ( relational_fv_a_b @ Phi ) )
         => ( ( Sigma @ N2 )
            = ( Sigma2 @ N2 ) ) )
     => ( ( relational_sat_a_b @ Phi @ I @ Sigma )
        = ( relational_sat_a_b @ Phi @ I @ Sigma2 ) ) ) ).

% sat_fv_cong
thf(fact_4_eval__deep__def,axiom,
    ( relational_eval_a_b
    = ( ^ [Q2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
          ( collect_list_a
          @ ^ [Ds: list_a] :
              ( ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q2 ) ) )
                = ( size_size_list_a @ Ds ) )
              & ? [Sigma3: nat > a] :
                  ( relational_sat_a_b @ Q2 @ I2
                  @ ( fold_P5280602285094830901_nat_a
                    @ ( produc2909000522608705447_nat_a
                      @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
                    @ ( zip_nat_a @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q2 ) ) @ Ds )
                    @ Sigma3 ) ) ) ) ) ) ).

% eval_deep_def
thf(fact_5_eval__on__def,axiom,
    ( relati8814510239606734169on_a_b
    = ( ^ [V: set_nat,Q2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a] :
          ( collect_list_a
          @ ^ [Ds: list_a] :
              ( ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ V ) )
                = ( size_size_list_a @ Ds ) )
              & ? [Sigma3: nat > a] :
                  ( relational_sat_a_b @ Q2 @ I2
                  @ ( fold_P5280602285094830901_nat_a
                    @ ( produc2909000522608705447_nat_a
                      @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
                    @ ( zip_nat_a @ ( linord2614967742042102400et_nat @ V ) @ Ds )
                    @ Sigma3 ) ) ) ) ) ) ).

% eval_on_def
thf(fact_6_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_7_finite__Collect__bounded__ex,axiom,
    ! [P: list_a > $o,Q: nat > list_a > $o] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: list_a] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: list_a] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_8_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: list_a > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [X: list_a] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [X: list_a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_9_finite__Collect__bounded__ex,axiom,
    ! [P: list_a > $o,Q: list_a > list_a > $o] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [X: list_a] :
              ? [Y: list_a] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: list_a] :
              ( ( P @ Y )
             => ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [X: list_a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_10_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: a > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_a
          @ ( collect_a
            @ ^ [X: a] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_a
                @ ( collect_a
                  @ ^ [X: a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_11_finite__Collect__bounded__ex,axiom,
    ! [P: a > $o,Q: nat > a > $o] :
      ( ( finite_finite_a @ ( collect_a @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: a] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: a] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_12_finite__Collect__bounded__ex,axiom,
    ! [P: a > $o,Q: a > a > $o] :
      ( ( finite_finite_a @ ( collect_a @ P ) )
     => ( ( finite_finite_a
          @ ( collect_a
            @ ^ [X: a] :
              ? [Y: a] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: a] :
              ( ( P @ Y )
             => ( finite_finite_a
                @ ( collect_a
                  @ ^ [X: a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_13_finite__Collect__bounded__ex,axiom,
    ! [P: list_a > $o,Q: a > list_a > $o] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_a
          @ ( collect_a
            @ ^ [X: a] :
              ? [Y: list_a] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: list_a] :
              ( ( P @ Y )
             => ( finite_finite_a
                @ ( collect_a
                  @ ^ [X: a] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_14_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: list_nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite8100373058378681591st_nat
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite8100373058378681591st_nat
                @ ( collect_list_nat
                  @ ^ [X: list_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_15_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: set_nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite1152437895449049373et_nat
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite1152437895449049373et_nat
                @ ( collect_set_nat
                  @ ^ [X: set_nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_16_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_17_finite__Collect__conjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ( finite_finite_a @ ( collect_a @ P ) )
        | ( finite_finite_a @ ( collect_a @ Q ) ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [X: a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_18_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_19_finite__Collect__conjI,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
        | ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X: list_list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_20_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_21_finite__Collect__conjI,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
        | ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X: set_list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_22_finite__Collect__conjI,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
        | ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_23_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_24_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_25_finite__Collect__disjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X: a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P ) )
        & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_26_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_27_finite__Collect__disjI,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X: list_list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
        & ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_28_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_29_finite__Collect__disjI,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X: set_list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
        & ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_30_finite__Collect__disjI,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
        & ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_31_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_32_fun__upd__upd,axiom,
    ! [F2: a > a,X2: a,Y2: a,Z: a] :
      ( ( fun_upd_a_a @ ( fun_upd_a_a @ F2 @ X2 @ Y2 ) @ X2 @ Z )
      = ( fun_upd_a_a @ F2 @ X2 @ Z ) ) ).

% fun_upd_upd
thf(fact_33_fun__upd__upd,axiom,
    ! [F2: a > nat,X2: a,Y2: nat,Z: nat] :
      ( ( fun_upd_a_nat @ ( fun_upd_a_nat @ F2 @ X2 @ Y2 ) @ X2 @ Z )
      = ( fun_upd_a_nat @ F2 @ X2 @ Z ) ) ).

% fun_upd_upd
thf(fact_34_fun__upd__upd,axiom,
    ! [F2: nat > nat,X2: nat,Y2: nat,Z: nat] :
      ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ F2 @ X2 @ Y2 ) @ X2 @ Z )
      = ( fun_upd_nat_nat @ F2 @ X2 @ Z ) ) ).

% fun_upd_upd
thf(fact_35_fun__upd__upd,axiom,
    ! [F2: nat > a,X2: nat,Y2: a,Z: a] :
      ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ F2 @ X2 @ Y2 ) @ X2 @ Z )
      = ( fun_upd_nat_a @ F2 @ X2 @ Z ) ) ).

% fun_upd_upd
thf(fact_36_fun__upd__triv,axiom,
    ! [F2: a > a,X2: a] :
      ( ( fun_upd_a_a @ F2 @ X2 @ ( F2 @ X2 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_37_fun__upd__triv,axiom,
    ! [F2: a > nat,X2: a] :
      ( ( fun_upd_a_nat @ F2 @ X2 @ ( F2 @ X2 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_38_fun__upd__triv,axiom,
    ! [F2: nat > nat,X2: nat] :
      ( ( fun_upd_nat_nat @ F2 @ X2 @ ( F2 @ X2 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_39_fun__upd__triv,axiom,
    ! [F2: nat > a,X2: nat] :
      ( ( fun_upd_nat_a @ F2 @ X2 @ ( F2 @ X2 ) )
      = F2 ) ).

% fun_upd_triv
thf(fact_40_fun__upd__apply,axiom,
    ( fun_upd_a_a
    = ( ^ [F: a > a,X: a,Y: a,Z2: a] : ( if_a @ ( Z2 = X ) @ Y @ ( F @ Z2 ) ) ) ) ).

% fun_upd_apply
thf(fact_41_fun__upd__apply,axiom,
    ( fun_upd_a_nat
    = ( ^ [F: a > nat,X: a,Y: nat,Z2: a] : ( if_nat @ ( Z2 = X ) @ Y @ ( F @ Z2 ) ) ) ) ).

% fun_upd_apply
thf(fact_42_fun__upd__apply,axiom,
    ( fun_upd_nat_nat
    = ( ^ [F: nat > nat,X: nat,Y: nat,Z2: nat] : ( if_nat @ ( Z2 = X ) @ Y @ ( F @ Z2 ) ) ) ) ).

% fun_upd_apply
thf(fact_43_fun__upd__apply,axiom,
    ( fun_upd_nat_a
    = ( ^ [F: nat > a,X: nat,Y: a,Z2: nat] : ( if_a @ ( Z2 = X ) @ Y @ ( F @ Z2 ) ) ) ) ).

% fun_upd_apply
thf(fact_44_case__prod__app,axiom,
    ( produc4481717121449037155_nat_a
    = ( ^ [F: nat > a > nat > a,X: product_prod_nat_a,Y: nat] :
          ( produc8288763603002351228at_a_a
          @ ^ [L: nat,R: a] : ( F @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_45_case__prod__app,axiom,
    ( produc7013214046051809481_a_nat
    = ( ^ [F: a > nat > ( a > nat ) > a > nat,X: product_prod_a_nat,Y: a > nat] :
          ( produc3427341041222504731_a_nat
          @ ^ [L: a,R: nat] : ( F @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_46_case__prod__app,axiom,
    ( produc2369190251411148053_a_a_a
    = ( ^ [F: a > a > ( a > a ) > a > a,X: product_prod_a_a,Y: a > a] :
          ( produc3909917034562049425_a_a_a
          @ ^ [L: a,R: a] : ( F @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_47_case__prod__app,axiom,
    ( produc2909000522608705447_nat_a
    = ( ^ [F: nat > a > ( nat > a ) > nat > a,X: product_prod_nat_a,Y: nat > a] :
          ( produc4481717121449037155_nat_a
          @ ^ [L: nat,R: a] : ( F @ L @ R @ Y )
          @ X ) ) ) ).

% case_prod_app
thf(fact_48_finite__update__induct,axiom,
    ! [F2: nat > nat,C: nat,P: ( nat > nat ) > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [A: nat] :
              ( ( F2 @ A )
             != C ) ) )
     => ( ( P
          @ ^ [A: nat] : C )
       => ( ! [A2: nat,B: nat,F3: nat > nat] :
              ( ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [C2: nat] :
                      ( ( F3 @ C2 )
                     != C ) ) )
             => ( ( ( F3 @ A2 )
                  = C )
               => ( ( B != C )
                 => ( ( P @ F3 )
                   => ( P @ ( fun_upd_nat_nat @ F3 @ A2 @ B ) ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_update_induct
thf(fact_49_finite__update__induct,axiom,
    ! [F2: a > a,C: a,P: ( a > a ) > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [A: a] :
              ( ( F2 @ A )
             != C ) ) )
     => ( ( P
          @ ^ [A: a] : C )
       => ( ! [A2: a,B: a,F3: a > a] :
              ( ( finite_finite_a
                @ ( collect_a
                  @ ^ [C2: a] :
                      ( ( F3 @ C2 )
                     != C ) ) )
             => ( ( ( F3 @ A2 )
                  = C )
               => ( ( B != C )
                 => ( ( P @ F3 )
                   => ( P @ ( fun_upd_a_a @ F3 @ A2 @ B ) ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_update_induct
thf(fact_50_finite__update__induct,axiom,
    ! [F2: a > nat,C: nat,P: ( a > nat ) > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [A: a] :
              ( ( F2 @ A )
             != C ) ) )
     => ( ( P
          @ ^ [A: a] : C )
       => ( ! [A2: a,B: nat,F3: a > nat] :
              ( ( finite_finite_a
                @ ( collect_a
                  @ ^ [C2: a] :
                      ( ( F3 @ C2 )
                     != C ) ) )
             => ( ( ( F3 @ A2 )
                  = C )
               => ( ( B != C )
                 => ( ( P @ F3 )
                   => ( P @ ( fun_upd_a_nat @ F3 @ A2 @ B ) ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_update_induct
thf(fact_51_finite__update__induct,axiom,
    ! [F2: nat > a,C: a,P: ( nat > a ) > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [A: nat] :
              ( ( F2 @ A )
             != C ) ) )
     => ( ( P
          @ ^ [A: nat] : C )
       => ( ! [A2: nat,B: a,F3: nat > a] :
              ( ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [C2: nat] :
                      ( ( F3 @ C2 )
                     != C ) ) )
             => ( ( ( F3 @ A2 )
                  = C )
               => ( ( B != C )
                 => ( ( P @ F3 )
                   => ( P @ ( fun_upd_nat_a @ F3 @ A2 @ B ) ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_update_induct
thf(fact_52_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_53_finite__image__set,axiom,
    ! [P: list_a > $o,F2: list_a > nat] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: list_a] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_54_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > list_a] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Uu: list_a] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_55_finite__image__set,axiom,
    ! [P: list_a > $o,F2: list_a > list_a] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Uu: list_a] :
            ? [X: list_a] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_56_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > a] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [Uu: a] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_57_finite__image__set,axiom,
    ! [P: a > $o,F2: a > nat] :
      ( ( finite_finite_a @ ( collect_a @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu: nat] :
            ? [X: a] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_58_finite__image__set,axiom,
    ! [P: a > $o,F2: a > a] :
      ( ( finite_finite_a @ ( collect_a @ P ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [Uu: a] :
            ? [X: a] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_59_finite__image__set,axiom,
    ! [P: list_a > $o,F2: list_a > a] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [Uu: a] :
            ? [X: list_a] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_60_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > list_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Uu: list_nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_61_finite__image__set,axiom,
    ! [P: nat > $o,F2: nat > set_nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [Uu: set_nat] :
            ? [X: nat] :
              ( ( Uu
                = ( F2 @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_62_Relational__Calculus_Oeval__def,axiom,
    ( relational_eval_a_b
    = ( ^ [Q2: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q2 ) @ Q2 ) ) ) ).

% Relational_Calculus.eval_def
thf(fact_63_finite__fv,axiom,
    ! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).

% finite_fv
thf(fact_64_finite__eval__on__Disj2D,axiom,
    ! [X3: set_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( finite_finite_nat @ X3 )
     => ( ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X3 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ I ) )
       => ( finite_finite_list_a @ ( relati8814510239606734169on_a_b @ X3 @ Q22 @ I ) ) ) ) ).

% finite_eval_on_Disj2D
thf(fact_65_fun__upd__idem__iff,axiom,
    ! [F2: a > a,X2: a,Y2: a] :
      ( ( ( fun_upd_a_a @ F2 @ X2 @ Y2 )
        = F2 )
      = ( ( F2 @ X2 )
        = Y2 ) ) ).

% fun_upd_idem_iff
thf(fact_66_fun__upd__idem__iff,axiom,
    ! [F2: a > nat,X2: a,Y2: nat] :
      ( ( ( fun_upd_a_nat @ F2 @ X2 @ Y2 )
        = F2 )
      = ( ( F2 @ X2 )
        = Y2 ) ) ).

% fun_upd_idem_iff
thf(fact_67_fun__upd__idem__iff,axiom,
    ! [F2: nat > nat,X2: nat,Y2: nat] :
      ( ( ( fun_upd_nat_nat @ F2 @ X2 @ Y2 )
        = F2 )
      = ( ( F2 @ X2 )
        = Y2 ) ) ).

% fun_upd_idem_iff
thf(fact_68_fun__upd__idem__iff,axiom,
    ! [F2: nat > a,X2: nat,Y2: a] :
      ( ( ( fun_upd_nat_a @ F2 @ X2 @ Y2 )
        = F2 )
      = ( ( F2 @ X2 )
        = Y2 ) ) ).

% fun_upd_idem_iff
thf(fact_69_fun__upd__twist,axiom,
    ! [A3: a,C: a,M: a > a,B2: a,D: a] :
      ( ( A3 != C )
     => ( ( fun_upd_a_a @ ( fun_upd_a_a @ M @ A3 @ B2 ) @ C @ D )
        = ( fun_upd_a_a @ ( fun_upd_a_a @ M @ C @ D ) @ A3 @ B2 ) ) ) ).

% fun_upd_twist
thf(fact_70_fun__upd__twist,axiom,
    ! [A3: a,C: a,M: a > nat,B2: nat,D: nat] :
      ( ( A3 != C )
     => ( ( fun_upd_a_nat @ ( fun_upd_a_nat @ M @ A3 @ B2 ) @ C @ D )
        = ( fun_upd_a_nat @ ( fun_upd_a_nat @ M @ C @ D ) @ A3 @ B2 ) ) ) ).

% fun_upd_twist
thf(fact_71_fun__upd__twist,axiom,
    ! [A3: nat,C: nat,M: nat > nat,B2: nat,D: nat] :
      ( ( A3 != C )
     => ( ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ A3 @ B2 ) @ C @ D )
        = ( fun_upd_nat_nat @ ( fun_upd_nat_nat @ M @ C @ D ) @ A3 @ B2 ) ) ) ).

% fun_upd_twist
thf(fact_72_fun__upd__twist,axiom,
    ! [A3: nat,C: nat,M: nat > a,B2: a,D: a] :
      ( ( A3 != C )
     => ( ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ A3 @ B2 ) @ C @ D )
        = ( fun_upd_nat_a @ ( fun_upd_nat_a @ M @ C @ D ) @ A3 @ B2 ) ) ) ).

% fun_upd_twist
thf(fact_73_fun__upd__other,axiom,
    ! [Z: a,X2: a,F2: a > a,Y2: a] :
      ( ( Z != X2 )
     => ( ( fun_upd_a_a @ F2 @ X2 @ Y2 @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_74_fun__upd__other,axiom,
    ! [Z: a,X2: a,F2: a > nat,Y2: nat] :
      ( ( Z != X2 )
     => ( ( fun_upd_a_nat @ F2 @ X2 @ Y2 @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_75_fun__upd__other,axiom,
    ! [Z: nat,X2: nat,F2: nat > nat,Y2: nat] :
      ( ( Z != X2 )
     => ( ( fun_upd_nat_nat @ F2 @ X2 @ Y2 @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_76_fun__upd__other,axiom,
    ! [Z: nat,X2: nat,F2: nat > a,Y2: a] :
      ( ( Z != X2 )
     => ( ( fun_upd_nat_a @ F2 @ X2 @ Y2 @ Z )
        = ( F2 @ Z ) ) ) ).

% fun_upd_other
thf(fact_77_fun__upd__same,axiom,
    ! [F2: a > a,X2: a,Y2: a] :
      ( ( fun_upd_a_a @ F2 @ X2 @ Y2 @ X2 )
      = Y2 ) ).

% fun_upd_same
thf(fact_78_fun__upd__same,axiom,
    ! [F2: a > nat,X2: a,Y2: nat] :
      ( ( fun_upd_a_nat @ F2 @ X2 @ Y2 @ X2 )
      = Y2 ) ).

% fun_upd_same
thf(fact_79_fun__upd__same,axiom,
    ! [F2: nat > nat,X2: nat,Y2: nat] :
      ( ( fun_upd_nat_nat @ F2 @ X2 @ Y2 @ X2 )
      = Y2 ) ).

% fun_upd_same
thf(fact_80_fun__upd__same,axiom,
    ! [F2: nat > a,X2: nat,Y2: a] :
      ( ( fun_upd_nat_a @ F2 @ X2 @ Y2 @ X2 )
      = Y2 ) ).

% fun_upd_same
thf(fact_81_fun__upd__idem,axiom,
    ! [F2: a > a,X2: a,Y2: a] :
      ( ( ( F2 @ X2 )
        = Y2 )
     => ( ( fun_upd_a_a @ F2 @ X2 @ Y2 )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_82_fun__upd__idem,axiom,
    ! [F2: a > nat,X2: a,Y2: nat] :
      ( ( ( F2 @ X2 )
        = Y2 )
     => ( ( fun_upd_a_nat @ F2 @ X2 @ Y2 )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_83_fun__upd__idem,axiom,
    ! [F2: nat > nat,X2: nat,Y2: nat] :
      ( ( ( F2 @ X2 )
        = Y2 )
     => ( ( fun_upd_nat_nat @ F2 @ X2 @ Y2 )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_84_fun__upd__idem,axiom,
    ! [F2: nat > a,X2: nat,Y2: a] :
      ( ( ( F2 @ X2 )
        = Y2 )
     => ( ( fun_upd_nat_a @ F2 @ X2 @ Y2 )
        = F2 ) ) ).

% fun_upd_idem
thf(fact_85_fun__upd__eqD,axiom,
    ! [F2: a > a,X2: a,Y2: a,G: a > a,Z: a] :
      ( ( ( fun_upd_a_a @ F2 @ X2 @ Y2 )
        = ( fun_upd_a_a @ G @ X2 @ Z ) )
     => ( Y2 = Z ) ) ).

% fun_upd_eqD
thf(fact_86_fun__upd__eqD,axiom,
    ! [F2: a > nat,X2: a,Y2: nat,G: a > nat,Z: nat] :
      ( ( ( fun_upd_a_nat @ F2 @ X2 @ Y2 )
        = ( fun_upd_a_nat @ G @ X2 @ Z ) )
     => ( Y2 = Z ) ) ).

% fun_upd_eqD
thf(fact_87_fun__upd__eqD,axiom,
    ! [F2: nat > nat,X2: nat,Y2: nat,G: nat > nat,Z: nat] :
      ( ( ( fun_upd_nat_nat @ F2 @ X2 @ Y2 )
        = ( fun_upd_nat_nat @ G @ X2 @ Z ) )
     => ( Y2 = Z ) ) ).

% fun_upd_eqD
thf(fact_88_fun__upd__eqD,axiom,
    ! [F2: nat > a,X2: nat,Y2: a,G: nat > a,Z: a] :
      ( ( ( fun_upd_nat_a @ F2 @ X2 @ Y2 )
        = ( fun_upd_nat_a @ G @ X2 @ Z ) )
     => ( Y2 = Z ) ) ).

% fun_upd_eqD
thf(fact_89_fun__upd__def,axiom,
    ( fun_upd_a_a
    = ( ^ [F: a > a,A: a,B3: a,X: a] : ( if_a @ ( X = A ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_90_fun__upd__def,axiom,
    ( fun_upd_a_nat
    = ( ^ [F: a > nat,A: a,B3: nat,X: a] : ( if_nat @ ( X = A ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_91_fun__upd__def,axiom,
    ( fun_upd_nat_nat
    = ( ^ [F: nat > nat,A: nat,B3: nat,X: nat] : ( if_nat @ ( X = A ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_92_fun__upd__def,axiom,
    ( fun_upd_nat_a
    = ( ^ [F: nat > a,A: nat,B3: a,X: nat] : ( if_a @ ( X = A ) @ B3 @ ( F @ X ) ) ) ) ).

% fun_upd_def
thf(fact_93_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_nat,R2: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_94_pigeonhole__infinite__rel,axiom,
    ! [A4: set_list_a,B4: set_nat,R2: list_a > nat > $o] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A: list_a] :
                        ( ( member_list_a @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_95_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_list_a,R2: nat > list_a > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_list_a @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_96_pigeonhole__infinite__rel,axiom,
    ! [A4: set_list_a,B4: set_list_a,R2: list_a > list_a > $o] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_list_a @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A4 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A: list_a] :
                        ( ( member_list_a @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_97_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_a,R2: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_a @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: a] :
              ( ( member_a @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_98_pigeonhole__infinite__rel,axiom,
    ! [A4: set_a,B4: set_nat,R2: a > nat > $o] :
      ( ~ ( finite_finite_a @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: a] :
              ( ( member_a @ X4 @ A4 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A: a] :
                        ( ( member_a @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_99_pigeonhole__infinite__rel,axiom,
    ! [A4: set_a,B4: set_a,R2: a > a > $o] :
      ( ~ ( finite_finite_a @ A4 )
     => ( ( finite_finite_a @ B4 )
       => ( ! [X4: a] :
              ( ( member_a @ X4 @ A4 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: a] :
              ( ( member_a @ X4 @ B4 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A: a] :
                        ( ( member_a @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_100_pigeonhole__infinite__rel,axiom,
    ! [A4: set_list_a,B4: set_a,R2: list_a > a > $o] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_a @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A4 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: a] :
              ( ( member_a @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A: list_a] :
                        ( ( member_list_a @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_101_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_list_nat,R2: nat > list_nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: list_nat] :
                  ( ( member_list_nat @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: list_nat] :
              ( ( member_list_nat @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_102_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B4: set_set_nat,R2: nat > set_nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ? [Xa: set_nat] :
                  ( ( member_set_nat @ Xa @ B4 )
                  & ( R2 @ X4 @ Xa ) ) )
         => ? [X4: set_nat] :
              ( ( member_set_nat @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A: nat] :
                        ( ( member_nat @ A @ A4 )
                        & ( R2 @ A @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_103_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_104_not__finite__existsD,axiom,
    ! [P: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P ) )
     => ? [X_1: a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_105_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_1: list_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_106_not__finite__existsD,axiom,
    ! [P: list_list_a > $o] :
      ( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
     => ? [X_1: list_list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_107_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_108_not__finite__existsD,axiom,
    ! [P: set_list_a > $o] :
      ( ~ ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
     => ? [X_1: set_list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_109_not__finite__existsD,axiom,
    ! [P: list_a > $o] :
      ( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ? [X_1: list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_110_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_111_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F2: nat > nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_112_finite__image__set2,axiom,
    ! [P: list_a > $o,Q: nat > $o,F2: list_a > nat > nat] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: list_a,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_113_finite__image__set2,axiom,
    ! [P: nat > $o,Q: list_a > $o,F2: nat > list_a > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_list_a @ ( collect_list_a @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: list_a] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_114_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F2: nat > nat > list_a] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [Uu: list_a] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_115_finite__image__set2,axiom,
    ! [P: list_a > $o,Q: list_a > $o,F2: list_a > list_a > nat] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_list_a @ ( collect_list_a @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: list_a,Y: list_a] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_116_finite__image__set2,axiom,
    ! [P: list_a > $o,Q: nat > $o,F2: list_a > nat > list_a] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [Uu: list_a] :
              ? [X: list_a,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_117_finite__image__set2,axiom,
    ! [P: nat > $o,Q: list_a > $o,F2: nat > list_a > list_a] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_list_a @ ( collect_list_a @ Q ) )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [Uu: list_a] :
              ? [X: nat,Y: list_a] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_118_finite__image__set2,axiom,
    ! [P: list_a > $o,Q: list_a > $o,F2: list_a > list_a > list_a] :
      ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ( ( finite_finite_list_a @ ( collect_list_a @ Q ) )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [Uu: list_a] :
              ? [X: list_a,Y: list_a] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_119_finite__image__set2,axiom,
    ! [P: nat > $o,Q: nat > $o,F2: nat > nat > a] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [Uu: a] :
              ? [X: nat,Y: nat] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_120_finite__image__set2,axiom,
    ! [P: nat > $o,Q: a > $o,F2: nat > a > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_a @ ( collect_a @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu: nat] :
              ? [X: nat,Y: a] :
                ( ( Uu
                  = ( F2 @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_121_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A4 )
        = ( linord2614967742042102400et_nat @ B4 ) )
     => ( ( finite_finite_nat @ A4 )
       => ( ( finite_finite_nat @ B4 )
         => ( A4 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_122_fun__upds__notin,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat,Sigma: nat > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P4014913454640143221at_nat
            @ ( produc8178142064113008363at_nat
              @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_123_fun__upds__notin,axiom,
    ! [Xs: list_a,Ys: list_nat,X2: a,Sigma: a > nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( ( fold_P3994820982079749301_a_nat
            @ ( produc7013214046051809481_a_nat
              @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
            @ ( zip_a_nat @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_124_fun__upds__notin,axiom,
    ! [Xs: list_a,Ys: list_a,X2: a,Sigma: a > a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( ( fold_P8422020818851269569_a_a_a
            @ ( produc2369190251411148053_a_a_a
              @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
            @ ( zip_a_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_125_fun__upds__notin,axiom,
    ! [Xs: list_nat,Ys: list_a,X2: nat,Sigma: nat > a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P5280602285094830901_nat_a
            @ ( produc2909000522608705447_nat_a
              @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
            @ ( zip_nat_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_126_fun__upds__notin,axiom,
    ! [Xs: list_set_nat,Ys: list_nat,X2: set_nat,Sigma: set_nat > nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
       => ( ( fold_P1940156743653614049at_nat
            @ ( produc1881937523299017485at_nat
              @ ^ [X: set_nat,Y: nat,F: set_nat > nat] : ( fun_upd_set_nat_nat @ F @ X @ Y ) )
            @ ( zip_set_nat_nat @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_127_fun__upds__notin,axiom,
    ! [Xs: list_set_nat,Ys: list_a,X2: set_nat,Sigma: set_nat > a] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
       => ( ( fold_P435287182084231349_nat_a
            @ ( produc8386917130950493169_nat_a
              @ ^ [X: set_nat,Y: a,F: set_nat > a] : ( fun_upd_set_nat_a @ F @ X @ Y ) )
            @ ( zip_set_nat_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_128_fun__upds__notin,axiom,
    ! [Xs: list_nat,Ys: list_list_a,X2: nat,Sigma: nat > list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P394168721639285173list_a
            @ ( produc3399763161783669037list_a
              @ ^ [X: nat,Y: list_a,F: nat > list_a] : ( fun_upd_nat_list_a @ F @ X @ Y ) )
            @ ( zip_nat_list_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_129_fun__upds__notin,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,X2: nat,Sigma: nat > list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P5635873918481876885st_nat
            @ ( produc5423136950428749467st_nat
              @ ^ [X: nat,Y: list_nat,F: nat > list_nat] : ( fun_upd_nat_list_nat @ F @ X @ Y ) )
            @ ( zip_nat_list_nat @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_130_fun__upds__notin,axiom,
    ! [Xs: list_nat,Ys: list_R6823256787227418703term_a,X2: nat,Sigma: nat > relational_term_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P6393280714071106933term_a
            @ ( produc3928451372858679218term_a
              @ ^ [X: nat,Y: relational_term_a,F: nat > relational_term_a] : ( fun_up808839731742462650term_a @ F @ X @ Y ) )
            @ ( zip_na8968355248959840256term_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_131_fun__upds__notin,axiom,
    ! [Xs: list_a,Ys: list_list_a,X2: a,Sigma: a > list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ( ( fold_P6355072832285262541list_a
            @ ( produc4092751348180932903list_a
              @ ^ [X: a,Y: list_a,F: a > list_a] : ( fun_upd_a_list_a @ F @ X @ Y ) )
            @ ( zip_a_list_a @ Xs @ Ys )
            @ Sigma
            @ X2 )
          = ( Sigma @ X2 ) ) ) ) ).

% fun_upds_notin
thf(fact_132_fun__upds__twist__apply,axiom,
    ! [Xs: list_nat,Ys: list_nat,A3: nat,B2: nat,Sigma: nat > nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P4014913454640143221at_nat
              @ ( produc8178142064113008363at_nat
                @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ Ys )
              @ ( fun_upd_nat_nat @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P4014913454640143221at_nat
              @ ( produc8178142064113008363at_nat
                @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_133_fun__upds__twist__apply,axiom,
    ! [Xs: list_a,Ys: list_nat,A3: a,B2: a,Sigma: a > nat,X2: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P3994820982079749301_a_nat
              @ ( produc7013214046051809481_a_nat
                @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
              @ ( zip_a_nat @ Xs @ Ys )
              @ ( fun_upd_a_nat @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P3994820982079749301_a_nat
              @ ( produc7013214046051809481_a_nat
                @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
              @ ( zip_a_nat @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_134_fun__upds__twist__apply,axiom,
    ! [Xs: list_a,Ys: list_a,A3: a,B2: a,Sigma: a > a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P8422020818851269569_a_a_a
              @ ( produc2369190251411148053_a_a_a
                @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
              @ ( zip_a_a @ Xs @ Ys )
              @ ( fun_upd_a_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P8422020818851269569_a_a_a
              @ ( produc2369190251411148053_a_a_a
                @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
              @ ( zip_a_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_135_fun__upds__twist__apply,axiom,
    ! [Xs: list_nat,Ys: list_a,A3: nat,B2: nat,Sigma: nat > a,X2: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P5280602285094830901_nat_a
              @ ( produc2909000522608705447_nat_a
                @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
              @ ( zip_nat_a @ Xs @ Ys )
              @ ( fun_upd_nat_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P5280602285094830901_nat_a
              @ ( produc2909000522608705447_nat_a
                @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
              @ ( zip_nat_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_136_fun__upds__twist__apply,axiom,
    ! [Xs: list_set_nat,Ys: list_nat,A3: set_nat,B2: set_nat,Sigma: set_nat > nat,X2: nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_set_nat @ A3 @ ( set_set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P1940156743653614049at_nat
              @ ( produc1881937523299017485at_nat
                @ ^ [X: set_nat,Y: nat,F: set_nat > nat] : ( fun_upd_set_nat_nat @ F @ X @ Y ) )
              @ ( zip_set_nat_nat @ Xs @ Ys )
              @ ( fun_upd_set_nat_nat @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P1940156743653614049at_nat
              @ ( produc1881937523299017485at_nat
                @ ^ [X: set_nat,Y: nat,F: set_nat > nat] : ( fun_upd_set_nat_nat @ F @ X @ Y ) )
              @ ( zip_set_nat_nat @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_137_fun__upds__twist__apply,axiom,
    ! [Xs: list_set_nat,Ys: list_a,A3: set_nat,B2: set_nat,Sigma: set_nat > a,X2: a] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_set_nat @ A3 @ ( set_set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P435287182084231349_nat_a
              @ ( produc8386917130950493169_nat_a
                @ ^ [X: set_nat,Y: a,F: set_nat > a] : ( fun_upd_set_nat_a @ F @ X @ Y ) )
              @ ( zip_set_nat_a @ Xs @ Ys )
              @ ( fun_upd_set_nat_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P435287182084231349_nat_a
              @ ( produc8386917130950493169_nat_a
                @ ^ [X: set_nat,Y: a,F: set_nat > a] : ( fun_upd_set_nat_a @ F @ X @ Y ) )
              @ ( zip_set_nat_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_138_fun__upds__twist__apply,axiom,
    ! [Xs: list_nat,Ys: list_list_a,A3: nat,B2: nat,Sigma: nat > list_a,X2: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P394168721639285173list_a
              @ ( produc3399763161783669037list_a
                @ ^ [X: nat,Y: list_a,F: nat > list_a] : ( fun_upd_nat_list_a @ F @ X @ Y ) )
              @ ( zip_nat_list_a @ Xs @ Ys )
              @ ( fun_upd_nat_list_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P394168721639285173list_a
              @ ( produc3399763161783669037list_a
                @ ^ [X: nat,Y: list_a,F: nat > list_a] : ( fun_upd_nat_list_a @ F @ X @ Y ) )
              @ ( zip_nat_list_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_139_fun__upds__twist__apply,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,A3: nat,B2: nat,Sigma: nat > list_nat,X2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P5635873918481876885st_nat
              @ ( produc5423136950428749467st_nat
                @ ^ [X: nat,Y: list_nat,F: nat > list_nat] : ( fun_upd_nat_list_nat @ F @ X @ Y ) )
              @ ( zip_nat_list_nat @ Xs @ Ys )
              @ ( fun_upd_nat_list_nat @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P5635873918481876885st_nat
              @ ( produc5423136950428749467st_nat
                @ ^ [X: nat,Y: list_nat,F: nat > list_nat] : ( fun_upd_nat_list_nat @ F @ X @ Y ) )
              @ ( zip_nat_list_nat @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_140_fun__upds__twist__apply,axiom,
    ! [Xs: list_nat,Ys: list_R6823256787227418703term_a,A3: nat,B2: nat,Sigma: nat > relational_term_a,X2: relational_term_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P6393280714071106933term_a
              @ ( produc3928451372858679218term_a
                @ ^ [X: nat,Y: relational_term_a,F: nat > relational_term_a] : ( fun_up808839731742462650term_a @ F @ X @ Y ) )
              @ ( zip_na8968355248959840256term_a @ Xs @ Ys )
              @ ( fun_up808839731742462650term_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P6393280714071106933term_a
              @ ( produc3928451372858679218term_a
                @ ^ [X: nat,Y: relational_term_a,F: nat > relational_term_a] : ( fun_up808839731742462650term_a @ F @ X @ Y ) )
              @ ( zip_na8968355248959840256term_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_141_fun__upds__twist__apply,axiom,
    ! [Xs: list_a,Ys: list_list_a,A3: a,B2: a,Sigma: a > list_a,X2: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( A3 != B2 )
         => ( ( fold_P6355072832285262541list_a
              @ ( produc4092751348180932903list_a
                @ ^ [X: a,Y: list_a,F: a > list_a] : ( fun_upd_a_list_a @ F @ X @ Y ) )
              @ ( zip_a_list_a @ Xs @ Ys )
              @ ( fun_upd_a_list_a @ Sigma @ A3 @ X2 )
              @ B2 )
            = ( fold_P6355072832285262541list_a
              @ ( produc4092751348180932903list_a
                @ ^ [X: a,Y: list_a,F: a > list_a] : ( fun_upd_a_list_a @ F @ X @ Y ) )
              @ ( zip_a_list_a @ Xs @ Ys )
              @ Sigma
              @ B2 ) ) ) ) ) ).

% fun_upds_twist_apply
thf(fact_142_fun__upds__twist,axiom,
    ! [Xs: list_nat,Ys: list_nat,A3: nat,Sigma: nat > nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P4014913454640143221at_nat
            @ ( produc8178142064113008363at_nat
              @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ ( fun_upd_nat_nat @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_nat_nat
            @ ( fold_P4014913454640143221at_nat
              @ ( produc8178142064113008363at_nat
                @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_143_fun__upds__twist,axiom,
    ! [Xs: list_a,Ys: list_nat,A3: a,Sigma: a > nat,X2: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( fold_P3994820982079749301_a_nat
            @ ( produc7013214046051809481_a_nat
              @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
            @ ( zip_a_nat @ Xs @ Ys )
            @ ( fun_upd_a_nat @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_a_nat
            @ ( fold_P3994820982079749301_a_nat
              @ ( produc7013214046051809481_a_nat
                @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
              @ ( zip_a_nat @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_144_fun__upds__twist,axiom,
    ! [Xs: list_a,Ys: list_a,A3: a,Sigma: a > a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( fold_P8422020818851269569_a_a_a
            @ ( produc2369190251411148053_a_a_a
              @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
            @ ( zip_a_a @ Xs @ Ys )
            @ ( fun_upd_a_a @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_a_a
            @ ( fold_P8422020818851269569_a_a_a
              @ ( produc2369190251411148053_a_a_a
                @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
              @ ( zip_a_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_145_fun__upds__twist,axiom,
    ! [Xs: list_nat,Ys: list_a,A3: nat,Sigma: nat > a,X2: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P5280602285094830901_nat_a
            @ ( produc2909000522608705447_nat_a
              @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
            @ ( zip_nat_a @ Xs @ Ys )
            @ ( fun_upd_nat_a @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_nat_a
            @ ( fold_P5280602285094830901_nat_a
              @ ( produc2909000522608705447_nat_a
                @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
              @ ( zip_nat_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_146_fun__upds__twist,axiom,
    ! [Xs: list_set_nat,Ys: list_nat,A3: set_nat,Sigma: set_nat > nat,X2: nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ~ ( member_set_nat @ A3 @ ( set_set_nat2 @ Xs ) )
       => ( ( fold_P1940156743653614049at_nat
            @ ( produc1881937523299017485at_nat
              @ ^ [X: set_nat,Y: nat,F: set_nat > nat] : ( fun_upd_set_nat_nat @ F @ X @ Y ) )
            @ ( zip_set_nat_nat @ Xs @ Ys )
            @ ( fun_upd_set_nat_nat @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_set_nat_nat
            @ ( fold_P1940156743653614049at_nat
              @ ( produc1881937523299017485at_nat
                @ ^ [X: set_nat,Y: nat,F: set_nat > nat] : ( fun_upd_set_nat_nat @ F @ X @ Y ) )
              @ ( zip_set_nat_nat @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_147_fun__upds__twist,axiom,
    ! [Xs: list_set_nat,Ys: list_a,A3: set_nat,Sigma: set_nat > a,X2: a] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ~ ( member_set_nat @ A3 @ ( set_set_nat2 @ Xs ) )
       => ( ( fold_P435287182084231349_nat_a
            @ ( produc8386917130950493169_nat_a
              @ ^ [X: set_nat,Y: a,F: set_nat > a] : ( fun_upd_set_nat_a @ F @ X @ Y ) )
            @ ( zip_set_nat_a @ Xs @ Ys )
            @ ( fun_upd_set_nat_a @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_set_nat_a
            @ ( fold_P435287182084231349_nat_a
              @ ( produc8386917130950493169_nat_a
                @ ^ [X: set_nat,Y: a,F: set_nat > a] : ( fun_upd_set_nat_a @ F @ X @ Y ) )
              @ ( zip_set_nat_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_148_fun__upds__twist,axiom,
    ! [Xs: list_nat,Ys: list_list_a,A3: nat,Sigma: nat > list_a,X2: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P394168721639285173list_a
            @ ( produc3399763161783669037list_a
              @ ^ [X: nat,Y: list_a,F: nat > list_a] : ( fun_upd_nat_list_a @ F @ X @ Y ) )
            @ ( zip_nat_list_a @ Xs @ Ys )
            @ ( fun_upd_nat_list_a @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_nat_list_a
            @ ( fold_P394168721639285173list_a
              @ ( produc3399763161783669037list_a
                @ ^ [X: nat,Y: list_a,F: nat > list_a] : ( fun_upd_nat_list_a @ F @ X @ Y ) )
              @ ( zip_nat_list_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_149_fun__upds__twist,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,A3: nat,Sigma: nat > list_nat,X2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P5635873918481876885st_nat
            @ ( produc5423136950428749467st_nat
              @ ^ [X: nat,Y: list_nat,F: nat > list_nat] : ( fun_upd_nat_list_nat @ F @ X @ Y ) )
            @ ( zip_nat_list_nat @ Xs @ Ys )
            @ ( fun_upd_nat_list_nat @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_nat_list_nat
            @ ( fold_P5635873918481876885st_nat
              @ ( produc5423136950428749467st_nat
                @ ^ [X: nat,Y: list_nat,F: nat > list_nat] : ( fun_upd_nat_list_nat @ F @ X @ Y ) )
              @ ( zip_nat_list_nat @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_150_fun__upds__twist,axiom,
    ! [Xs: list_nat,Ys: list_R6823256787227418703term_a,A3: nat,Sigma: nat > relational_term_a,X2: relational_term_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) )
     => ( ~ ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
       => ( ( fold_P6393280714071106933term_a
            @ ( produc3928451372858679218term_a
              @ ^ [X: nat,Y: relational_term_a,F: nat > relational_term_a] : ( fun_up808839731742462650term_a @ F @ X @ Y ) )
            @ ( zip_na8968355248959840256term_a @ Xs @ Ys )
            @ ( fun_up808839731742462650term_a @ Sigma @ A3 @ X2 ) )
          = ( fun_up808839731742462650term_a
            @ ( fold_P6393280714071106933term_a
              @ ( produc3928451372858679218term_a
                @ ^ [X: nat,Y: relational_term_a,F: nat > relational_term_a] : ( fun_up808839731742462650term_a @ F @ X @ Y ) )
              @ ( zip_na8968355248959840256term_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_151_fun__upds__twist,axiom,
    ! [Xs: list_a,Ys: list_list_a,A3: a,Sigma: a > list_a,X2: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ~ ( member_a @ A3 @ ( set_a2 @ Xs ) )
       => ( ( fold_P6355072832285262541list_a
            @ ( produc4092751348180932903list_a
              @ ^ [X: a,Y: list_a,F: a > list_a] : ( fun_upd_a_list_a @ F @ X @ Y ) )
            @ ( zip_a_list_a @ Xs @ Ys )
            @ ( fun_upd_a_list_a @ Sigma @ A3 @ X2 ) )
          = ( fun_upd_a_list_a
            @ ( fold_P6355072832285262541list_a
              @ ( produc4092751348180932903list_a
                @ ^ [X: a,Y: list_a,F: a > list_a] : ( fun_upd_a_list_a @ F @ X @ Y ) )
              @ ( zip_a_list_a @ Xs @ Ys )
              @ Sigma )
            @ A3
            @ X2 ) ) ) ) ).

% fun_upds_twist
thf(fact_152_prod_Ocase__distrib,axiom,
    ! [H: ( ( nat > a ) > nat > a ) > ( nat > a ) > nat > a,F2: nat > a > ( nat > a ) > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc2909000522608705447_nat_a @ F2 @ Prod ) )
      = ( produc2909000522608705447_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_153_prod_Ocase__distrib,axiom,
    ! [H: product_prod_a_nat > product_prod_a_nat,F2: nat > a > product_prod_a_nat,Prod: product_prod_nat_a] :
      ( ( H @ ( produc732906326552059263_a_nat @ F2 @ Prod ) )
      = ( produc732906326552059263_a_nat
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_154_prod_Ocase__distrib,axiom,
    ! [H: product_prod_a_nat > nat > a,F2: nat > a > product_prod_a_nat,Prod: product_prod_nat_a] :
      ( ( H @ ( produc732906326552059263_a_nat @ F2 @ Prod ) )
      = ( produc4481717121449037155_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_155_prod_Ocase__distrib,axiom,
    ! [H: ( nat > a ) > product_prod_a_nat,F2: nat > a > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc4481717121449037155_nat_a @ F2 @ Prod ) )
      = ( produc732906326552059263_a_nat
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_156_prod_Ocase__distrib,axiom,
    ! [H: ( nat > a ) > nat > a,F2: nat > a > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc4481717121449037155_nat_a @ F2 @ Prod ) )
      = ( produc4481717121449037155_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_157_prod_Ocase__distrib,axiom,
    ! [H: product_prod_nat_a > product_prod_nat_a,F2: a > nat > product_prod_nat_a,Prod: product_prod_a_nat] :
      ( ( H @ ( produc2753474147469666119_nat_a @ F2 @ Prod ) )
      = ( produc2753474147469666119_nat_a
        @ ^ [X1: a,X22: nat] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_158_prod_Ocase__distrib,axiom,
    ! [H: ( ( nat > a ) > nat > a ) > product_prod_a_nat,F2: nat > a > ( nat > a ) > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc2909000522608705447_nat_a @ F2 @ Prod ) )
      = ( produc732906326552059263_a_nat
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_159_prod_Ocase__distrib,axiom,
    ! [H: ( ( nat > a ) > nat > a ) > nat > a,F2: nat > a > ( nat > a ) > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc2909000522608705447_nat_a @ F2 @ Prod ) )
      = ( produc4481717121449037155_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_160_prod_Ocase__distrib,axiom,
    ! [H: product_prod_a_nat > ( nat > a ) > nat > a,F2: nat > a > product_prod_a_nat,Prod: product_prod_nat_a] :
      ( ( H @ ( produc732906326552059263_a_nat @ F2 @ Prod ) )
      = ( produc2909000522608705447_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_161_prod_Ocase__distrib,axiom,
    ! [H: ( nat > a ) > ( nat > a ) > nat > a,F2: nat > a > nat > a,Prod: product_prod_nat_a] :
      ( ( H @ ( produc4481717121449037155_nat_a @ F2 @ Prod ) )
      = ( produc2909000522608705447_nat_a
        @ ^ [X1: nat,X22: a] : ( H @ ( F2 @ X1 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_162_fun__upds__map__self,axiom,
    ! [Xs: list_nat,Sigma: nat > product_prod_nat_a] :
      ( ( fold_P2735308026484458651_nat_a
        @ ( produc3622568479822136292_nat_a
          @ ^ [X: nat,Y: product_prod_nat_a,F: nat > product_prod_nat_a] : ( fun_up4291488353418121970_nat_a @ F @ X @ Y ) )
        @ ( zip_na7849663319851687980_nat_a @ Xs @ ( map_na7918145567968794425_nat_a @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_163_fun__upds__map__self,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Sigma: product_prod_a_nat > product_prod_nat_a] :
      ( ( fold_P3343710452709354869_nat_a
        @ ( produc8511647682417282411_nat_a
          @ ^ [X: product_prod_a_nat,Y: product_prod_nat_a,F: product_prod_a_nat > product_prod_nat_a] : ( fun_up7371703469266106015_nat_a @ F @ X @ Y ) )
        @ ( zip_Pr5063037930166122969_nat_a @ Xs @ ( map_Pr990148473491725286_nat_a @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_164_fun__upds__map__self,axiom,
    ! [Xs: list_nat,Sigma: nat > nat] :
      ( ( fold_P4014913454640143221at_nat
        @ ( produc8178142064113008363at_nat
          @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
        @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_165_fun__upds__map__self,axiom,
    ! [Xs: list_a,Sigma: a > nat] :
      ( ( fold_P3994820982079749301_a_nat
        @ ( produc7013214046051809481_a_nat
          @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
        @ ( zip_a_nat @ Xs @ ( map_a_nat @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_166_fun__upds__map__self,axiom,
    ! [Xs: list_a,Sigma: a > a] :
      ( ( fold_P8422020818851269569_a_a_a
        @ ( produc2369190251411148053_a_a_a
          @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
        @ ( zip_a_a @ Xs @ ( map_a_a @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_167_fun__upds__map__self,axiom,
    ! [Xs: list_nat,Sigma: nat > a] :
      ( ( fold_P5280602285094830901_nat_a
        @ ( produc2909000522608705447_nat_a
          @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
        @ ( zip_nat_a @ Xs @ ( map_nat_a @ Sigma @ Xs ) )
        @ Sigma )
      = Sigma ) ).

% fun_upds_map_self
thf(fact_168_neq__if__length__neq,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ Xs )
       != ( size_s349497388124573686list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_169_neq__if__length__neq,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ Xs )
       != ( size_s3023201423986296836st_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_170_neq__if__length__neq,axiom,
    ! [Xs: list_R6823256787227418703term_a,Ys: list_R6823256787227418703term_a] :
      ( ( ( size_s88622898042387131term_a @ Xs )
       != ( size_s88622898042387131term_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_171_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_172_neq__if__length__neq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
       != ( size_size_list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_173_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_list_a] :
      ( ( size_s349497388124573686list_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_174_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_list_nat] :
      ( ( size_s3023201423986296836st_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_175_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_R6823256787227418703term_a] :
      ( ( size_s88622898042387131term_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_176_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_177_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_a] :
      ( ( size_size_list_a @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_178_size__neq__size__imp__neq,axiom,
    ! [X2: list_list_a,Y2: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ X2 )
       != ( size_s349497388124573686list_a @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_179_size__neq__size__imp__neq,axiom,
    ! [X2: list_list_nat,Y2: list_list_nat] :
      ( ( ( size_s3023201423986296836st_nat @ X2 )
       != ( size_s3023201423986296836st_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_180_size__neq__size__imp__neq,axiom,
    ! [X2: list_R6823256787227418703term_a,Y2: list_R6823256787227418703term_a] :
      ( ( ( size_s88622898042387131term_a @ X2 )
       != ( size_s88622898042387131term_a @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_181_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_182_size__neq__size__imp__neq,axiom,
    ! [X2: list_a,Y2: list_a] :
      ( ( ( size_size_list_a @ X2 )
       != ( size_size_list_a @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_183_sr__Disj,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( relational_fv_a_b @ Q1 )
        = ( relational_fv_a_b @ Q22 ) )
     => ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
        = ( ( relational_sr_a_b @ Q1 )
          & ( relational_sr_a_b @ Q22 ) ) ) ) ).

% sr_Disj
thf(fact_184_sat__exists,axiom,
    ! [N: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relati3989891337220013914ts_a_b @ N @ Q ) @ I @ Sigma )
      = ( ? [X: a] : ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ N @ X ) ) ) ) ).

% sat_exists
thf(fact_185_sat__subst,axiom,
    ! [Q: relational_fmla_a_b,X2: nat,Y2: nat,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_subst_a_b @ Q @ X2 @ Y2 ) @ I @ Sigma )
      = ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma @ X2 @ ( Sigma @ Y2 ) ) ) ) ).

% sat_subst
thf(fact_186_size__subst,axiom,
    ! [Q: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( size_s453432777765377587la_a_b @ ( relational_subst_a_b @ Q @ X2 @ Y2 ) )
      = ( size_s453432777765377587la_a_b @ Q ) ) ).

% size_subst
thf(fact_187_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs3: list_nat] : Xs3 ) ) ).

% map_ident
thf(fact_188_map__ident,axiom,
    ( ( map_a_a
      @ ^ [X: a] : X )
    = ( ^ [Xs3: list_a] : Xs3 ) ) ).

% map_ident
thf(fact_189_mem__Collect__eq,axiom,
    ! [A3: ( nat > a ) > nat > a,P: ( ( nat > a ) > nat > a ) > $o] :
      ( ( member_nat_a_nat_a @ A3 @ ( collect_nat_a_nat_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_190_mem__Collect__eq,axiom,
    ! [A3: a,P: a > $o] :
      ( ( member_a @ A3 @ ( collect_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_191_mem__Collect__eq,axiom,
    ! [A3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_192_mem__Collect__eq,axiom,
    ! [A3: list_list_a,P: list_list_a > $o] :
      ( ( member_list_list_a @ A3 @ ( collect_list_list_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
    ! [A3: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A3 @ ( collect_list_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
    ! [A3: set_list_a,P: set_list_a > $o] :
      ( ( member_set_list_a @ A3 @ ( collect_set_list_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A3 @ ( collect_set_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_196_mem__Collect__eq,axiom,
    ! [A3: list_a,P: list_a > $o] :
      ( ( member_list_a @ A3 @ ( collect_list_a @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_197_mem__Collect__eq,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( member_nat @ A3 @ ( collect_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_198_Collect__mem__eq,axiom,
    ! [A4: set_nat_a_nat_a] :
      ( ( collect_nat_a_nat_a
        @ ^ [X: ( nat > a ) > nat > a] : ( member_nat_a_nat_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_200_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_201_Collect__mem__eq,axiom,
    ! [A4: set_list_list_a] :
      ( ( collect_list_list_a
        @ ^ [X: list_list_a] : ( member_list_list_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_202_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_203_Collect__mem__eq,axiom,
    ! [A4: set_set_list_a] :
      ( ( collect_set_list_a
        @ ^ [X: set_list_a] : ( member_set_list_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_204_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_205_Collect__mem__eq,axiom,
    ! [A4: set_list_a] :
      ( ( collect_list_a
        @ ^ [X: list_a] : ( member_list_a @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_206_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_207_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_208_Collect__cong,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ! [X4: list_list_a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_list_a @ P )
        = ( collect_list_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_209_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X4: list_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_210_Collect__cong,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ! [X4: set_list_a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_list_a @ P )
        = ( collect_set_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_211_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_212_Collect__cong,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_a @ P )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_213_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_214_List_Ofinite__set,axiom,
    ! [Xs: list_P1396940483166286381od_a_a] : ( finite6544458595007987280od_a_a @ ( set_Product_prod_a_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_215_List_Ofinite__set,axiom,
    ! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_216_List_Ofinite__set,axiom,
    ! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_217_List_Ofinite__set,axiom,
    ! [Xs: list_list_list_a] : ( finite1660835950917165235list_a @ ( set_list_list_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_218_List_Ofinite__set,axiom,
    ! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_219_List_Ofinite__set,axiom,
    ! [Xs: list_set_list_a] : ( finite5282473924520328461list_a @ ( set_set_list_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_220_List_Ofinite__set,axiom,
    ! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_221_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_222_map__eq__conv,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a,Xs: list_P3592885314253461005_a_nat,G: product_prod_a_nat > product_prod_nat_a] :
      ( ( ( map_Pr990148473491725286_nat_a @ F2 @ Xs )
        = ( map_Pr990148473491725286_nat_a @ G @ Xs ) )
      = ( ! [X: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X @ ( set_Pr924983374503034536_a_nat @ Xs ) )
           => ( ( F2 @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_223_map__eq__conv,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F2 @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_224_map__eq__conv,axiom,
    ! [F2: nat > a,Xs: list_nat,G: nat > a] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( map_nat_a @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F2 @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_225_map__eq__conv,axiom,
    ! [F2: a > nat,Xs: list_a,G: a > nat] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = ( map_a_nat @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F2 @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_226_map__eq__conv,axiom,
    ! [F2: a > a,Xs: list_a,G: a > a] :
      ( ( ( map_a_a @ F2 @ Xs )
        = ( map_a_a @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F2 @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_227_length__map,axiom,
    ! [F2: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_228_length__map,axiom,
    ! [F2: a > nat,Xs: list_a] :
      ( ( size_size_list_nat @ ( map_a_nat @ F2 @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_229_length__map,axiom,
    ! [F2: nat > a,Xs: list_nat] :
      ( ( size_size_list_a @ ( map_nat_a @ F2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_230_length__map,axiom,
    ! [F2: a > a,Xs: list_a] :
      ( ( size_size_list_a @ ( map_a_a @ F2 @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_231_length__map,axiom,
    ! [F2: list_a > nat,Xs: list_list_a] :
      ( ( size_size_list_nat @ ( map_list_a_nat @ F2 @ Xs ) )
      = ( size_s349497388124573686list_a @ Xs ) ) ).

% length_map
thf(fact_232_length__map,axiom,
    ! [F2: list_nat > nat,Xs: list_list_nat] :
      ( ( size_size_list_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
      = ( size_s3023201423986296836st_nat @ Xs ) ) ).

% length_map
thf(fact_233_length__map,axiom,
    ! [F2: relational_term_a > nat,Xs: list_R6823256787227418703term_a] :
      ( ( size_size_list_nat @ ( map_Re700991392905337813_a_nat @ F2 @ Xs ) )
      = ( size_s88622898042387131term_a @ Xs ) ) ).

% length_map
thf(fact_234_length__map,axiom,
    ! [F2: list_a > a,Xs: list_list_a] :
      ( ( size_size_list_a @ ( map_list_a_a @ F2 @ Xs ) )
      = ( size_s349497388124573686list_a @ Xs ) ) ).

% length_map
thf(fact_235_length__map,axiom,
    ! [F2: list_nat > a,Xs: list_list_nat] :
      ( ( size_size_list_a @ ( map_list_nat_a @ F2 @ Xs ) )
      = ( size_s3023201423986296836st_nat @ Xs ) ) ).

% length_map
thf(fact_236_length__map,axiom,
    ! [F2: relational_term_a > a,Xs: list_R6823256787227418703term_a] :
      ( ( size_size_list_a @ ( map_Re419313091343012409rm_a_a @ F2 @ Xs ) )
      = ( size_s88622898042387131term_a @ Xs ) ) ).

% length_map
thf(fact_237_map__fun__upd,axiom,
    ! [Y2: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,V2: product_prod_nat_a] :
      ( ~ ( member5724188588386418708_a_nat @ Y2 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
     => ( ( map_Pr990148473491725286_nat_a @ ( fun_up7371703469266106015_nat_a @ F2 @ Y2 @ V2 ) @ Xs )
        = ( map_Pr990148473491725286_nat_a @ F2 @ Xs ) ) ) ).

% map_fun_upd
thf(fact_238_map__fun__upd,axiom,
    ! [Y2: nat,Xs: list_nat,F2: nat > nat,V2: nat] :
      ( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
     => ( ( map_nat_nat @ ( fun_upd_nat_nat @ F2 @ Y2 @ V2 ) @ Xs )
        = ( map_nat_nat @ F2 @ Xs ) ) ) ).

% map_fun_upd
thf(fact_239_map__fun__upd,axiom,
    ! [Y2: a,Xs: list_a,F2: a > nat,V2: nat] :
      ( ~ ( member_a @ Y2 @ ( set_a2 @ Xs ) )
     => ( ( map_a_nat @ ( fun_upd_a_nat @ F2 @ Y2 @ V2 ) @ Xs )
        = ( map_a_nat @ F2 @ Xs ) ) ) ).

% map_fun_upd
thf(fact_240_map__fun__upd,axiom,
    ! [Y2: a,Xs: list_a,F2: a > a,V2: a] :
      ( ~ ( member_a @ Y2 @ ( set_a2 @ Xs ) )
     => ( ( map_a_a @ ( fun_upd_a_a @ F2 @ Y2 @ V2 ) @ Xs )
        = ( map_a_a @ F2 @ Xs ) ) ) ).

% map_fun_upd
thf(fact_241_map__fun__upd,axiom,
    ! [Y2: nat,Xs: list_nat,F2: nat > a,V2: a] :
      ( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
     => ( ( map_nat_a @ ( fun_upd_nat_a @ F2 @ Y2 @ V2 ) @ Xs )
        = ( map_nat_a @ F2 @ Xs ) ) ) ).

% map_fun_upd
thf(fact_242_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A4 ) )
        = A4 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_243_ex__map__conv,axiom,
    ! [Ys: list_P2851791750731487283_nat_a,F2: product_prod_a_nat > product_prod_nat_a] :
      ( ( ? [Xs3: list_P3592885314253461005_a_nat] :
            ( Ys
            = ( map_Pr990148473491725286_nat_a @ F2 @ Xs3 ) ) )
      = ( ! [X: product_prod_nat_a] :
            ( ( member8962352052110095674_nat_a @ X @ ( set_Pr4163146838226711502_nat_a @ Ys ) )
           => ? [Y: product_prod_a_nat] :
                ( X
                = ( F2 @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_244_ex__map__conv,axiom,
    ! [Ys: list_nat,F2: nat > nat] :
      ( ( ? [Xs3: list_nat] :
            ( Ys
            = ( map_nat_nat @ F2 @ Xs3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F2 @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_245_ex__map__conv,axiom,
    ! [Ys: list_nat,F2: a > nat] :
      ( ( ? [Xs3: list_a] :
            ( Ys
            = ( map_a_nat @ F2 @ Xs3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: a] :
                ( X
                = ( F2 @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_246_ex__map__conv,axiom,
    ! [Ys: list_a,F2: nat > a] :
      ( ( ? [Xs3: list_nat] :
            ( Ys
            = ( map_nat_a @ F2 @ Xs3 ) ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F2 @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_247_ex__map__conv,axiom,
    ! [Ys: list_a,F2: a > a] :
      ( ( ? [Xs3: list_a] :
            ( Ys
            = ( map_a_a @ F2 @ Xs3 ) ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys ) )
           => ? [Y: a] :
                ( X
                = ( F2 @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_248_map__cong,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,G: product_prod_a_nat > product_prod_nat_a] :
      ( ( Xs = Ys )
     => ( ! [X4: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_Pr990148473491725286_nat_a @ F2 @ Xs )
          = ( map_Pr990148473491725286_nat_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_249_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F2: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_nat_nat @ F2 @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_250_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F2: nat > a,G: nat > a] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_nat_a @ F2 @ Xs )
          = ( map_nat_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_251_map__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F2: a > nat,G: a > nat] :
      ( ( Xs = Ys )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_a_nat @ F2 @ Xs )
          = ( map_a_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_252_map__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F2: a > a,G: a > a] :
      ( ( Xs = Ys )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_a_a @ F2 @ Xs )
          = ( map_a_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_253_map__idI,axiom,
    ! [Xs: list_set_list_a,F2: set_list_a > set_list_a] :
      ( ! [X4: set_list_a] :
          ( ( member_set_list_a @ X4 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_se2668659675339852484list_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_254_map__idI,axiom,
    ! [Xs: list_set_nat,F2: set_nat > set_nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_set_nat_set_nat @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_255_map__idI,axiom,
    ! [Xs: list_nat_a_nat_a,F2: ( ( nat > a ) > nat > a ) > ( nat > a ) > nat > a] :
      ( ! [X4: ( nat > a ) > nat > a] :
          ( ( member_nat_a_nat_a @ X4 @ ( set_nat_a_nat_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_na3828533370298643216_nat_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_256_map__idI,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,F2: product_prod_a_a > product_prod_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_Pr7904243085458786820od_a_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_257_map__idI,axiom,
    ! [Xs: list_list_nat,F2: list_nat > list_nat] :
      ( ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_258_map__idI,axiom,
    ! [Xs: list_list_a,F2: list_a > list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_list_a_list_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_259_map__idI,axiom,
    ! [Xs: list_a,F2: a > a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_a_a @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_260_map__idI,axiom,
    ! [Xs: list_nat,F2: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_nat_nat @ F2 @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_261_map__ext,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,G: product_prod_a_nat > product_prod_nat_a] :
      ( ! [X4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_Pr990148473491725286_nat_a @ F2 @ Xs )
        = ( map_Pr990148473491725286_nat_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_262_map__ext,axiom,
    ! [Xs: list_nat,F2: nat > nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_263_map__ext,axiom,
    ! [Xs: list_nat,F2: nat > a,G: nat > a] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_nat_a @ F2 @ Xs )
        = ( map_nat_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_264_map__ext,axiom,
    ! [Xs: list_a,F2: a > nat,G: a > nat] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_a_nat @ F2 @ Xs )
        = ( map_a_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_265_map__ext,axiom,
    ! [Xs: list_a,F2: a > a,G: a > a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_a_a @ F2 @ Xs )
        = ( map_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_266_list_Omap__ident__strong,axiom,
    ! [T: list_set_list_a,F2: set_list_a > set_list_a] :
      ( ! [Z3: set_list_a] :
          ( ( member_set_list_a @ Z3 @ ( set_set_list_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_se2668659675339852484list_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_267_list_Omap__ident__strong,axiom,
    ! [T: list_set_nat,F2: set_nat > set_nat] :
      ( ! [Z3: set_nat] :
          ( ( member_set_nat @ Z3 @ ( set_set_nat2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_set_nat_set_nat @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_268_list_Omap__ident__strong,axiom,
    ! [T: list_nat_a_nat_a,F2: ( ( nat > a ) > nat > a ) > ( nat > a ) > nat > a] :
      ( ! [Z3: ( nat > a ) > nat > a] :
          ( ( member_nat_a_nat_a @ Z3 @ ( set_nat_a_nat_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_na3828533370298643216_nat_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_269_list_Omap__ident__strong,axiom,
    ! [T: list_P1396940483166286381od_a_a,F2: product_prod_a_a > product_prod_a_a] :
      ( ! [Z3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ Z3 @ ( set_Product_prod_a_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_Pr7904243085458786820od_a_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_270_list_Omap__ident__strong,axiom,
    ! [T: list_list_nat,F2: list_nat > list_nat] :
      ( ! [Z3: list_nat] :
          ( ( member_list_nat @ Z3 @ ( set_list_nat2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_li7225945977422193158st_nat @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_271_list_Omap__ident__strong,axiom,
    ! [T: list_list_a,F2: list_a > list_a] :
      ( ! [Z3: list_a] :
          ( ( member_list_a @ Z3 @ ( set_list_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_list_a_list_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_272_list_Omap__ident__strong,axiom,
    ! [T: list_a,F2: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_a_a @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_273_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F2: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
         => ( ( F2 @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F2 @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_274_list_Oinj__map__strong,axiom,
    ! [X2: list_P3592885314253461005_a_nat,Xa2: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,Fa: product_prod_a_nat > product_prod_nat_a] :
      ( ! [Z3: product_prod_a_nat,Za: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ Z3 @ ( set_Pr924983374503034536_a_nat @ X2 ) )
         => ( ( member5724188588386418708_a_nat @ Za @ ( set_Pr924983374503034536_a_nat @ Xa2 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_Pr990148473491725286_nat_a @ F2 @ X2 )
          = ( map_Pr990148473491725286_nat_a @ Fa @ Xa2 ) )
       => ( X2 = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_275_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa2: list_nat,F2: nat > nat,Fa: nat > nat] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F2 @ X2 )
          = ( map_nat_nat @ Fa @ Xa2 ) )
       => ( X2 = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_276_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa2: list_nat,F2: nat > a,Fa: nat > a] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_a @ F2 @ X2 )
          = ( map_nat_a @ Fa @ Xa2 ) )
       => ( X2 = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_277_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa2: list_a,F2: a > nat,Fa: a > nat] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_a_nat @ F2 @ X2 )
          = ( map_a_nat @ Fa @ Xa2 ) )
       => ( X2 = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_278_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa2: list_a,F2: a > a,Fa: a > a] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
           => ( ( ( F2 @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_a_a @ F2 @ X2 )
          = ( map_a_a @ Fa @ Xa2 ) )
       => ( X2 = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_279_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_280_list_Omap__ident,axiom,
    ! [T: list_a] :
      ( ( map_a_a
        @ ^ [X: a] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_281_list_Omap__cong0,axiom,
    ! [X2: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,G: product_prod_a_nat > product_prod_nat_a] :
      ( ! [Z3: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ Z3 @ ( set_Pr924983374503034536_a_nat @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_Pr990148473491725286_nat_a @ F2 @ X2 )
        = ( map_Pr990148473491725286_nat_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_282_list_Omap__cong0,axiom,
    ! [X2: list_nat,F2: nat > nat,G: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_nat @ F2 @ X2 )
        = ( map_nat_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_283_list_Omap__cong0,axiom,
    ! [X2: list_nat,F2: nat > a,G: nat > a] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_a @ F2 @ X2 )
        = ( map_nat_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_284_list_Omap__cong0,axiom,
    ! [X2: list_a,F2: a > nat,G: a > nat] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_a_nat @ F2 @ X2 )
        = ( map_a_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_285_list_Omap__cong0,axiom,
    ! [X2: list_a,F2: a > a,G: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X2 ) )
         => ( ( F2 @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_a_a @ F2 @ X2 )
        = ( map_a_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_286_list_Omap__cong,axiom,
    ! [X2: list_P3592885314253461005_a_nat,Ya: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a,G: product_prod_a_nat > product_prod_nat_a] :
      ( ( X2 = Ya )
     => ( ! [Z3: product_prod_a_nat] :
            ( ( member5724188588386418708_a_nat @ Z3 @ ( set_Pr924983374503034536_a_nat @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_Pr990148473491725286_nat_a @ F2 @ X2 )
          = ( map_Pr990148473491725286_nat_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_287_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F2: nat > nat,G: nat > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_nat @ F2 @ X2 )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_288_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F2: nat > a,G: nat > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_a @ F2 @ X2 )
          = ( map_nat_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_289_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F2: a > nat,G: a > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_a_nat @ F2 @ X2 )
          = ( map_a_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_290_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F2: a > a,G: a > a] :
      ( ( X2 = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
           => ( ( F2 @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_a_a @ F2 @ X2 )
          = ( map_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_291_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_292_map__eq__imp__length__eq,axiom,
    ! [F2: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_293_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_a_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_294_map__eq__imp__length__eq,axiom,
    ! [F2: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( map_a_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_295_map__eq__imp__length__eq,axiom,
    ! [F2: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_296_map__eq__imp__length__eq,axiom,
    ! [F2: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
      ( ( ( map_a_a @ F2 @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_297_map__eq__imp__length__eq,axiom,
    ! [F2: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
      ( ( ( map_a_nat @ F2 @ Xs )
        = ( map_a_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_298_map__eq__imp__length__eq,axiom,
    ! [F2: a > a,Xs: list_a,G: a > a,Ys: list_a] :
      ( ( ( map_a_a @ F2 @ Xs )
        = ( map_a_a @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_299_map__eq__imp__length__eq,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: list_a > nat,Ys: list_list_a] :
      ( ( ( map_nat_nat @ F2 @ Xs )
        = ( map_list_a_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_300_map__eq__imp__length__eq,axiom,
    ! [F2: nat > a,Xs: list_nat,G: list_a > a,Ys: list_list_a] :
      ( ( ( map_nat_a @ F2 @ Xs )
        = ( map_list_a_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_301_finite__list,axiom,
    ! [A4: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A4 )
     => ? [Xs2: list_P1396940483166286381od_a_a] :
          ( ( set_Product_prod_a_a2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_302_finite__list,axiom,
    ! [A4: set_a] :
      ( ( finite_finite_a @ A4 )
     => ? [Xs2: list_a] :
          ( ( set_a2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_303_finite__list,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ? [Xs2: list_list_nat] :
          ( ( set_list_nat2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_304_finite__list,axiom,
    ! [A4: set_list_list_a] :
      ( ( finite1660835950917165235list_a @ A4 )
     => ? [Xs2: list_list_list_a] :
          ( ( set_list_list_a2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_305_finite__list,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ? [Xs2: list_set_nat] :
          ( ( set_set_nat2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_306_finite__list,axiom,
    ! [A4: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ? [Xs2: list_set_list_a] :
          ( ( set_set_list_a2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_307_finite__list,axiom,
    ! [A4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ? [Xs2: list_list_a] :
          ( ( set_list_a2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_308_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A4 ) ) ).

% finite_list
thf(fact_309_List_Ofold__cong,axiom,
    ! [A3: a > nat,B2: a > nat,Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > ( a > nat ) > a > nat,G: product_prod_a_nat > ( a > nat ) > a > nat] :
      ( ( A3 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: product_prod_a_nat] :
              ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
             => ( ( F2 @ X4 )
                = ( G @ X4 ) ) )
         => ( ( fold_P3994820982079749301_a_nat @ F2 @ Xs @ A3 )
            = ( fold_P3994820982079749301_a_nat @ G @ Ys @ B2 ) ) ) ) ) ).

% List.fold_cong
thf(fact_310_List_Ofold__cong,axiom,
    ! [A3: nat > nat,B2: nat > nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > ( nat > nat ) > nat > nat,G: product_prod_nat_nat > ( nat > nat ) > nat > nat] :
      ( ( A3 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
             => ( ( F2 @ X4 )
                = ( G @ X4 ) ) )
         => ( ( fold_P4014913454640143221at_nat @ F2 @ Xs @ A3 )
            = ( fold_P4014913454640143221at_nat @ G @ Ys @ B2 ) ) ) ) ) ).

% List.fold_cong
thf(fact_311_List_Ofold__cong,axiom,
    ! [A3: relational_fmla_a_b,B2: relational_fmla_a_b,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b,G: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( A3 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
             => ( ( F2 @ X4 )
                = ( G @ X4 ) ) )
         => ( ( fold_P7970104616371074773la_a_b @ F2 @ Xs @ A3 )
            = ( fold_P7970104616371074773la_a_b @ G @ Ys @ B2 ) ) ) ) ) ).

% List.fold_cong
thf(fact_312_List_Ofold__cong,axiom,
    ! [A3: a > a,B2: a > a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,F2: product_prod_a_a > ( a > a ) > a > a,G: product_prod_a_a > ( a > a ) > a > a] :
      ( ( A3 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
             => ( ( F2 @ X4 )
                = ( G @ X4 ) ) )
         => ( ( fold_P8422020818851269569_a_a_a @ F2 @ Xs @ A3 )
            = ( fold_P8422020818851269569_a_a_a @ G @ Ys @ B2 ) ) ) ) ) ).

% List.fold_cong
thf(fact_313_List_Ofold__cong,axiom,
    ! [A3: nat > a,B2: nat > a,Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a,F2: product_prod_nat_a > ( nat > a ) > nat > a,G: product_prod_nat_a > ( nat > a ) > nat > a] :
      ( ( A3 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X4: product_prod_nat_a] :
              ( ( member8962352052110095674_nat_a @ X4 @ ( set_Pr4163146838226711502_nat_a @ Xs ) )
             => ( ( F2 @ X4 )
                = ( G @ X4 ) ) )
         => ( ( fold_P5280602285094830901_nat_a @ F2 @ Xs @ A3 )
            = ( fold_P5280602285094830901_nat_a @ G @ Ys @ B2 ) ) ) ) ) ).

% List.fold_cong
thf(fact_314_fold__invariant,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Q: product_prod_a_nat > $o,P: ( a > nat ) > $o,S: a > nat,F2: product_prod_a_nat > ( a > nat ) > a > nat] :
      ( ! [X4: product_prod_a_nat] :
          ( ( member5724188588386418708_a_nat @ X4 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
         => ( Q @ X4 ) )
     => ( ( P @ S )
       => ( ! [X4: product_prod_a_nat,S2: a > nat] :
              ( ( Q @ X4 )
             => ( ( P @ S2 )
               => ( P @ ( F2 @ X4 @ S2 ) ) ) )
         => ( P @ ( fold_P3994820982079749301_a_nat @ F2 @ Xs @ S ) ) ) ) ) ).

% fold_invariant
thf(fact_315_fold__invariant,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Q: product_prod_nat_nat > $o,P: ( nat > nat ) > $o,S: nat > nat,F2: product_prod_nat_nat > ( nat > nat ) > nat > nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( Q @ X4 ) )
     => ( ( P @ S )
       => ( ! [X4: product_prod_nat_nat,S2: nat > nat] :
              ( ( Q @ X4 )
             => ( ( P @ S2 )
               => ( P @ ( F2 @ X4 @ S2 ) ) ) )
         => ( P @ ( fold_P4014913454640143221at_nat @ F2 @ Xs @ S ) ) ) ) ) ).

% fold_invariant
thf(fact_316_fold__invariant,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Q: product_prod_nat_nat > $o,P: relational_fmla_a_b > $o,S: relational_fmla_a_b,F2: product_prod_nat_nat > relational_fmla_a_b > relational_fmla_a_b] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( Q @ X4 ) )
     => ( ( P @ S )
       => ( ! [X4: product_prod_nat_nat,S2: relational_fmla_a_b] :
              ( ( Q @ X4 )
             => ( ( P @ S2 )
               => ( P @ ( F2 @ X4 @ S2 ) ) ) )
         => ( P @ ( fold_P7970104616371074773la_a_b @ F2 @ Xs @ S ) ) ) ) ) ).

% fold_invariant
thf(fact_317_fold__invariant,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,Q: product_prod_a_a > $o,P: ( a > a ) > $o,S: a > a,F2: product_prod_a_a > ( a > a ) > a > a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
         => ( Q @ X4 ) )
     => ( ( P @ S )
       => ( ! [X4: product_prod_a_a,S2: a > a] :
              ( ( Q @ X4 )
             => ( ( P @ S2 )
               => ( P @ ( F2 @ X4 @ S2 ) ) ) )
         => ( P @ ( fold_P8422020818851269569_a_a_a @ F2 @ Xs @ S ) ) ) ) ) ).

% fold_invariant
thf(fact_318_fold__invariant,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Q: product_prod_nat_a > $o,P: ( nat > a ) > $o,S: nat > a,F2: product_prod_nat_a > ( nat > a ) > nat > a] :
      ( ! [X4: product_prod_nat_a] :
          ( ( member8962352052110095674_nat_a @ X4 @ ( set_Pr4163146838226711502_nat_a @ Xs ) )
         => ( Q @ X4 ) )
     => ( ( P @ S )
       => ( ! [X4: product_prod_nat_a,S2: nat > a] :
              ( ( Q @ X4 )
             => ( ( P @ S2 )
               => ( P @ ( F2 @ X4 @ S2 ) ) ) )
         => ( P @ ( fold_P5280602285094830901_nat_a @ F2 @ Xs @ S ) ) ) ) ) ).

% fold_invariant
thf(fact_319_map2__map__map,axiom,
    ! [H: nat > nat > nat,F2: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_320_map2__map__map,axiom,
    ! [H: nat > nat > a,F2: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr5244471862779271681_nat_a @ ( produc3276484115406849584_nat_a @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_321_map2__map__map,axiom,
    ! [H: nat > a > nat,F2: nat > nat,Xs: list_nat,G: nat > a] :
      ( ( map_Pr7638933597999785017_a_nat @ ( produc8467924651083338962_a_nat @ H ) @ ( zip_nat_a @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_322_map2__map__map,axiom,
    ! [H: nat > a > a,F2: nat > nat,Xs: list_nat,G: nat > a] :
      ( ( map_Pr3654303163821447893at_a_a @ ( produc8288763603002351228at_a_a @ H ) @ ( zip_nat_a @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_323_map2__map__map,axiom,
    ! [H: a > nat > nat,F2: nat > a,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr87709696777291539at_nat @ ( produc6967628985812163444at_nat @ H ) @ ( zip_a_nat @ ( map_nat_a @ F2 @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_324_map2__map__map,axiom,
    ! [H: a > nat > a,F2: nat > a,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr6606814258838092411_nat_a @ ( produc7884564006109396058_nat_a @ H ) @ ( zip_a_nat @ ( map_nat_a @ F2 @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_325_map2__map__map,axiom,
    ! [H: a > a > nat,F2: nat > a,Xs: list_nat,G: nat > a] :
      ( ( map_Pr9001275994058605747_a_nat @ ( produc3852632504931109628_a_nat @ H ) @ ( zip_a_a @ ( map_nat_a @ F2 @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_326_map2__map__map,axiom,
    ! [H: a > a > a,F2: nat > a,Xs: list_nat,G: nat > a] :
      ( ( map_Pr3897371633210041563_a_a_a @ ( produc8815886927560695506_a_a_a @ H ) @ ( zip_a_a @ ( map_nat_a @ F2 @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X: nat] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_327_map2__map__map,axiom,
    ! [H: nat > nat > nat,F2: a > nat,Xs: list_a,G: a > nat] :
      ( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_a_nat @ F2 @ Xs ) @ ( map_a_nat @ G @ Xs ) ) )
      = ( map_a_nat
        @ ^ [X: a] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_328_map2__map__map,axiom,
    ! [H: nat > nat > a,F2: a > nat,Xs: list_a,G: a > nat] :
      ( ( map_Pr5244471862779271681_nat_a @ ( produc3276484115406849584_nat_a @ H ) @ ( zip_nat_nat @ ( map_a_nat @ F2 @ Xs ) @ ( map_a_nat @ G @ Xs ) ) )
      = ( map_a_a
        @ ^ [X: a] : ( H @ ( F2 @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_329_subst_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X2 @ Y2 )
      = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X2 @ Y2 ) @ ( relational_subst_a_b @ Q22 @ X2 @ Y2 ) ) ) ).

% subst.simps(6)
thf(fact_330_fun__upds__single,axiom,
    ! [Xs: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat,Sigma: product_prod_a_nat > product_prod_nat_a,D: product_prod_nat_a] :
      ( ( distin4161512318331733700_a_nat @ Xs )
     => ( ( ( member5724188588386418708_a_nat @ Y2 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
         => ( ( fold_P3343710452709354869_nat_a
              @ ( produc8511647682417282411_nat_a
                @ ^ [X: product_prod_a_nat,Y: product_prod_nat_a,F: product_prod_a_nat > product_prod_nat_a] : ( fun_up7371703469266106015_nat_a @ F @ X @ Y ) )
              @ ( zip_Pr5063037930166122969_nat_a @ Xs @ ( map_Pr990148473491725286_nat_a @ ( fun_up7371703469266106015_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_up7371703469266106015_nat_a @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member5724188588386418708_a_nat @ Y2 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
         => ( ( fold_P3343710452709354869_nat_a
              @ ( produc8511647682417282411_nat_a
                @ ^ [X: product_prod_a_nat,Y: product_prod_nat_a,F: product_prod_a_nat > product_prod_nat_a] : ( fun_up7371703469266106015_nat_a @ F @ X @ Y ) )
              @ ( zip_Pr5063037930166122969_nat_a @ Xs @ ( map_Pr990148473491725286_nat_a @ ( fun_up7371703469266106015_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_331_fun__upds__single,axiom,
    ! [Xs: list_nat,Y2: nat,Sigma: nat > product_prod_nat_a,D: product_prod_nat_a] :
      ( ( distinct_nat @ Xs )
     => ( ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P2735308026484458651_nat_a
              @ ( produc3622568479822136292_nat_a
                @ ^ [X: nat,Y: product_prod_nat_a,F: nat > product_prod_nat_a] : ( fun_up4291488353418121970_nat_a @ F @ X @ Y ) )
              @ ( zip_na7849663319851687980_nat_a @ Xs @ ( map_na7918145567968794425_nat_a @ ( fun_up4291488353418121970_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_up4291488353418121970_nat_a @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P2735308026484458651_nat_a
              @ ( produc3622568479822136292_nat_a
                @ ^ [X: nat,Y: product_prod_nat_a,F: nat > product_prod_nat_a] : ( fun_up4291488353418121970_nat_a @ F @ X @ Y ) )
              @ ( zip_na7849663319851687980_nat_a @ Xs @ ( map_na7918145567968794425_nat_a @ ( fun_up4291488353418121970_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_332_fun__upds__single,axiom,
    ! [Xs: list_nat,Y2: nat,Sigma: nat > nat,D: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P4014913454640143221at_nat
              @ ( produc8178142064113008363at_nat
                @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ ( fun_upd_nat_nat @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_upd_nat_nat @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P4014913454640143221at_nat
              @ ( produc8178142064113008363at_nat
                @ ^ [X: nat,Y: nat,F: nat > nat] : ( fun_upd_nat_nat @ F @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ ( map_nat_nat @ ( fun_upd_nat_nat @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_333_fun__upds__single,axiom,
    ! [Xs: list_a,Y2: a,Sigma: a > nat,D: nat] :
      ( ( distinct_a @ Xs )
     => ( ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
         => ( ( fold_P3994820982079749301_a_nat
              @ ( produc7013214046051809481_a_nat
                @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
              @ ( zip_a_nat @ Xs @ ( map_a_nat @ ( fun_upd_a_nat @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_upd_a_nat @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member_a @ Y2 @ ( set_a2 @ Xs ) )
         => ( ( fold_P3994820982079749301_a_nat
              @ ( produc7013214046051809481_a_nat
                @ ^ [X: a,Y: nat,F: a > nat] : ( fun_upd_a_nat @ F @ X @ Y ) )
              @ ( zip_a_nat @ Xs @ ( map_a_nat @ ( fun_upd_a_nat @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_334_fun__upds__single,axiom,
    ! [Xs: list_a,Y2: a,Sigma: a > a,D: a] :
      ( ( distinct_a @ Xs )
     => ( ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
         => ( ( fold_P8422020818851269569_a_a_a
              @ ( produc2369190251411148053_a_a_a
                @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
              @ ( zip_a_a @ Xs @ ( map_a_a @ ( fun_upd_a_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_upd_a_a @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member_a @ Y2 @ ( set_a2 @ Xs ) )
         => ( ( fold_P8422020818851269569_a_a_a
              @ ( produc2369190251411148053_a_a_a
                @ ^ [X: a,Y: a,F: a > a] : ( fun_upd_a_a @ F @ X @ Y ) )
              @ ( zip_a_a @ Xs @ ( map_a_a @ ( fun_upd_a_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_335_fun__upds__single,axiom,
    ! [Xs: list_nat,Y2: nat,Sigma: nat > a,D: a] :
      ( ( distinct_nat @ Xs )
     => ( ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P5280602285094830901_nat_a
              @ ( produc2909000522608705447_nat_a
                @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
              @ ( zip_nat_a @ Xs @ ( map_nat_a @ ( fun_upd_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = ( fun_upd_nat_a @ Sigma @ Y2 @ D ) ) )
        & ( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
         => ( ( fold_P5280602285094830901_nat_a
              @ ( produc2909000522608705447_nat_a
                @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
              @ ( zip_nat_a @ Xs @ ( map_nat_a @ ( fun_upd_nat_a @ Sigma @ Y2 @ D ) @ Xs ) )
              @ Sigma )
            = Sigma ) ) ) ) ).

% fun_upds_single
thf(fact_336_sorted__list__of__set__def,axiom,
    ( linord2614967742042102400et_nat
    = ( linord1089935798310486446at_nat
      @ ^ [X: nat] : X ) ) ).

% sorted_list_of_set_def
thf(fact_337_lookup__map,axiom,
    ! [X2: product_prod_a_nat,Xs: list_P3592885314253461005_a_nat,F2: product_prod_a_nat > product_prod_nat_a] :
      ( ( member5724188588386418708_a_nat @ X2 @ ( set_Pr924983374503034536_a_nat @ Xs ) )
     => ( ( lookup6839019746583965316_nat_a @ Xs @ ( map_Pr990148473491725286_nat_a @ F2 @ Xs ) @ X2 )
        = ( F2 @ X2 ) ) ) ).

% lookup_map
thf(fact_338_lookup__map,axiom,
    ! [X2: nat,Xs: list_nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( lookup_nat_nat @ Xs @ ( map_nat_nat @ F2 @ Xs ) @ X2 )
        = ( F2 @ X2 ) ) ) ).

% lookup_map
thf(fact_339_lookup__map,axiom,
    ! [X2: nat,Xs: list_nat,F2: nat > a] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( lookup_nat_a @ Xs @ ( map_nat_a @ F2 @ Xs ) @ X2 )
        = ( F2 @ X2 ) ) ) ).

% lookup_map
thf(fact_340_lookup__map,axiom,
    ! [X2: a,Xs: list_a,F2: a > nat] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( lookup_a_nat @ Xs @ ( map_a_nat @ F2 @ Xs ) @ X2 )
        = ( F2 @ X2 ) ) ) ).

% lookup_map
thf(fact_341_lookup__map,axiom,
    ! [X2: a,Xs: list_a,F2: a > a] :
      ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
     => ( ( lookup_a_a @ Xs @ ( map_a_a @ F2 @ Xs ) @ X2 )
        = ( F2 @ X2 ) ) ) ).

% lookup_map
thf(fact_342_finite__lists__length__eq,axiom,
    ! [A4: set_Product_prod_a_a,N: nat] :
      ( ( finite6544458595007987280od_a_a @ A4 )
     => ( finite3567696607326427862od_a_a
        @ ( collec2774578310585404696od_a_a
          @ ^ [Xs3: list_P1396940483166286381od_a_a] :
              ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs3 ) @ A4 )
              & ( ( size_s3885678630836030617od_a_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_343_finite__lists__length__eq,axiom,
    ! [A4: set_list_list_a,N: nat] :
      ( ( finite1660835950917165235list_a @ A4 )
     => ( finite2085839564794461113list_a
        @ ( collec1292721268053437947list_a
          @ ^ [Xs3: list_list_list_a] :
              ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs3 ) @ A4 )
              & ( ( size_s2403821588304063868list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_344_finite__lists__length__eq,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( finite8170528100393595399st_nat
        @ ( collec5989764272469232197st_nat
          @ ^ [Xs3: list_list_nat] :
              ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A4 )
              & ( ( size_s3023201423986296836st_nat @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_345_finite__lists__length__eq,axiom,
    ! [A4: set_Re5178783185447174953term_a,N: nat] :
      ( ( finite1361000192780721522term_a @ A4 )
     => ( finite8994012911387560184term_a
        @ ( collec8200894614646537018term_a
          @ ^ [Xs3: list_R6823256787227418703term_a] :
              ( ( ord_le6173019262470748873term_a @ ( set_Re3569617851344498910term_a @ Xs3 ) @ A4 )
              & ( ( size_s88622898042387131term_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_346_finite__lists__length__eq,axiom,
    ! [A4: set_set_list_a,N: nat] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( finite3202133031812794515list_a
        @ ( collec5381118732811369429list_a
          @ ^ [Xs3: list_set_list_a] :
              ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs3 ) @ A4 )
              & ( ( size_s1991367317912710102list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_347_finite__lists__length__eq,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( finite1091814263879798189et_nat
        @ ( collect_list_set_nat
          @ ^ [Xs3: list_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs3 ) @ A4 )
              & ( ( size_s3254054031482475050et_nat @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_348_finite__lists__length__eq,axiom,
    ! [A4: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A4 )
              & ( ( size_s349497388124573686list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_349_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
              & ( ( size_size_list_nat @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_350_finite__lists__length__eq,axiom,
    ! [A4: set_a,N: nat] :
      ( ( finite_finite_a @ A4 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A4 )
              & ( ( size_size_list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_351_to__nat__list__def,axiom,
    infini6278850185503974415list_a = size_s349497388124573686list_a ).

% to_nat_list_def
thf(fact_352_to__nat__list__def,axiom,
    infini6049765617473507243st_nat = size_s3023201423986296836st_nat ).

% to_nat_list_def
thf(fact_353_to__nat__list__def,axiom,
    infini3862824544962047572term_a = size_s88622898042387131term_a ).

% to_nat_list_def
thf(fact_354_to__nat__list__def,axiom,
    infini5756963555370273947st_nat = size_size_list_nat ).

% to_nat_list_def
thf(fact_355_to__nat__list__def,axiom,
    infini4950935752332860041list_a = size_size_list_a ).

% to_nat_list_def
thf(fact_356_map__eq__map__tailrec,axiom,
    map_Pr990148473491725286_nat_a = map_ta6611357153064403760_nat_a ).

% map_eq_map_tailrec
thf(fact_357_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_358_map__eq__map__tailrec,axiom,
    map_nat_a = map_tailrec_nat_a ).

% map_eq_map_tailrec
thf(fact_359_map__eq__map__tailrec,axiom,
    map_a_nat = map_tailrec_a_nat ).

% map_eq_map_tailrec
thf(fact_360_map__eq__map__tailrec,axiom,
    map_a_a = map_tailrec_a_a ).

% map_eq_map_tailrec
thf(fact_361_internal__case__prod__def,axiom,
    produc3668125112861867305_a_nat = produc732906326552059263_a_nat ).

% internal_case_prod_def
thf(fact_362_internal__case__prod__def,axiom,
    produc6526265228523360569_nat_a = produc4481717121449037155_nat_a ).

% internal_case_prod_def
thf(fact_363_internal__case__prod__def,axiom,
    produc5688692933779474161_nat_a = produc2753474147469666119_nat_a ).

% internal_case_prod_def
thf(fact_364_internal__case__prod__def,axiom,
    produc3592179021021687923_a_nat = produc7013214046051809481_a_nat ).

% internal_case_prod_def
thf(fact_365_internal__case__prod__def,axiom,
    produc8734483249530697963_a_a_a = produc2369190251411148053_a_a_a ).

% internal_case_prod_def
thf(fact_366_internal__case__prod__def,axiom,
    produc8711337534433359697_nat_a = produc2909000522608705447_nat_a ).

% internal_case_prod_def
thf(fact_367_subst__exists,axiom,
    ! [Z: nat,Q: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
       => ( ( ( X2 = Z )
           => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X2 @ Y2 )
              = ( relati3989891337220013914ts_a_b @ X2 @ Q ) ) )
          & ( ( X2 != Z )
           => ( ( ( Z = Y2 )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X2 @ Y2 )
                  = ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X2 @ Y2 @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z @ ( relati2677767559083392098h2_a_b @ X2 @ Y2 @ Q ) ) @ X2 @ Y2 ) ) ) )
              & ( ( Z != Y2 )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X2 @ Y2 )
                  = ( relati3989891337220013914ts_a_b @ Z @ ( relational_subst_a_b @ Q @ X2 @ Y2 ) ) ) ) ) ) ) )
      & ( ~ ( member_nat @ Z @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z @ Q ) @ X2 @ Y2 )
          = ( relational_subst_a_b @ Q @ X2 @ Y2 ) ) ) ) ).

% subst_exists
thf(fact_368_substs__Disj,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
        = ( relational_Disj_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q1 )
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q22 ) ) ) ) ).

% substs_Disj
thf(fact_369_sr__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ! [X4: relational_fmla_a_b] :
          ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
         => ( ( relational_fv_a_b @ X4 )
            = ( relational_fv_a_b @ Q ) ) )
     => ( ( relational_sr_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
        = ( ! [X: relational_fmla_a_b] :
              ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Qs ) )
             => ( relational_sr_a_b @ X ) )
          & ( relational_sr_a_b @ Q ) ) ) ) ).

% sr_foldr_Disj
thf(fact_370_map__zip__map,axiom,
    ! [F2: product_prod_nat_a > ( nat > a ) > nat > a,G: nat > nat,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr2159937840280155150_nat_a @ F2 @ ( zip_nat_a @ ( map_nat_nat @ G @ Xs ) @ Ys ) )
      = ( map_Pr2159937840280155150_nat_a
        @ ( produc2909000522608705447_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_371_map__zip__map,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a,G: nat > a,Xs: list_nat,Ys: list_nat] :
      ( ( map_Pr990148473491725286_nat_a @ F2 @ ( zip_a_nat @ ( map_nat_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr7092700573198925024_nat_a
        @ ( produc7553103386638219761_nat_a
          @ ^ [X: nat,Y: nat] : ( F2 @ ( product_Pair_a_nat @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_372_map__zip__map,axiom,
    ! [F2: product_prod_nat_a > product_prod_a_nat,G: nat > nat,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr3415083616219229670_a_nat @ F2 @ ( zip_nat_a @ ( map_nat_nat @ G @ Xs ) @ Ys ) )
      = ( map_Pr3415083616219229670_a_nat
        @ ( produc732906326552059263_a_nat
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_373_map__zip__map,axiom,
    ! [F2: product_prod_a_a > product_prod_a_nat,G: nat > a,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr2104877469212525664_a_nat @ F2 @ ( zip_a_a @ ( map_nat_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr3415083616219229670_a_nat
        @ ( produc732906326552059263_a_nat
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_a_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_374_map__zip__map,axiom,
    ! [F2: product_prod_nat_a > nat > a,G: nat > nat,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr7512474648352264252_nat_a @ F2 @ ( zip_nat_a @ ( map_nat_nat @ G @ Xs ) @ Ys ) )
      = ( map_Pr7512474648352264252_nat_a
        @ ( produc4481717121449037155_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_375_map__zip__map,axiom,
    ! [F2: product_prod_a_a > nat > a,G: nat > a,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr5024032959400171842_nat_a @ F2 @ ( zip_a_a @ ( map_nat_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr7512474648352264252_nat_a
        @ ( produc4481717121449037155_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_a_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_376_map__zip__map,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_a,G: a > nat,Xs: list_a,Ys: list_nat] :
      ( ( map_Pr7092700573198925024_nat_a @ F2 @ ( zip_nat_nat @ ( map_a_nat @ G @ Xs ) @ Ys ) )
      = ( map_Pr990148473491725286_nat_a
        @ ( produc2753474147469666119_nat_a
          @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_nat_nat @ ( G @ X ) @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_377_map__zip__map,axiom,
    ! [F2: product_prod_b_nat > product_prod_nat_a,G: a > b,Xs: list_a,Ys: list_nat] :
      ( ( map_Pr521579360530524903_nat_a @ F2 @ ( zip_b_nat @ ( map_a_b @ G @ Xs ) @ Ys ) )
      = ( map_Pr990148473491725286_nat_a
        @ ( produc2753474147469666119_nat_a
          @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_b_nat @ ( G @ X ) @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_378_map__zip__map,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a,G: a > a,Xs: list_a,Ys: list_nat] :
      ( ( map_Pr990148473491725286_nat_a @ F2 @ ( zip_a_nat @ ( map_a_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr990148473491725286_nat_a
        @ ( produc2753474147469666119_nat_a
          @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_a_nat @ ( G @ X ) @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_379_map__zip__map,axiom,
    ! [F2: product_prod_a_a > ( nat > a ) > nat > a,G: nat > a,Xs: list_nat,Ys: list_a] :
      ( ( map_Pr7631421235243360904_nat_a @ F2 @ ( zip_a_a @ ( map_nat_a @ G @ Xs ) @ Ys ) )
      = ( map_Pr2159937840280155150_nat_a
        @ ( produc2909000522608705447_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_a_a @ ( G @ X ) @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map
thf(fact_380_old_Oprod_Oinject,axiom,
    ! [A3: nat,B2: a,A5: nat,B5: a] :
      ( ( ( product_Pair_nat_a @ A3 @ B2 )
        = ( product_Pair_nat_a @ A5 @ B5 ) )
      = ( ( A3 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_381_old_Oprod_Oinject,axiom,
    ! [A3: product_prod_b_nat > set_list_a,B2: nat > a,A5: product_prod_b_nat > set_list_a,B5: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ A3 @ B2 )
        = ( produc2895298938842563487_nat_a @ A5 @ B5 ) )
      = ( ( A3 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_382_old_Oprod_Oinject,axiom,
    ! [A3: b,B2: nat,A5: b,B5: nat] :
      ( ( ( product_Pair_b_nat @ A3 @ B2 )
        = ( product_Pair_b_nat @ A5 @ B5 ) )
      = ( ( A3 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_383_old_Oprod_Oinject,axiom,
    ! [A3: a,B2: nat,A5: a,B5: nat] :
      ( ( ( product_Pair_a_nat @ A3 @ B2 )
        = ( product_Pair_a_nat @ A5 @ B5 ) )
      = ( ( A3 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_384_old_Oprod_Oinject,axiom,
    ! [A3: a,B2: a,A5: a,B5: a] :
      ( ( ( product_Pair_a_a @ A3 @ B2 )
        = ( product_Pair_a_a @ A5 @ B5 ) )
      = ( ( A3 = A5 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
thf(fact_385_prod_Oinject,axiom,
    ! [X12: nat,X23: a,Y1: nat,Y22: a] :
      ( ( ( product_Pair_nat_a @ X12 @ X23 )
        = ( product_Pair_nat_a @ Y1 @ Y22 ) )
      = ( ( X12 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_386_prod_Oinject,axiom,
    ! [X12: product_prod_b_nat > set_list_a,X23: nat > a,Y1: product_prod_b_nat > set_list_a,Y22: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ X12 @ X23 )
        = ( produc2895298938842563487_nat_a @ Y1 @ Y22 ) )
      = ( ( X12 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_387_prod_Oinject,axiom,
    ! [X12: b,X23: nat,Y1: b,Y22: nat] :
      ( ( ( product_Pair_b_nat @ X12 @ X23 )
        = ( product_Pair_b_nat @ Y1 @ Y22 ) )
      = ( ( X12 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_388_prod_Oinject,axiom,
    ! [X12: a,X23: nat,Y1: a,Y22: nat] :
      ( ( ( product_Pair_a_nat @ X12 @ X23 )
        = ( product_Pair_a_nat @ Y1 @ Y22 ) )
      = ( ( X12 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_389_prod_Oinject,axiom,
    ! [X12: a,X23: a,Y1: a,Y22: a] :
      ( ( ( product_Pair_a_a @ X12 @ X23 )
        = ( product_Pair_a_a @ Y1 @ Y22 ) )
      = ( ( X12 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_390_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_a,Z: nat,C: nat > a > set_nat] :
      ( ! [A2: nat,B: a] :
          ( ( P2
            = ( product_Pair_nat_a @ A2 @ B ) )
         => ( member_nat @ Z @ ( C @ A2 @ B ) ) )
     => ( member_nat @ Z @ ( produc4363688908303775880et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_391_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_a,Z: a,C: nat > a > set_a] :
      ( ! [A2: nat,B: a] :
          ( ( P2
            = ( product_Pair_nat_a @ A2 @ B ) )
         => ( member_a @ Z @ ( C @ A2 @ B ) ) )
     => ( member_a @ Z @ ( produc3640957956001042396_set_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_392_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_nat,Z: nat,C: b > nat > set_nat] :
      ( ! [A2: b,B: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A2 @ B ) )
         => ( member_nat @ Z @ ( C @ A2 @ B ) ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_393_mem__case__prodI2,axiom,
    ! [P2: product_prod_b_nat,Z: a,C: b > nat > set_a] :
      ( ! [A2: b,B: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A2 @ B ) )
         => ( member_a @ Z @ ( C @ A2 @ B ) ) )
     => ( member_a @ Z @ ( produc6760141347230400505_set_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_394_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_nat,Z: nat,C: a > nat > set_nat] :
      ( ! [A2: a,B: nat] :
          ( ( P2
            = ( product_Pair_a_nat @ A2 @ B ) )
         => ( member_nat @ Z @ ( C @ A2 @ B ) ) )
     => ( member_nat @ Z @ ( produc1217782509110736682et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_395_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_nat,Z: a,C: a > nat > set_a] :
      ( ! [A2: a,B: nat] :
          ( ( P2
            = ( product_Pair_a_nat @ A2 @ B ) )
         => ( member_a @ Z @ ( C @ A2 @ B ) ) )
     => ( member_a @ Z @ ( produc3842549568706409914_set_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_396_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: nat,C: a > a > set_nat] :
      ( ! [A2: a,B: a] :
          ( ( P2
            = ( product_Pair_a_a @ A2 @ B ) )
         => ( member_nat @ Z @ ( C @ A2 @ B ) ) )
     => ( member_nat @ Z @ ( produc153843693180602034et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_397_mem__case__prodI2,axiom,
    ! [P2: product_prod_a_a,Z: a,C: a > a > set_a] :
      ( ! [A2: a,B: a] :
          ( ( P2
            = ( product_Pair_a_a @ A2 @ B ) )
         => ( member_a @ Z @ ( C @ A2 @ B ) ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_398_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_a,Z: list_a,C: nat > a > set_list_a] :
      ( ! [A2: nat,B: a] :
          ( ( P2
            = ( product_Pair_nat_a @ A2 @ B ) )
         => ( member_list_a @ Z @ ( C @ A2 @ B ) ) )
     => ( member_list_a @ Z @ ( produc2518682296622388194list_a @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_399_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_a,Z: set_nat,C: nat > a > set_set_nat] :
      ( ! [A2: nat,B: a] :
          ( ( P2
            = ( product_Pair_nat_a @ A2 @ B ) )
         => ( member_set_nat @ Z @ ( C @ A2 @ B ) ) )
     => ( member_set_nat @ Z @ ( produc742995856587207998et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_400_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > a > set_nat,A3: nat,B2: a] :
      ( ( member_nat @ Z @ ( C @ A3 @ B2 ) )
     => ( member_nat @ Z @ ( produc4363688908303775880et_nat @ C @ ( product_Pair_nat_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_401_mem__case__prodI,axiom,
    ! [Z: a,C: nat > a > set_a,A3: nat,B2: a] :
      ( ( member_a @ Z @ ( C @ A3 @ B2 ) )
     => ( member_a @ Z @ ( produc3640957956001042396_set_a @ C @ ( product_Pair_nat_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_402_mem__case__prodI,axiom,
    ! [Z: nat,C: b > nat > set_nat,A3: b,B2: nat] :
      ( ( member_nat @ Z @ ( C @ A3 @ B2 ) )
     => ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ ( product_Pair_b_nat @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_403_mem__case__prodI,axiom,
    ! [Z: a,C: b > nat > set_a,A3: b,B2: nat] :
      ( ( member_a @ Z @ ( C @ A3 @ B2 ) )
     => ( member_a @ Z @ ( produc6760141347230400505_set_a @ C @ ( product_Pair_b_nat @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_404_mem__case__prodI,axiom,
    ! [Z: nat,C: a > nat > set_nat,A3: a,B2: nat] :
      ( ( member_nat @ Z @ ( C @ A3 @ B2 ) )
     => ( member_nat @ Z @ ( produc1217782509110736682et_nat @ C @ ( product_Pair_a_nat @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_405_mem__case__prodI,axiom,
    ! [Z: a,C: a > nat > set_a,A3: a,B2: nat] :
      ( ( member_a @ Z @ ( C @ A3 @ B2 ) )
     => ( member_a @ Z @ ( produc3842549568706409914_set_a @ C @ ( product_Pair_a_nat @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_406_mem__case__prodI,axiom,
    ! [Z: nat,C: a > a > set_nat,A3: a,B2: a] :
      ( ( member_nat @ Z @ ( C @ A3 @ B2 ) )
     => ( member_nat @ Z @ ( produc153843693180602034et_nat @ C @ ( product_Pair_a_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_407_mem__case__prodI,axiom,
    ! [Z: a,C: a > a > set_a,A3: a,B2: a] :
      ( ( member_a @ Z @ ( C @ A3 @ B2 ) )
     => ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ ( product_Pair_a_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_408_mem__case__prodI,axiom,
    ! [Z: list_a,C: nat > a > set_list_a,A3: nat,B2: a] :
      ( ( member_list_a @ Z @ ( C @ A3 @ B2 ) )
     => ( member_list_a @ Z @ ( produc2518682296622388194list_a @ C @ ( product_Pair_nat_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_409_mem__case__prodI,axiom,
    ! [Z: set_nat,C: nat > a > set_set_nat,A3: nat,B2: a] :
      ( ( member_set_nat @ Z @ ( C @ A3 @ B2 ) )
     => ( member_set_nat @ Z @ ( produc742995856587207998et_nat @ C @ ( product_Pair_nat_a @ A3 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_410_case__prodI2,axiom,
    ! [P2: product_prod_nat_a,C: nat > a > $o] :
      ( ! [A2: nat,B: a] :
          ( ( P2
            = ( product_Pair_nat_a @ A2 @ B ) )
         => ( C @ A2 @ B ) )
     => ( produc2746933349376800278at_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_411_case__prodI2,axiom,
    ! [P2: produc5835360497134304175_nat_a,C: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o] :
      ( ! [A2: product_prod_b_nat > set_list_a,B: nat > a] :
          ( ( P2
            = ( produc2895298938842563487_nat_a @ A2 @ B ) )
         => ( C @ A2 @ B ) )
     => ( produc7664446012336723172at_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_412_case__prodI2,axiom,
    ! [P2: product_prod_b_nat,C: b > nat > $o] :
      ( ! [A2: b,B: nat] :
          ( ( P2
            = ( product_Pair_b_nat @ A2 @ B ) )
         => ( C @ A2 @ B ) )
     => ( produc795641402153621683_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_413_case__prodI2,axiom,
    ! [P2: product_prod_a_nat,C: a > nat > $o] :
      ( ! [A2: a,B: nat] :
          ( ( P2
            = ( product_Pair_a_nat @ A2 @ B ) )
         => ( C @ A2 @ B ) )
     => ( produc3680711911437148916_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_414_case__prodI2,axiom,
    ! [P2: product_prod_a_a,C: a > a > $o] :
      ( ! [A2: a,B: a] :
          ( ( P2
            = ( product_Pair_a_a @ A2 @ B ) )
         => ( C @ A2 @ B ) )
     => ( produc6436628058953941356_a_a_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_415_case__prodI,axiom,
    ! [F2: nat > a > $o,A3: nat,B2: a] :
      ( ( F2 @ A3 @ B2 )
     => ( produc2746933349376800278at_a_o @ F2 @ ( product_Pair_nat_a @ A3 @ B2 ) ) ) ).

% case_prodI
thf(fact_416_case__prodI,axiom,
    ! [F2: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,A3: product_prod_b_nat > set_list_a,B2: nat > a] :
      ( ( F2 @ A3 @ B2 )
     => ( produc7664446012336723172at_a_o @ F2 @ ( produc2895298938842563487_nat_a @ A3 @ B2 ) ) ) ).

% case_prodI
thf(fact_417_case__prodI,axiom,
    ! [F2: b > nat > $o,A3: b,B2: nat] :
      ( ( F2 @ A3 @ B2 )
     => ( produc795641402153621683_nat_o @ F2 @ ( product_Pair_b_nat @ A3 @ B2 ) ) ) ).

% case_prodI
thf(fact_418_case__prodI,axiom,
    ! [F2: a > nat > $o,A3: a,B2: nat] :
      ( ( F2 @ A3 @ B2 )
     => ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ A3 @ B2 ) ) ) ).

% case_prodI
thf(fact_419_case__prodI,axiom,
    ! [F2: a > a > $o,A3: a,B2: a] :
      ( ( F2 @ A3 @ B2 )
     => ( produc6436628058953941356_a_a_o @ F2 @ ( product_Pair_a_a @ A3 @ B2 ) ) ) ).

% case_prodI
thf(fact_420_finite__Collect__subsets,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( finite7047420756378620717st_nat
        @ ( collect_set_list_nat
          @ ^ [B6: set_list_nat] : ( ord_le6045566169113846134st_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_421_finite__Collect__subsets,axiom,
    ! [A4: set_list_list_a] :
      ( ( finite1660835950917165235list_a @ A4 )
     => ( finite7235877257443484435list_a
        @ ( collec191490921587283541list_a
          @ ^ [B6: set_list_list_a] : ( ord_le8488217952732425610list_a @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_422_finite__Collect__subsets,axiom,
    ! [A4: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( finite6594153429226962157list_a
        @ ( collec3809296942973202735list_a
          @ ^ [B6: set_set_list_a] : ( ord_le8877086941679407844list_a @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_423_finite__Collect__subsets,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( finite6739761609112101331et_nat
        @ ( collect_set_set_nat
          @ ^ [B6: set_set_nat] : ( ord_le6893508408891458716et_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_424_finite__Collect__subsets,axiom,
    ! [A4: set_a] :
      ( ( finite_finite_a @ A4 )
     => ( finite_finite_set_a
        @ ( collect_set_a
          @ ^ [B6: set_a] : ( ord_less_eq_set_a @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_425_finite__Collect__subsets,axiom,
    ! [A4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [B6: set_list_a] : ( ord_le8861187494160871172list_a @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_426_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_427_case__prod__conv,axiom,
    ! [F2: nat > a > product_prod_a_nat,A3: nat,B2: a] :
      ( ( produc732906326552059263_a_nat @ F2 @ ( product_Pair_nat_a @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_428_case__prod__conv,axiom,
    ! [F2: nat > a > nat > a,A3: nat,B2: a] :
      ( ( produc4481717121449037155_nat_a @ F2 @ ( product_Pair_nat_a @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_429_case__prod__conv,axiom,
    ! [F2: a > nat > product_prod_nat_a,A3: a,B2: nat] :
      ( ( produc2753474147469666119_nat_a @ F2 @ ( product_Pair_a_nat @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_430_case__prod__conv,axiom,
    ! [F2: a > nat > ( a > nat ) > a > nat,A3: a,B2: nat] :
      ( ( produc7013214046051809481_a_nat @ F2 @ ( product_Pair_a_nat @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_431_case__prod__conv,axiom,
    ! [F2: a > a > ( a > a ) > a > a,A3: a,B2: a] :
      ( ( produc2369190251411148053_a_a_a @ F2 @ ( product_Pair_a_a @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_432_case__prod__conv,axiom,
    ! [F2: nat > a > ( nat > a ) > nat > a,A3: nat,B2: a] :
      ( ( produc2909000522608705447_nat_a @ F2 @ ( product_Pair_nat_a @ A3 @ B2 ) )
      = ( F2 @ A3 @ B2 ) ) ).

% case_prod_conv
thf(fact_433_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_Pr4193341848836149977_nat_a,N: nat] :
      ( ( finite659689790015031866_nat_a @ A4 )
     => ( finite6757528592613115722_nat_a
        @ ( collec4933398170921330824_nat_a
          @ ^ [Xs3: list_P2851791750731487283_nat_a] :
              ( ( ( size_s243904063682394823_nat_a @ Xs3 )
                = N )
              & ( distin7399675782055410666_nat_a @ Xs3 )
              & ( ord_le7924913712489149241_nat_a @ ( set_Pr4163146838226711502_nat_a @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_434_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_Product_prod_a_a,N: nat] :
      ( ( finite6544458595007987280od_a_a @ A4 )
     => ( finite3567696607326427862od_a_a
        @ ( collec2774578310585404696od_a_a
          @ ^ [Xs3: list_P1396940483166286381od_a_a] :
              ( ( ( size_s3885678630836030617od_a_a @ Xs3 )
                = N )
              & ( distin132333870042060960od_a_a @ Xs3 )
              & ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_435_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_list_list_a,N: nat] :
      ( ( finite1660835950917165235list_a @ A4 )
     => ( finite2085839564794461113list_a
        @ ( collec1292721268053437947list_a
          @ ^ [Xs3: list_list_list_a] :
              ( ( ( size_s2403821588304063868list_a @ Xs3 )
                = N )
              & ( distinct_list_list_a @ Xs3 )
              & ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_436_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( finite8170528100393595399st_nat
        @ ( collec5989764272469232197st_nat
          @ ^ [Xs3: list_list_nat] :
              ( ( ( size_s3023201423986296836st_nat @ Xs3 )
                = N )
              & ( distinct_list_nat @ Xs3 )
              & ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_437_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_Re5178783185447174953term_a,N: nat] :
      ( ( finite1361000192780721522term_a @ A4 )
     => ( finite8994012911387560184term_a
        @ ( collec8200894614646537018term_a
          @ ^ [Xs3: list_R6823256787227418703term_a] :
              ( ( ( size_s88622898042387131term_a @ Xs3 )
                = N )
              & ( distin4172247504669571010term_a @ Xs3 )
              & ( ord_le6173019262470748873term_a @ ( set_Re3569617851344498910term_a @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_438_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_set_list_a,N: nat] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( finite3202133031812794515list_a
        @ ( collec5381118732811369429list_a
          @ ^ [Xs3: list_set_list_a] :
              ( ( ( size_s1991367317912710102list_a @ Xs3 )
                = N )
              & ( distinct_set_list_a @ Xs3 )
              & ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_439_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( finite1091814263879798189et_nat
        @ ( collect_list_set_nat
          @ ^ [Xs3: list_set_nat] :
              ( ( ( size_s3254054031482475050et_nat @ Xs3 )
                = N )
              & ( distinct_set_nat @ Xs3 )
              & ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_440_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ( size_s349497388124573686list_a @ Xs3 )
                = N )
              & ( distinct_list_a @ Xs3 )
              & ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_441_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = N )
              & ( distinct_nat @ Xs3 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_442_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_a,N: nat] :
      ( ( finite_finite_a @ A4 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ( size_size_list_a @ Xs3 )
                = N )
              & ( distinct_a @ Xs3 )
              & ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_443_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > a > set_nat,P2: product_prod_nat_a] :
      ( ( member_nat @ Z @ ( produc4363688908303775880et_nat @ C @ P2 ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( P2
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_444_mem__case__prodE,axiom,
    ! [Z: a,C: nat > a > set_a,P2: product_prod_nat_a] :
      ( ( member_a @ Z @ ( produc3640957956001042396_set_a @ C @ P2 ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( P2
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_445_mem__case__prodE,axiom,
    ! [Z: nat,C: b > nat > set_nat,P2: product_prod_b_nat] :
      ( ( member_nat @ Z @ ( produc7337630463249427243et_nat @ C @ P2 ) )
     => ~ ! [X4: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_446_mem__case__prodE,axiom,
    ! [Z: a,C: b > nat > set_a,P2: product_prod_b_nat] :
      ( ( member_a @ Z @ ( produc6760141347230400505_set_a @ C @ P2 ) )
     => ~ ! [X4: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X4 @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_447_mem__case__prodE,axiom,
    ! [Z: nat,C: a > nat > set_nat,P2: product_prod_a_nat] :
      ( ( member_nat @ Z @ ( produc1217782509110736682et_nat @ C @ P2 ) )
     => ~ ! [X4: a,Y3: nat] :
            ( ( P2
              = ( product_Pair_a_nat @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_448_mem__case__prodE,axiom,
    ! [Z: a,C: a > nat > set_a,P2: product_prod_a_nat] :
      ( ( member_a @ Z @ ( produc3842549568706409914_set_a @ C @ P2 ) )
     => ~ ! [X4: a,Y3: nat] :
            ( ( P2
              = ( product_Pair_a_nat @ X4 @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_449_mem__case__prodE,axiom,
    ! [Z: nat,C: a > a > set_nat,P2: product_prod_a_a] :
      ( ( member_nat @ Z @ ( produc153843693180602034et_nat @ C @ P2 ) )
     => ~ ! [X4: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X4 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_450_mem__case__prodE,axiom,
    ! [Z: a,C: a > a > set_a,P2: product_prod_a_a] :
      ( ( member_a @ Z @ ( produc9217457822752978994_set_a @ C @ P2 ) )
     => ~ ! [X4: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X4 @ Y3 ) )
           => ~ ( member_a @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_451_mem__case__prodE,axiom,
    ! [Z: list_a,C: nat > a > set_list_a,P2: product_prod_nat_a] :
      ( ( member_list_a @ Z @ ( produc2518682296622388194list_a @ C @ P2 ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( P2
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( member_list_a @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_452_mem__case__prodE,axiom,
    ! [Z: set_nat,C: nat > a > set_set_nat,P2: product_prod_nat_a] :
      ( ( member_set_nat @ Z @ ( produc742995856587207998et_nat @ C @ P2 ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( P2
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( member_set_nat @ Z @ ( C @ X4 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_453_internal__case__prod__conv,axiom,
    ! [C: nat > a > ( nat > a ) > nat > a,A3: nat,B2: a] :
      ( ( produc8711337534433359697_nat_a @ C @ ( product_Pair_nat_a @ A3 @ B2 ) )
      = ( C @ A3 @ B2 ) ) ).

% internal_case_prod_conv
thf(fact_454_Pair__inject,axiom,
    ! [A3: nat,B2: a,A5: nat,B5: a] :
      ( ( ( product_Pair_nat_a @ A3 @ B2 )
        = ( product_Pair_nat_a @ A5 @ B5 ) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_455_Pair__inject,axiom,
    ! [A3: product_prod_b_nat > set_list_a,B2: nat > a,A5: product_prod_b_nat > set_list_a,B5: nat > a] :
      ( ( ( produc2895298938842563487_nat_a @ A3 @ B2 )
        = ( produc2895298938842563487_nat_a @ A5 @ B5 ) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_456_Pair__inject,axiom,
    ! [A3: b,B2: nat,A5: b,B5: nat] :
      ( ( ( product_Pair_b_nat @ A3 @ B2 )
        = ( product_Pair_b_nat @ A5 @ B5 ) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_457_Pair__inject,axiom,
    ! [A3: a,B2: nat,A5: a,B5: nat] :
      ( ( ( product_Pair_a_nat @ A3 @ B2 )
        = ( product_Pair_a_nat @ A5 @ B5 ) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_458_Pair__inject,axiom,
    ! [A3: a,B2: a,A5: a,B5: a] :
      ( ( ( product_Pair_a_a @ A3 @ B2 )
        = ( product_Pair_a_a @ A5 @ B5 ) )
     => ~ ( ( A3 = A5 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
thf(fact_459_prod__cases,axiom,
    ! [P: product_prod_nat_a > $o,P2: product_prod_nat_a] :
      ( ! [A2: nat,B: a] : ( P @ ( product_Pair_nat_a @ A2 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_460_prod__cases,axiom,
    ! [P: produc5835360497134304175_nat_a > $o,P2: produc5835360497134304175_nat_a] :
      ( ! [A2: product_prod_b_nat > set_list_a,B: nat > a] : ( P @ ( produc2895298938842563487_nat_a @ A2 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_461_prod__cases,axiom,
    ! [P: product_prod_b_nat > $o,P2: product_prod_b_nat] :
      ( ! [A2: b,B: nat] : ( P @ ( product_Pair_b_nat @ A2 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_462_prod__cases,axiom,
    ! [P: product_prod_a_nat > $o,P2: product_prod_a_nat] :
      ( ! [A2: a,B: nat] : ( P @ ( product_Pair_a_nat @ A2 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_463_prod__cases,axiom,
    ! [P: product_prod_a_a > $o,P2: product_prod_a_a] :
      ( ! [A2: a,B: a] : ( P @ ( product_Pair_a_a @ A2 @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_464_surj__pair,axiom,
    ! [P2: product_prod_nat_a] :
    ? [X4: nat,Y3: a] :
      ( P2
      = ( product_Pair_nat_a @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_465_surj__pair,axiom,
    ! [P2: produc5835360497134304175_nat_a] :
    ? [X4: product_prod_b_nat > set_list_a,Y3: nat > a] :
      ( P2
      = ( produc2895298938842563487_nat_a @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_466_surj__pair,axiom,
    ! [P2: product_prod_b_nat] :
    ? [X4: b,Y3: nat] :
      ( P2
      = ( product_Pair_b_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_467_surj__pair,axiom,
    ! [P2: product_prod_a_nat] :
    ? [X4: a,Y3: nat] :
      ( P2
      = ( product_Pair_a_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_468_surj__pair,axiom,
    ! [P2: product_prod_a_a] :
    ? [X4: a,Y3: a] :
      ( P2
      = ( product_Pair_a_a @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_469_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_470_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_a] :
      ~ ! [A2: nat,B: a] :
          ( Y2
         != ( product_Pair_nat_a @ A2 @ B ) ) ).

% old.prod.exhaust
thf(fact_471_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc5835360497134304175_nat_a] :
      ~ ! [A2: product_prod_b_nat > set_list_a,B: nat > a] :
          ( Y2
         != ( produc2895298938842563487_nat_a @ A2 @ B ) ) ).

% old.prod.exhaust
thf(fact_472_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_b_nat] :
      ~ ! [A2: b,B: nat] :
          ( Y2
         != ( product_Pair_b_nat @ A2 @ B ) ) ).

% old.prod.exhaust
thf(fact_473_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_a_nat] :
      ~ ! [A2: a,B: nat] :
          ( Y2
         != ( product_Pair_a_nat @ A2 @ B ) ) ).

% old.prod.exhaust
thf(fact_474_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_a_a] :
      ~ ! [A2: a,B: a] :
          ( Y2
         != ( product_Pair_a_a @ A2 @ B ) ) ).

% old.prod.exhaust
thf(fact_475_fresh2_I2_J,axiom,
    ! [Y2: nat,X2: nat,Q: relational_fmla_a_b] :
      ( Y2
     != ( relati2677767559083392098h2_a_b @ X2 @ Y2 @ Q ) ) ).

% fresh2(2)
thf(fact_476_fresh2_I1_J,axiom,
    ! [X2: nat,Y2: nat,Q: relational_fmla_a_b] :
      ( X2
     != ( relati2677767559083392098h2_a_b @ X2 @ Y2 @ Q ) ) ).

% fresh2(1)
thf(fact_477_distinct__zipI2,axiom,
    ! [Ys: list_nat,Xs: list_a] :
      ( ( distinct_nat @ Ys )
     => ( distin4161512318331733700_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_478_distinct__zipI2,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( distinct_nat @ Ys )
     => ( distin6923225563576452346at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_479_distinct__zipI2,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( distinct_a @ Ys )
     => ( distin132333870042060960od_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_480_distinct__zipI2,axiom,
    ! [Ys: list_P2851791750731487283_nat_a,Xs: list_nat] :
      ( ( distin7399675782055410666_nat_a @ Ys )
     => ( distin4551915876771988557_nat_a @ ( zip_na7849663319851687980_nat_a @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_481_distinct__zipI2,axiom,
    ! [Ys: list_a,Xs: list_nat] :
      ( ( distinct_a @ Ys )
     => ( distin7399675782055410666_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_482_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distin6923225563576452346at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_483_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_P2851791750731487283_nat_a] :
      ( ( distinct_nat @ Xs )
     => ( distin4551915876771988557_nat_a @ ( zip_na7849663319851687980_nat_a @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_484_distinct__zipI1,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( distinct_a @ Xs )
     => ( distin132333870042060960od_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_485_distinct__zipI1,axiom,
    ! [Xs: list_a,Ys: list_nat] :
      ( ( distinct_a @ Xs )
     => ( distin4161512318331733700_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_486_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( distinct_nat @ Xs )
     => ( distin7399675782055410666_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_487_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_a] :
      ( ( produc3971069790275736229_nat_a @ product_Pair_nat_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_488_case__prod__Pair__iden,axiom,
    ! [P2: produc5835360497134304175_nat_a] :
      ( ( produc5978674056842407163_nat_a @ produc2895298938842563487_nat_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_489_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_b_nat] :
      ( ( produc282185899741183267_b_nat @ product_Pair_b_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_490_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_a_nat] :
      ( ( produc8738682720600764961_a_nat @ product_Pair_a_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_491_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_a_a] :
      ( ( produc408267641121961211od_a_a @ product_Pair_a_a @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_492_case__prodE,axiom,
    ! [C: nat > a > $o,P2: product_prod_nat_a] :
      ( ( produc2746933349376800278at_a_o @ C @ P2 )
     => ~ ! [X4: nat,Y3: a] :
            ( ( P2
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( C @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_493_case__prodE,axiom,
    ! [C: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,P2: produc5835360497134304175_nat_a] :
      ( ( produc7664446012336723172at_a_o @ C @ P2 )
     => ~ ! [X4: product_prod_b_nat > set_list_a,Y3: nat > a] :
            ( ( P2
              = ( produc2895298938842563487_nat_a @ X4 @ Y3 ) )
           => ~ ( C @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_494_case__prodE,axiom,
    ! [C: b > nat > $o,P2: product_prod_b_nat] :
      ( ( produc795641402153621683_nat_o @ C @ P2 )
     => ~ ! [X4: b,Y3: nat] :
            ( ( P2
              = ( product_Pair_b_nat @ X4 @ Y3 ) )
           => ~ ( C @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_495_case__prodE,axiom,
    ! [C: a > nat > $o,P2: product_prod_a_nat] :
      ( ( produc3680711911437148916_nat_o @ C @ P2 )
     => ~ ! [X4: a,Y3: nat] :
            ( ( P2
              = ( product_Pair_a_nat @ X4 @ Y3 ) )
           => ~ ( C @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_496_case__prodE,axiom,
    ! [C: a > a > $o,P2: product_prod_a_a] :
      ( ( produc6436628058953941356_a_a_o @ C @ P2 )
     => ~ ! [X4: a,Y3: a] :
            ( ( P2
              = ( product_Pair_a_a @ X4 @ Y3 ) )
           => ~ ( C @ X4 @ Y3 ) ) ) ).

% case_prodE
thf(fact_497_case__prodD,axiom,
    ! [F2: nat > a > $o,A3: nat,B2: a] :
      ( ( produc2746933349376800278at_a_o @ F2 @ ( product_Pair_nat_a @ A3 @ B2 ) )
     => ( F2 @ A3 @ B2 ) ) ).

% case_prodD
thf(fact_498_case__prodD,axiom,
    ! [F2: ( product_prod_b_nat > set_list_a ) > ( nat > a ) > $o,A3: product_prod_b_nat > set_list_a,B2: nat > a] :
      ( ( produc7664446012336723172at_a_o @ F2 @ ( produc2895298938842563487_nat_a @ A3 @ B2 ) )
     => ( F2 @ A3 @ B2 ) ) ).

% case_prodD
thf(fact_499_case__prodD,axiom,
    ! [F2: b > nat > $o,A3: b,B2: nat] :
      ( ( produc795641402153621683_nat_o @ F2 @ ( product_Pair_b_nat @ A3 @ B2 ) )
     => ( F2 @ A3 @ B2 ) ) ).

% case_prodD
thf(fact_500_case__prodD,axiom,
    ! [F2: a > nat > $o,A3: a,B2: nat] :
      ( ( produc3680711911437148916_nat_o @ F2 @ ( product_Pair_a_nat @ A3 @ B2 ) )
     => ( F2 @ A3 @ B2 ) ) ).

% case_prodD
thf(fact_501_case__prodD,axiom,
    ! [F2: a > a > $o,A3: a,B2: a] :
      ( ( produc6436628058953941356_a_a_o @ F2 @ ( product_Pair_a_a @ A3 @ B2 ) )
     => ( F2 @ A3 @ B2 ) ) ).

% case_prodD
thf(fact_502_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_a] :
      ( ( zip_na7849663319851687980_nat_a @ Xs @ ( zip_nat_a @ Ys @ Zs ) )
      = ( map_Pr2333666522789686724_nat_a
        @ ( produc1387242609909889523_nat_a
          @ ^ [Y: nat] :
              ( produc2545732676826019794_nat_a
              @ ^ [X: nat,Z2: a] : ( produc3386236975552542988_nat_a @ X @ ( product_Pair_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_na7849663319851687980_nat_a @ Ys @ ( zip_nat_a @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_503_zip__left__commute,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_nat] :
      ( ( zip_a_4627834845020498874at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr4315978473967884992at_nat
        @ ( produc8870274859882306031at_nat
          @ ^ [Y: nat] :
              ( produc2856186189040280070at_nat
              @ ^ [X: a,Z2: nat] : ( produc164408500721353882at_nat @ X @ ( product_Pair_nat_nat @ Y @ Z2 ) ) ) )
        @ ( zip_na4611499856128011014_a_nat @ Ys @ ( zip_a_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_504_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr6261813372141627026at_nat
        @ ( produc968775922737392939at_nat
          @ ^ [Y: nat] :
              ( produc9083241971206738548at_nat
              @ ^ [X: nat,Z2: nat] : ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y @ Z2 ) ) ) )
        @ ( zip_na1006125974040638520at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_505_zip__left__commute,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_a] :
      ( ( zip_a_3617590485355791658_nat_a @ Xs @ ( zip_nat_a @ Ys @ Zs ) )
      = ( map_Pr8661482707881699810_nat_a
        @ ( produc49806113092040955_nat_a
          @ ^ [Y: nat] :
              ( produc5195428540743832060_nat_a
              @ ^ [X: a,Z2: a] : ( produc5715557436965102666_nat_a @ X @ ( product_Pair_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_na4928441614332330206od_a_a @ Ys @ ( zip_a_a @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_506_zip__left__commute,axiom,
    ! [Xs: list_a,Ys: list_b,Zs: list_nat] :
      ( ( zip_a_1614871350588509189_b_nat @ Xs @ ( zip_b_nat @ Ys @ Zs ) )
      = ( map_Pr7367901545490869660_b_nat
        @ ( produc7324804376757242493_b_nat
          @ ^ [Y: b] :
              ( produc1845523455684344783_b_nat
              @ ^ [X: a,Z2: nat] : ( produc3712838302197820197_b_nat @ X @ ( product_Pair_b_nat @ Y @ Z2 ) ) ) )
        @ ( zip_b_7486371000496415493_a_nat @ Ys @ ( zip_a_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_507_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_b,Zs: list_nat] :
      ( ( zip_na5846944185084405511_b_nat @ Xs @ ( zip_b_nat @ Ys @ Zs ) )
      = ( map_Pr6582810724067715530_b_nat
        @ ( produc1256931253707451841_b_nat
          @ ^ [Y: b] :
              ( produc1217974579069012705_b_nat
              @ ^ [X: nat,Z2: nat] : ( produc1383517840785260519_b_nat @ X @ ( product_Pair_b_nat @ Y @ Z2 ) ) ) )
        @ ( zip_b_4974252998908922489at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_508_zip__left__commute,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_nat] :
      ( ( zip_a_379427021632114692_a_nat @ Xs @ ( zip_a_nat @ Ys @ Zs ) )
      = ( map_Pr7340712779209073946_a_nat
        @ ( produc7297615610475446779_a_nat
          @ ^ [Y: a] :
              ( produc5472676877350989902_a_nat
              @ ^ [X: a,Z2: nat] : ( produc2477393973241425700_a_nat @ X @ ( product_Pair_a_nat @ Y @ Z2 ) ) ) )
        @ ( zip_a_379427021632114692_a_nat @ Ys @ ( zip_a_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_509_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_nat] :
      ( ( zip_na4611499856128011014_a_nat @ Xs @ ( zip_a_nat @ Ys @ Zs ) )
      = ( map_Pr4591565078579541192_a_nat
        @ ( produc8489057645074053311_a_nat
          @ ^ [Y: a] :
              ( produc4845128000735657824_a_nat
              @ ^ [X: nat,Z2: nat] : ( produc148073511828866022_a_nat @ X @ ( product_Pair_a_nat @ Y @ Z2 ) ) ) )
        @ ( zip_a_4627834845020498874at_nat @ Ys @ ( zip_nat_nat @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_510_zip__left__commute,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_a] :
      ( ( zip_na4928441614332330206od_a_a @ Xs @ ( zip_a_a @ Ys @ Zs ) )
      = ( map_Pr7857667300534112738od_a_a
        @ ( produc8960266415599428035od_a_a
          @ ^ [Y: a] :
              ( produc132582535617199182od_a_a
              @ ^ [X: nat,Z2: a] : ( produc7026408565941641214od_a_a @ X @ ( product_Pair_a_a @ Y @ Z2 ) ) ) )
        @ ( zip_a_3617590485355791658_nat_a @ Ys @ ( zip_nat_a @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_511_zip__left__commute,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( zip_a_3848334447339249504od_a_a @ Xs @ ( zip_a_a @ Ys @ Zs ) )
      = ( map_Pr243255708951746116od_a_a
        @ ( produc5605962277697872315od_a_a
          @ ^ [Y: a] :
              ( produc182479327980443812od_a_a
              @ ^ [X: a,Z2: a] : ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y @ Z2 ) ) ) )
        @ ( zip_a_3848334447339249504od_a_a @ Ys @ ( zip_a_a @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_512_zip__assoc,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_nat] :
      ( ( zip_a_4627834845020498874at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr3093682038575878976at_nat
        @ ( produc3829246639187309807at_nat
          @ ( produc8268367250359847917at_nat
            @ ^ [X: a,Y: nat,Z2: nat] : ( produc164408500721353882at_nat @ X @ ( product_Pair_nat_nat @ Y @ Z2 ) ) ) )
        @ ( zip_Pr19227448660185094at_nat @ ( zip_a_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_513_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( zip_na1006125974040638520at_nat @ Xs @ ( zip_nat_nat @ Ys @ Zs ) )
      = ( map_Pr1164872866496146796at_nat
        @ ( produc3206169289476954189at_nat
          @ ( produc7810592499157111267at_nat
            @ ^ [X: nat,Y: nat,Z2: nat] : ( produc487386426758144856at_nat @ X @ ( product_Pair_nat_nat @ Y @ Z2 ) ) ) )
        @ ( zip_Pr6869450617852699226at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_514_zip__assoc,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_P2851791750731487283_nat_a] :
      ( ( zip_a_2833505244495620877_nat_a @ Xs @ ( zip_na7849663319851687980_nat_a @ Ys @ Zs ) )
      = ( map_Pr5285739362723675136_nat_a
        @ ( produc1241749739936913711_nat_a
          @ ( produc8507952538743350445_nat_a
            @ ^ [X: a,Y: nat,Z2: product_prod_nat_a] : ( produc3035341915875714029_nat_a @ X @ ( produc3386236975552542988_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr5063037930166122969_nat_a @ ( zip_a_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_515_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_P2851791750731487283_nat_a] :
      ( ( zip_na5177785549967607179_nat_a @ Xs @ ( zip_na7849663319851687980_nat_a @ Ys @ Zs ) )
      = ( map_Pr5692667193431781650_nat_a
        @ ( produc2197140174377443443_nat_a
          @ ( produc76103943688722057_nat_a
            @ ^ [X: nat,Y: nat,Z2: product_prod_nat_a] : ( produc5355117898495153323_nat_a @ X @ ( produc3386236975552542988_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr961747059092150317_nat_a @ ( zip_nat_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_516_zip__assoc,axiom,
    ! [Xs: list_a,Ys: list_nat,Zs: list_a] :
      ( ( zip_a_3617590485355791658_nat_a @ Xs @ ( zip_nat_a @ Ys @ Zs ) )
      = ( map_Pr4399194344325520844_nat_a
        @ ( produc5590375493161346149_nat_a
          @ ( produc1955241629584076827_nat_a
            @ ^ [X: a,Y: nat,Z2: a] : ( produc5715557436965102666_nat_a @ X @ ( product_Pair_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr1991336852937410632_nat_a @ ( zip_a_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_517_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_a] :
      ( ( zip_na7849663319851687980_nat_a @ Xs @ ( zip_nat_a @ Ys @ Zs ) )
      = ( map_Pr3956365365643586996_nat_a
        @ ( produc2641136667158398251_nat_a
          @ ( produc2634911726474885897_nat_a
            @ ^ [X: nat,Y: nat,Z2: a] : ( produc3386236975552542988_nat_a @ X @ ( product_Pair_nat_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr5175989614662165236_nat_a @ ( zip_nat_nat @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_518_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_nat] :
      ( ( zip_na4611499856128011014_a_nat @ Xs @ ( zip_a_nat @ Ys @ Zs ) )
      = ( map_Pr2487136454648023326_a_nat
        @ ( produc3222701055259454157_a_nat
          @ ( produc7396323926689596051_a_nat
            @ ^ [X: nat,Y: a,Z2: nat] : ( produc148073511828866022_a_nat @ X @ ( product_Pair_a_nat @ Y @ Z2 ) ) ) )
        @ ( zip_Pr7570451349882678572_a_nat @ ( zip_nat_a @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_519_zip__assoc,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_nat] :
      ( ( zip_a_379427021632114692_a_nat @ Xs @ ( zip_a_nat @ Ys @ Zs ) )
      = ( map_Pr5453640069514183190_a_nat
        @ ( produc4424194328353004663_a_nat
          @ ( produc8625940116382029381_a_nat
            @ ^ [X: a,Y: a,Z2: nat] : ( produc2477393973241425700_a_nat @ X @ ( product_Pair_a_nat @ Y @ Z2 ) ) ) )
        @ ( zip_Pr4385798588157923968_a_nat @ ( zip_a_a @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_520_zip__assoc,axiom,
    ! [Xs: list_nat,Ys: list_a,Zs: list_a] :
      ( ( zip_na4928441614332330206od_a_a @ Xs @ ( zip_a_a @ Ys @ Zs ) )
      = ( map_Pr1604178888947773090od_a_a
        @ ( produc2795360037783598395od_a_a
          @ ( produc3716946133456241081od_a_a
            @ ^ [X: nat,Y: a,Z2: a] : ( produc7026408565941641214od_a_a @ X @ ( product_Pair_a_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr8262197794775541922at_a_a @ ( zip_nat_a @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_521_zip__assoc,axiom,
    ! [Xs: list_a,Ys: list_a,Zs: list_a] :
      ( ( zip_a_3848334447339249504od_a_a @ Xs @ ( zip_a_a @ Ys @ Zs ) )
      = ( map_Pr3785664189271781462od_a_a
        @ ( produc5524653410268046413od_a_a
          @ ( produc8901291747929497187od_a_a
            @ ^ [X: a,Y: a,Z2: a] : ( produc431845341423274048od_a_a @ X @ ( product_Pair_a_a @ Y @ Z2 ) ) ) )
        @ ( zip_Pr9109025954503487566_a_a_a @ ( zip_a_a @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_522_finite__has__minimal2,axiom,
    ! [A4: set_set_list_a,A3: set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( ( member_set_list_a @ A3 @ A4 )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A4 )
            & ( ord_le8861187494160871172list_a @ X4 @ A3 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A4 )
               => ( ( ord_le8861187494160871172list_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_523_finite__has__minimal2,axiom,
    ! [A4: set_set_set_list_a,A3: set_set_list_a] :
      ( ( finite6594153429226962157list_a @ A4 )
     => ( ( member8857465052274545133list_a @ A3 @ A4 )
       => ? [X4: set_set_list_a] :
            ( ( member8857465052274545133list_a @ X4 @ A4 )
            & ( ord_le8877086941679407844list_a @ X4 @ A3 )
            & ! [Xa: set_set_list_a] :
                ( ( member8857465052274545133list_a @ Xa @ A4 )
               => ( ( ord_le8877086941679407844list_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_524_finite__has__minimal2,axiom,
    ! [A4: set_set_set_nat,A3: set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A4 )
     => ( ( member_set_set_nat @ A3 @ A4 )
       => ? [X4: set_set_nat] :
            ( ( member_set_set_nat @ X4 @ A4 )
            & ( ord_le6893508408891458716et_nat @ X4 @ A3 )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A4 )
               => ( ( ord_le6893508408891458716et_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_525_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A3 @ A4 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( ord_less_eq_set_nat @ X4 @ A3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_526_finite__has__minimal2,axiom,
    ! [A4: set_set_a,A3: set_a] :
      ( ( finite_finite_set_a @ A4 )
     => ( ( member_set_a @ A3 @ A4 )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A4 )
            & ( ord_less_eq_set_a @ X4 @ A3 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A4 )
               => ( ( ord_less_eq_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_527_finite__has__minimal2,axiom,
    ! [A4: set_nat_o,A3: nat > $o] :
      ( ( finite_finite_nat_o @ A4 )
     => ( ( member_nat_o @ A3 @ A4 )
       => ? [X4: nat > $o] :
            ( ( member_nat_o @ X4 @ A4 )
            & ( ord_less_eq_nat_o @ X4 @ A3 )
            & ! [Xa: nat > $o] :
                ( ( member_nat_o @ Xa @ A4 )
               => ( ( ord_less_eq_nat_o @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_528_finite__has__minimal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ X4 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_529_finite__has__maximal2,axiom,
    ! [A4: set_set_list_a,A3: set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( ( member_set_list_a @ A3 @ A4 )
       => ? [X4: set_list_a] :
            ( ( member_set_list_a @ X4 @ A4 )
            & ( ord_le8861187494160871172list_a @ A3 @ X4 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A4 )
               => ( ( ord_le8861187494160871172list_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_530_finite__has__maximal2,axiom,
    ! [A4: set_set_set_list_a,A3: set_set_list_a] :
      ( ( finite6594153429226962157list_a @ A4 )
     => ( ( member8857465052274545133list_a @ A3 @ A4 )
       => ? [X4: set_set_list_a] :
            ( ( member8857465052274545133list_a @ X4 @ A4 )
            & ( ord_le8877086941679407844list_a @ A3 @ X4 )
            & ! [Xa: set_set_list_a] :
                ( ( member8857465052274545133list_a @ Xa @ A4 )
               => ( ( ord_le8877086941679407844list_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_531_finite__has__maximal2,axiom,
    ! [A4: set_set_set_nat,A3: set_set_nat] :
      ( ( finite6739761609112101331et_nat @ A4 )
     => ( ( member_set_set_nat @ A3 @ A4 )
       => ? [X4: set_set_nat] :
            ( ( member_set_set_nat @ X4 @ A4 )
            & ( ord_le6893508408891458716et_nat @ A3 @ X4 )
            & ! [Xa: set_set_nat] :
                ( ( member_set_set_nat @ Xa @ A4 )
               => ( ( ord_le6893508408891458716et_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_532_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A3 @ A4 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( ord_less_eq_set_nat @ A3 @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_533_finite__has__maximal2,axiom,
    ! [A4: set_set_a,A3: set_a] :
      ( ( finite_finite_set_a @ A4 )
     => ( ( member_set_a @ A3 @ A4 )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A4 )
            & ( ord_less_eq_set_a @ A3 @ X4 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A4 )
               => ( ( ord_less_eq_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_534_finite__has__maximal2,axiom,
    ! [A4: set_nat_o,A3: nat > $o] :
      ( ( finite_finite_nat_o @ A4 )
     => ( ( member_nat_o @ A3 @ A4 )
       => ? [X4: nat > $o] :
            ( ( member_nat_o @ X4 @ A4 )
            & ( ord_less_eq_nat_o @ A3 @ X4 )
            & ! [Xa: nat > $o] :
                ( ( member_nat_o @ Xa @ A4 )
               => ( ( ord_less_eq_nat_o @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_535_finite__has__maximal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( ord_less_eq_nat @ A3 @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_536_old_Oprod_Ocase,axiom,
    ! [F2: nat > a > product_prod_a_nat,X12: nat,X23: a] :
      ( ( produc732906326552059263_a_nat @ F2 @ ( product_Pair_nat_a @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_537_old_Oprod_Ocase,axiom,
    ! [F2: nat > a > nat > a,X12: nat,X23: a] :
      ( ( produc4481717121449037155_nat_a @ F2 @ ( product_Pair_nat_a @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_538_old_Oprod_Ocase,axiom,
    ! [F2: a > nat > product_prod_nat_a,X12: a,X23: nat] :
      ( ( produc2753474147469666119_nat_a @ F2 @ ( product_Pair_a_nat @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_539_old_Oprod_Ocase,axiom,
    ! [F2: a > nat > ( a > nat ) > a > nat,X12: a,X23: nat] :
      ( ( produc7013214046051809481_a_nat @ F2 @ ( product_Pair_a_nat @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_540_old_Oprod_Ocase,axiom,
    ! [F2: a > a > ( a > a ) > a > a,X12: a,X23: a] :
      ( ( produc2369190251411148053_a_a_a @ F2 @ ( product_Pair_a_a @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_541_old_Oprod_Ocase,axiom,
    ! [F2: nat > a > ( nat > a ) > nat > a,X12: nat,X23: a] :
      ( ( produc2909000522608705447_nat_a @ F2 @ ( product_Pair_nat_a @ X12 @ X23 ) )
      = ( F2 @ X12 @ X23 ) ) ).

% old.prod.case
thf(fact_542_rev__finite__subset,axiom,
    ! [B4: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B4 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
       => ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_543_rev__finite__subset,axiom,
    ! [B4: set_list_list_a,A4: set_list_list_a] :
      ( ( finite1660835950917165235list_a @ B4 )
     => ( ( ord_le8488217952732425610list_a @ A4 @ B4 )
       => ( finite1660835950917165235list_a @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_544_rev__finite__subset,axiom,
    ! [B4: set_set_list_a,A4: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ B4 )
     => ( ( ord_le8877086941679407844list_a @ A4 @ B4 )
       => ( finite5282473924520328461list_a @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_545_rev__finite__subset,axiom,
    ! [B4: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
       => ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_546_rev__finite__subset,axiom,
    ! [B4: set_a,A4: set_a] :
      ( ( finite_finite_a @ B4 )
     => ( ( ord_less_eq_set_a @ A4 @ B4 )
       => ( finite_finite_a @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_547_rev__finite__subset,axiom,
    ! [B4: set_list_a,A4: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
       => ( finite_finite_list_a @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_548_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_549_infinite__super,axiom,
    ! [S3: set_list_nat,T2: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ S3 @ T2 )
     => ( ~ ( finite8100373058378681591st_nat @ S3 )
       => ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_550_infinite__super,axiom,
    ! [S3: set_list_list_a,T2: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ S3 @ T2 )
     => ( ~ ( finite1660835950917165235list_a @ S3 )
       => ~ ( finite1660835950917165235list_a @ T2 ) ) ) ).

% infinite_super
thf(fact_551_infinite__super,axiom,
    ! [S3: set_set_list_a,T2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ S3 @ T2 )
     => ( ~ ( finite5282473924520328461list_a @ S3 )
       => ~ ( finite5282473924520328461list_a @ T2 ) ) ) ).

% infinite_super
thf(fact_552_infinite__super,axiom,
    ! [S3: set_set_nat,T2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ S3 @ T2 )
     => ( ~ ( finite1152437895449049373et_nat @ S3 )
       => ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_553_infinite__super,axiom,
    ! [S3: set_a,T2: set_a] :
      ( ( ord_less_eq_set_a @ S3 @ T2 )
     => ( ~ ( finite_finite_a @ S3 )
       => ~ ( finite_finite_a @ T2 ) ) ) ).

% infinite_super
thf(fact_554_infinite__super,axiom,
    ! [S3: set_list_a,T2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S3 @ T2 )
     => ( ~ ( finite_finite_list_a @ S3 )
       => ~ ( finite_finite_list_a @ T2 ) ) ) ).

% infinite_super
thf(fact_555_infinite__super,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T2 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_556_finite__subset,axiom,
    ! [A4: set_list_nat,B4: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
     => ( ( finite8100373058378681591st_nat @ B4 )
       => ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_557_finite__subset,axiom,
    ! [A4: set_list_list_a,B4: set_list_list_a] :
      ( ( ord_le8488217952732425610list_a @ A4 @ B4 )
     => ( ( finite1660835950917165235list_a @ B4 )
       => ( finite1660835950917165235list_a @ A4 ) ) ) ).

% finite_subset
thf(fact_558_finite__subset,axiom,
    ! [A4: set_set_list_a,B4: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A4 @ B4 )
     => ( ( finite5282473924520328461list_a @ B4 )
       => ( finite5282473924520328461list_a @ A4 ) ) ) ).

% finite_subset
thf(fact_559_finite__subset,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
     => ( ( finite1152437895449049373et_nat @ B4 )
       => ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_560_finite__subset,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( finite_finite_a @ B4 )
       => ( finite_finite_a @ A4 ) ) ) ).

% finite_subset
thf(fact_561_finite__subset,axiom,
    ! [A4: set_list_a,B4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A4 @ B4 )
     => ( ( finite_finite_list_a @ B4 )
       => ( finite_finite_list_a @ A4 ) ) ) ).

% finite_subset
thf(fact_562_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_563_subset__code_I1_J,axiom,
    ! [Xs: list_nat_a_nat_a,B4: set_nat_a_nat_a] :
      ( ( ord_le7151205609328483131_nat_a @ ( set_nat_a_nat_a2 @ Xs ) @ B4 )
      = ( ! [X: ( nat > a ) > nat > a] :
            ( ( member_nat_a_nat_a @ X @ ( set_nat_a_nat_a2 @ Xs ) )
           => ( member_nat_a_nat_a @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_564_subset__code_I1_J,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,B4: set_Product_prod_a_a] :
      ( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B4 )
      = ( ! [X: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
           => ( member1426531477525435216od_a_a @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_565_subset__code_I1_J,axiom,
    ! [Xs: list_list_nat,B4: set_list_nat] :
      ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B4 )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
           => ( member_list_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_566_subset__code_I1_J,axiom,
    ! [Xs: list_list_a,B4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B4 )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
           => ( member_list_a @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_567_subset__code_I1_J,axiom,
    ! [Xs: list_set_list_a,B4: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ B4 )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( member_set_list_a @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_568_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_569_subset__code_I1_J,axiom,
    ! [Xs: list_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( member_a @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_570_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_571_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A4: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_572_fresh2_I3_J,axiom,
    ! [X2: nat,Y2: nat,Q: relational_fmla_a_b] :
      ~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X2 @ Y2 @ Q ) @ ( relational_fv_a_b @ Q ) ) ).

% fresh2(3)
thf(fact_573_foldr__cong,axiom,
    ! [A3: relational_fmla_a_b,B2: relational_fmla_a_b,L2: list_R8263082107343818799la_a_b,K: list_R8263082107343818799la_a_b,F2: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b,G: relational_fmla_a_b > relational_fmla_a_b > relational_fmla_a_b] :
      ( ( A3 = B2 )
     => ( ( L2 = K )
       => ( ! [A2: relational_fmla_a_b,X4: relational_fmla_a_b] :
              ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ L2 ) )
             => ( ( F2 @ X4 @ A2 )
                = ( G @ X4 @ A2 ) ) )
         => ( ( foldr_789212930732525799la_a_b @ F2 @ L2 @ A3 )
            = ( foldr_789212930732525799la_a_b @ G @ K @ B2 ) ) ) ) ) ).

% foldr_cong
thf(fact_574_cond__case__prod__eta,axiom,
    ! [F2: nat > a > product_prod_a_nat,G: product_prod_nat_a > product_prod_a_nat] :
      ( ! [X4: nat,Y3: a] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_nat_a @ X4 @ Y3 ) ) )
     => ( ( produc732906326552059263_a_nat @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_575_cond__case__prod__eta,axiom,
    ! [F2: nat > a > nat > a,G: product_prod_nat_a > nat > a] :
      ( ! [X4: nat,Y3: a] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_nat_a @ X4 @ Y3 ) ) )
     => ( ( produc4481717121449037155_nat_a @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_576_cond__case__prod__eta,axiom,
    ! [F2: a > nat > product_prod_nat_a,G: product_prod_a_nat > product_prod_nat_a] :
      ( ! [X4: a,Y3: nat] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_a_nat @ X4 @ Y3 ) ) )
     => ( ( produc2753474147469666119_nat_a @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_577_cond__case__prod__eta,axiom,
    ! [F2: a > nat > ( a > nat ) > a > nat,G: product_prod_a_nat > ( a > nat ) > a > nat] :
      ( ! [X4: a,Y3: nat] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_a_nat @ X4 @ Y3 ) ) )
     => ( ( produc7013214046051809481_a_nat @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_578_cond__case__prod__eta,axiom,
    ! [F2: a > a > ( a > a ) > a > a,G: product_prod_a_a > ( a > a ) > a > a] :
      ( ! [X4: a,Y3: a] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_a_a @ X4 @ Y3 ) ) )
     => ( ( produc2369190251411148053_a_a_a @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_579_cond__case__prod__eta,axiom,
    ! [F2: nat > a > ( nat > a ) > nat > a,G: product_prod_nat_a > ( nat > a ) > nat > a] :
      ( ! [X4: nat,Y3: a] :
          ( ( F2 @ X4 @ Y3 )
          = ( G @ ( product_Pair_nat_a @ X4 @ Y3 ) ) )
     => ( ( produc2909000522608705447_nat_a @ F2 )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_580_case__prod__eta,axiom,
    ! [F2: product_prod_nat_a > product_prod_a_nat] :
      ( ( produc732906326552059263_a_nat
        @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_581_case__prod__eta,axiom,
    ! [F2: product_prod_nat_a > nat > a] :
      ( ( produc4481717121449037155_nat_a
        @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_582_case__prod__eta,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a] :
      ( ( produc2753474147469666119_nat_a
        @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_a_nat @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_583_case__prod__eta,axiom,
    ! [F2: product_prod_a_nat > ( a > nat ) > a > nat] :
      ( ( produc7013214046051809481_a_nat
        @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_a_nat @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_584_case__prod__eta,axiom,
    ! [F2: product_prod_a_a > ( a > a ) > a > a] :
      ( ( produc2369190251411148053_a_a_a
        @ ^ [X: a,Y: a] : ( F2 @ ( product_Pair_a_a @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_585_case__prod__eta,axiom,
    ! [F2: product_prod_nat_a > ( nat > a ) > nat > a] :
      ( ( produc2909000522608705447_nat_a
        @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ Y ) ) )
      = F2 ) ).

% case_prod_eta
thf(fact_586_case__prodE2,axiom,
    ! [Q: product_prod_a_nat > $o,P: nat > a > product_prod_a_nat,Z: product_prod_nat_a] :
      ( ( Q @ ( produc732906326552059263_a_nat @ P @ Z ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( Z
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_587_case__prodE2,axiom,
    ! [Q: ( nat > a ) > $o,P: nat > a > nat > a,Z: product_prod_nat_a] :
      ( ( Q @ ( produc4481717121449037155_nat_a @ P @ Z ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( Z
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_588_case__prodE2,axiom,
    ! [Q: product_prod_nat_a > $o,P: a > nat > product_prod_nat_a,Z: product_prod_a_nat] :
      ( ( Q @ ( produc2753474147469666119_nat_a @ P @ Z ) )
     => ~ ! [X4: a,Y3: nat] :
            ( ( Z
              = ( product_Pair_a_nat @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_589_case__prodE2,axiom,
    ! [Q: ( ( a > nat ) > a > nat ) > $o,P: a > nat > ( a > nat ) > a > nat,Z: product_prod_a_nat] :
      ( ( Q @ ( produc7013214046051809481_a_nat @ P @ Z ) )
     => ~ ! [X4: a,Y3: nat] :
            ( ( Z
              = ( product_Pair_a_nat @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_590_case__prodE2,axiom,
    ! [Q: ( ( a > a ) > a > a ) > $o,P: a > a > ( a > a ) > a > a,Z: product_prod_a_a] :
      ( ( Q @ ( produc2369190251411148053_a_a_a @ P @ Z ) )
     => ~ ! [X4: a,Y3: a] :
            ( ( Z
              = ( product_Pair_a_a @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_591_case__prodE2,axiom,
    ! [Q: ( ( nat > a ) > nat > a ) > $o,P: nat > a > ( nat > a ) > nat > a,Z: product_prod_nat_a] :
      ( ( Q @ ( produc2909000522608705447_nat_a @ P @ Z ) )
     => ~ ! [X4: nat,Y3: a] :
            ( ( Z
              = ( product_Pair_nat_a @ X4 @ Y3 ) )
           => ~ ( Q @ ( P @ X4 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_592_zip__commute,axiom,
    ( zip_Pr7570451349882678572_a_nat
    = ( ^ [Xs3: list_P2851791750731487283_nat_a,Ys2: list_nat] :
          ( map_Pr2492294159274913860_a_nat
          @ ( produc1545870246395116659_a_nat
            @ ^ [X: nat,Y: product_prod_nat_a] : ( produc3107025005583533580_a_nat @ Y @ X ) )
          @ ( zip_na7849663319851687980_nat_a @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_593_zip__commute,axiom,
    ( zip_nat_nat
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( map_Pr8058819605623181956at_nat
          @ ( produc2626176000494625587at_nat
            @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
          @ ( zip_nat_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_594_zip__commute,axiom,
    ( zip_na7849663319851687980_nat_a
    = ( ^ [Xs3: list_nat,Ys2: list_P2851791750731487283_nat_a] :
          ( map_Pr1746042891126049604_nat_a
          @ ( produc2481607491737480435_nat_a
            @ ^ [X: product_prod_nat_a,Y: nat] : ( produc3386236975552542988_nat_a @ Y @ X ) )
          @ ( zip_Pr7570451349882678572_a_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_595_zip__commute,axiom,
    ( zip_Pr5387723086728981183_nat_a
    = ( ^ [Xs3: list_P6049048235159712035list_a,Ys2: list_nat_a] :
          ( map_Pr8478378587642173844_nat_a
          @ ( produc4690939927037265291_nat_a
            @ ^ [X: nat > a,Y: product_prod_b_nat > set_list_a] : ( produc2895298938842563487_nat_a @ Y @ X ) )
          @ ( zip_na3962188320132642351list_a @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_596_zip__commute,axiom,
    ( zip_b_nat
    = ( ^ [Xs3: list_b,Ys2: list_nat] :
          ( map_Pr8590476531800460392_b_nat
          @ ( produc9075294634372754561_b_nat
            @ ^ [X: nat,Y: b] : ( product_Pair_b_nat @ Y @ X ) )
          @ ( zip_nat_b @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_597_zip__commute,axiom,
    ( zip_a_a
    = ( ^ [Xs3: list_a,Ys2: list_a] :
          ( map_Pr7904243085458786820od_a_a
          @ ( produc408267641121961211od_a_a
            @ ^ [X: a,Y: a] : ( product_Pair_a_a @ Y @ X ) )
          @ ( zip_a_a @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_598_zip__commute,axiom,
    ( zip_a_nat
    = ( ^ [Xs3: list_a,Ys2: list_nat] :
          ( map_Pr3415083616219229670_a_nat
          @ ( produc732906326552059263_a_nat
            @ ^ [X: nat,Y: a] : ( product_Pair_a_nat @ Y @ X ) )
          @ ( zip_nat_a @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_599_zip__commute,axiom,
    ( zip_nat_a
    = ( ^ [Xs3: list_nat,Ys2: list_a] :
          ( map_Pr990148473491725286_nat_a
          @ ( produc2753474147469666119_nat_a
            @ ^ [X: a,Y: nat] : ( product_Pair_nat_a @ Y @ X ) )
          @ ( zip_a_nat @ Ys2 @ Xs3 ) ) ) ) ).

% zip_commute
thf(fact_600_zip__same__conv__map,axiom,
    ! [Xs: list_nat] :
      ( ( zip_nat_nat @ Xs @ Xs )
      = ( map_na7298421622053143531at_nat
        @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_601_zip__same__conv__map,axiom,
    ! [Xs: list_a] :
      ( ( zip_a_a @ Xs @ Xs )
      = ( map_a_7860052162900579309od_a_a
        @ ^ [X: a] : ( product_Pair_a_a @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_602_finite__distinct__list,axiom,
    ! [A4: set_Pr4193341848836149977_nat_a] :
      ( ( finite659689790015031866_nat_a @ A4 )
     => ? [Xs2: list_P2851791750731487283_nat_a] :
          ( ( ( set_Pr4163146838226711502_nat_a @ Xs2 )
            = A4 )
          & ( distin7399675782055410666_nat_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_603_finite__distinct__list,axiom,
    ! [A4: set_Product_prod_a_a] :
      ( ( finite6544458595007987280od_a_a @ A4 )
     => ? [Xs2: list_P1396940483166286381od_a_a] :
          ( ( ( set_Product_prod_a_a2 @ Xs2 )
            = A4 )
          & ( distin132333870042060960od_a_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_604_finite__distinct__list,axiom,
    ! [A4: set_a] :
      ( ( finite_finite_a @ A4 )
     => ? [Xs2: list_a] :
          ( ( ( set_a2 @ Xs2 )
            = A4 )
          & ( distinct_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_605_finite__distinct__list,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ? [Xs2: list_list_nat] :
          ( ( ( set_list_nat2 @ Xs2 )
            = A4 )
          & ( distinct_list_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_606_finite__distinct__list,axiom,
    ! [A4: set_list_list_a] :
      ( ( finite1660835950917165235list_a @ A4 )
     => ? [Xs2: list_list_list_a] :
          ( ( ( set_list_list_a2 @ Xs2 )
            = A4 )
          & ( distinct_list_list_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_607_finite__distinct__list,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ? [Xs2: list_set_nat] :
          ( ( ( set_set_nat2 @ Xs2 )
            = A4 )
          & ( distinct_set_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_608_finite__distinct__list,axiom,
    ! [A4: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ? [Xs2: list_set_list_a] :
          ( ( ( set_set_list_a2 @ Xs2 )
            = A4 )
          & ( distinct_set_list_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_609_finite__distinct__list,axiom,
    ! [A4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ? [Xs2: list_list_a] :
          ( ( ( set_list_a2 @ Xs2 )
            = A4 )
          & ( distinct_list_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_610_finite__distinct__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs2: list_nat] :
          ( ( ( set_nat2 @ Xs2 )
            = A4 )
          & ( distinct_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_611_set__zip__rightD,axiom,
    ! [X2: nat,Y2: product_prod_nat_a,Xs: list_nat,Ys: list_P2851791750731487283_nat_a] :
      ( ( member2321988868818003709_nat_a @ ( produc3386236975552542988_nat_a @ X2 @ Y2 ) @ ( set_Pr8046525447620763497_nat_a @ ( zip_na7849663319851687980_nat_a @ Xs @ Ys ) ) )
     => ( member8962352052110095674_nat_a @ Y2 @ ( set_Pr4163146838226711502_nat_a @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_612_set__zip__rightD,axiom,
    ! [X2: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_613_set__zip__rightD,axiom,
    ! [X2: product_prod_b_nat > set_list_a,Y2: nat > a,Xs: list_P6049048235159712035list_a,Ys: list_nat_a] :
      ( ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ X2 @ Y2 ) @ ( set_Pr4376371411171465668_nat_a @ ( zip_Pr5387723086728981183_nat_a @ Xs @ Ys ) ) )
     => ( member_nat_a @ Y2 @ ( set_nat_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_614_set__zip__rightD,axiom,
    ! [X2: b,Y2: nat,Xs: list_b,Ys: list_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X2 @ Y2 ) @ ( set_Pr2160427703459429033_b_nat @ ( zip_b_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_615_set__zip__rightD,axiom,
    ! [X2: a,Y2: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_616_set__zip__rightD,axiom,
    ! [X2: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_617_set__zip__rightD,axiom,
    ! [X2: nat,Y2: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X2 @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_618_set__zip__leftD,axiom,
    ! [X2: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_619_set__zip__leftD,axiom,
    ! [X2: nat,Y2: product_prod_nat_a,Xs: list_nat,Ys: list_P2851791750731487283_nat_a] :
      ( ( member2321988868818003709_nat_a @ ( produc3386236975552542988_nat_a @ X2 @ Y2 ) @ ( set_Pr8046525447620763497_nat_a @ ( zip_na7849663319851687980_nat_a @ Xs @ Ys ) ) )
     => ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_620_set__zip__leftD,axiom,
    ! [X2: product_prod_b_nat > set_list_a,Y2: nat > a,Xs: list_P6049048235159712035list_a,Ys: list_nat_a] :
      ( ( member9198066416134578520_nat_a @ ( produc2895298938842563487_nat_a @ X2 @ Y2 ) @ ( set_Pr4376371411171465668_nat_a @ ( zip_Pr5387723086728981183_nat_a @ Xs @ Ys ) ) )
     => ( member8404886500659538246list_a @ X2 @ ( set_Pr6937066619593490354list_a @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_621_set__zip__leftD,axiom,
    ! [X2: b,Y2: nat,Xs: list_b,Ys: list_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X2 @ Y2 ) @ ( set_Pr2160427703459429033_b_nat @ ( zip_b_nat @ Xs @ Ys ) ) )
     => ( member_b @ X2 @ ( set_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_622_set__zip__leftD,axiom,
    ! [X2: a,Y2: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_623_set__zip__leftD,axiom,
    ! [X2: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ X2 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_624_set__zip__leftD,axiom,
    ! [X2: nat,Y2: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X2 @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_625_in__set__zipE,axiom,
    ! [X2: nat,Y2: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X2 @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_626_in__set__zipE,axiom,
    ! [X2: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_627_in__set__zipE,axiom,
    ! [X2: b,Y2: nat,Xs: list_b,Ys: list_nat] :
      ( ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X2 @ Y2 ) @ ( set_Pr2160427703459429033_b_nat @ ( zip_b_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_628_in__set__zipE,axiom,
    ! [X2: a,Y2: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_629_in__set__zipE,axiom,
    ! [X2: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_630_in__set__zipE,axiom,
    ! [X2: set_nat,Y2: nat,Xs: list_set_nat,Ys: list_nat] :
      ( ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ X2 @ Y2 ) @ ( set_Pr2055882785421719500at_nat @ ( zip_set_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_631_in__set__zipE,axiom,
    ! [X2: set_nat,Y2: a,Xs: list_set_nat,Ys: list_a] :
      ( ( member5257926396607048452_nat_a @ ( produc6206495014686954143_nat_a @ X2 @ Y2 ) @ ( set_Pr6887806534782065048_nat_a @ ( zip_set_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_632_in__set__zipE,axiom,
    ! [X2: list_nat,Y2: nat,Xs: list_list_nat,Ys: list_nat] :
      ( ( member7820552544826472890at_nat @ ( produc1298395424260782409at_nat @ X2 @ Y2 ) @ ( set_Pr7715406932454250534at_nat @ ( zip_list_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_633_in__set__zipE,axiom,
    ! [X2: list_nat,Y2: a,Xs: list_list_nat,Ys: list_a] :
      ( ( member6908590373215729450_nat_a @ ( produc1615968307924265413_nat_a @ X2 @ Y2 ) @ ( set_Pr7185350859094612926_nat_a @ ( zip_list_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_634_in__set__zipE,axiom,
    ! [X2: list_a,Y2: nat,Xs: list_list_a,Ys: list_nat] :
      ( ( member873448109036076442_a_nat @ ( produc4010430043144778749_a_nat @ X2 @ Y2 ) @ ( set_Pr1150208594914959918_a_nat @ ( zip_list_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_635_zip__same,axiom,
    ! [A3: set_list_a,B2: set_list_a,Xs: list_set_list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A3 @ B2 ) @ ( set_Pr3117605934882692796list_a @ ( zip_se6500254529988017719list_a @ Xs @ Xs ) ) )
      = ( ( member_set_list_a @ A3 @ ( set_set_list_a2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_636_zip__same,axiom,
    ! [A3: set_nat,B2: set_nat,Xs: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B2 ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
      = ( ( member_set_nat @ A3 @ ( set_set_nat2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_637_zip__same,axiom,
    ! [A3: ( nat > a ) > nat > a,B2: ( nat > a ) > nat > a,Xs: list_nat_a_nat_a] :
      ( ( member5591254206257022164_nat_a @ ( produc3289167689110286819_nat_a @ A3 @ B2 ) @ ( set_Pr9157543853878792512_nat_a @ ( zip_na1212149644256180483_nat_a @ Xs @ Xs ) ) )
      = ( ( member_nat_a_nat_a @ A3 @ ( set_nat_a_nat_a2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_638_zip__same,axiom,
    ! [A3: product_prod_a_a,B2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A3 @ B2 ) @ ( set_Pr6493056779096584572od_a_a @ ( zip_Pr1329156589218441847od_a_a @ Xs @ Xs ) ) )
      = ( ( member1426531477525435216od_a_a @ A3 @ ( set_Product_prod_a_a2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_639_zip__same,axiom,
    ! [A3: list_nat,B2: list_nat,Xs: list_list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A3 @ B2 ) @ ( set_Pr3842133991353686454st_nat @ ( zip_li7157463729305086713st_nat @ Xs @ Xs ) ) )
      = ( ( member_list_nat @ A3 @ ( set_list_nat2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_640_zip__same,axiom,
    ! [A3: list_a,B2: list_a,Xs: list_list_a] :
      ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A3 @ B2 ) @ ( set_Pr2906193453920572092list_a @ ( zip_list_a_list_a @ Xs @ Xs ) ) )
      = ( ( member_list_a @ A3 @ ( set_list_a2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_641_zip__same,axiom,
    ! [A3: a,B2: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A3 @ B2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A3 @ ( set_a2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_642_zip__same,axiom,
    ! [A3: nat,B2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
        & ( A3 = B2 ) ) ) ).

% zip_same
thf(fact_643_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( zip_nat_a @ ( map_nat_nat @ F2 @ Xs ) @ ( map_a_a @ G @ Ys ) )
      = ( map_Pr6653247079942906636_nat_a
        @ ( produc3971069790275736229_nat_a
          @ ^ [X: nat,Y: a] : ( product_Pair_nat_a @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_644_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_645_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_a_nat @ G @ Ys ) )
      = ( map_Pr4626788704701184344at_nat
        @ ( produc4331274911807030655at_nat
          @ ^ [X: nat,Y: a] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_646_zip__map__map,axiom,
    ! [F2: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_a_nat @ F2 @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr6893346637286776318at_nat
        @ ( produc4793729866244303965at_nat
          @ ^ [X: a,Y: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_647_zip__map__map,axiom,
    ! [F2: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
      ( ( zip_nat_nat @ ( map_a_nat @ F2 @ Xs ) @ ( map_a_nat @ G @ Ys ) )
      = ( map_Pr7747608641848760414at_nat
        @ ( produc8755017709493252821at_nat
          @ ^ [X: a,Y: a] : ( product_Pair_nat_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_648_zip__map__map,axiom,
    ! [F2: nat > nat,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( zip_nat_a @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_a @ G @ Ys ) )
      = ( map_Pr7092700573198925024_nat_a
        @ ( produc7553103386638219761_nat_a
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_a @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_649_zip__map__map,axiom,
    ! [F2: a > nat,Xs: list_a,G: a > a,Ys: list_a] :
      ( ( zip_nat_a @ ( map_a_nat @ F2 @ Xs ) @ ( map_a_a @ G @ Ys ) )
      = ( map_Pr5343040932936202630_nat_a
        @ ( produc5537466922496691919_nat_a
          @ ^ [X: a,Y: a] : ( product_Pair_nat_a @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_650_zip__map__map,axiom,
    ! [F2: a > b,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_a_b @ F2 @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr8210801375579218625_b_nat
        @ ( produc750755012702383650_b_nat
          @ ^ [X: a,Y: nat] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_651_zip__map__map,axiom,
    ! [F2: nat > b,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr5089981438431642555_b_nat
        @ ( produc5550384251870937292_b_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_652_zip__map__map,axiom,
    ! [F2: nat > b,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ ( map_a_nat @ G @ Ys ) )
      = ( map_Pr4650527945175624167_b_nat
        @ ( produc1968350655508453760_b_nat
          @ ^ [X: nat,Y: a] : ( product_Pair_b_nat @ ( F2 @ X ) @ ( G @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_653_zip__map2,axiom,
    ! [Xs: list_nat,F2: a > a,Ys: list_a] :
      ( ( zip_nat_a @ Xs @ ( map_a_a @ F2 @ Ys ) )
      = ( map_Pr6653247079942906636_nat_a
        @ ( produc3971069790275736229_nat_a
          @ ^ [X: nat,Y: a] : ( product_Pair_nat_a @ X @ ( F2 @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_654_zip__map2,axiom,
    ! [Xs: list_nat,F2: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_655_zip__map2,axiom,
    ! [Xs: list_nat,F2: a > nat,Ys: list_a] :
      ( ( zip_nat_nat @ Xs @ ( map_a_nat @ F2 @ Ys ) )
      = ( map_Pr4626788704701184344at_nat
        @ ( produc4331274911807030655at_nat
          @ ^ [X: nat,Y: a] : ( product_Pair_nat_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_656_zip__map2,axiom,
    ! [Xs: list_nat,F2: nat > a,Ys: list_nat] :
      ( ( zip_nat_a @ Xs @ ( map_nat_a @ F2 @ Ys ) )
      = ( map_Pr7092700573198925024_nat_a
        @ ( produc7553103386638219761_nat_a
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_a @ X @ ( F2 @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_657_zip__map2,axiom,
    ! [Xs: list_b,F2: nat > nat,Ys: list_nat] :
      ( ( zip_b_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr7742232262618018242_b_nat
        @ ( produc282185899741183267_b_nat
          @ ^ [X: b,Y: nat] : ( product_Pair_b_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_b_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_658_zip__map2,axiom,
    ! [Xs: list_b,F2: a > nat,Ys: list_a] :
      ( ( zip_b_nat @ Xs @ ( map_a_nat @ F2 @ Ys ) )
      = ( map_Pr4581446971666788256_b_nat
        @ ( produc4775872961227277545_b_nat
          @ ^ [X: b,Y: a] : ( product_Pair_b_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_b_a @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_659_zip__map2,axiom,
    ! [Xs: list_a,F2: nat > nat,Ys: list_nat] :
      ( ( zip_a_nat @ Xs @ ( map_nat_nat @ F2 @ Ys ) )
      = ( map_Pr6975357046622824128_a_nat
        @ ( produc8738682720600764961_a_nat
          @ ^ [X: a,Y: nat] : ( product_Pair_a_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_660_zip__map2,axiom,
    ! [Xs: list_a,F2: a > nat,Ys: list_a] :
      ( ( zip_a_nat @ Xs @ ( map_a_nat @ F2 @ Ys ) )
      = ( map_Pr2104877469212525664_a_nat
        @ ( produc2299303458773014953_a_nat
          @ ^ [X: a,Y: a] : ( product_Pair_a_nat @ X @ ( F2 @ Y ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_661_zip__map2,axiom,
    ! [Xs: list_a,F2: nat > a,Ys: list_nat] :
      ( ( zip_a_a @ Xs @ ( map_nat_a @ F2 @ Ys ) )
      = ( map_Pr1850710106576109476od_a_a
        @ ( produc6848318051473230467od_a_a
          @ ^ [X: a,Y: nat] : ( product_Pair_a_a @ X @ ( F2 @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_662_zip__map2,axiom,
    ! [Xs: list_a,F2: a > a,Ys: list_a] :
      ( ( zip_a_a @ Xs @ ( map_a_a @ F2 @ Ys ) )
      = ( map_Pr7904243085458786820od_a_a
        @ ( produc408267641121961211od_a_a
          @ ^ [X: a,Y: a] : ( product_Pair_a_a @ X @ ( F2 @ Y ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_663_zip__map1,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_a] :
      ( ( zip_nat_a @ ( map_nat_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr6653247079942906636_nat_a
        @ ( produc3971069790275736229_nat_a
          @ ^ [X: nat] : ( product_Pair_nat_a @ ( F2 @ X ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_664_zip__map1,axiom,
    ! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_665_zip__map1,axiom,
    ! [F2: a > nat,Xs: list_a,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_a_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr6893346637286776318at_nat
        @ ( produc4793729866244303965at_nat
          @ ^ [X: a] : ( product_Pair_nat_nat @ ( F2 @ X ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_666_zip__map1,axiom,
    ! [F2: a > nat,Xs: list_a,Ys: list_a] :
      ( ( zip_nat_a @ ( map_a_nat @ F2 @ Xs ) @ Ys )
      = ( map_Pr5343040932936202630_nat_a
        @ ( produc5537466922496691919_nat_a
          @ ^ [X: a] : ( product_Pair_nat_a @ ( F2 @ X ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_667_zip__map1,axiom,
    ! [F2: a > b,Xs: list_a,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_a_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr8210801375579218625_b_nat
        @ ( produc750755012702383650_b_nat
          @ ^ [X: a] : ( product_Pair_b_nat @ ( F2 @ X ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_668_zip__map1,axiom,
    ! [F2: nat > b,Xs: list_nat,Ys: list_nat] :
      ( ( zip_b_nat @ ( map_nat_b @ F2 @ Xs ) @ Ys )
      = ( map_Pr5089981438431642555_b_nat
        @ ( produc5550384251870937292_b_nat
          @ ^ [X: nat] : ( product_Pair_b_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_669_zip__map1,axiom,
    ! [F2: nat > a,Xs: list_nat,Ys: list_nat] :
      ( ( zip_a_nat @ ( map_nat_a @ F2 @ Xs ) @ Ys )
      = ( map_Pr3854537109475248058_a_nat
        @ ( produc4314939922914542795_a_nat
          @ ^ [X: nat] : ( product_Pair_a_nat @ ( F2 @ X ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_670_zip__map1,axiom,
    ! [F2: a > a,Xs: list_a,Ys: list_nat] :
      ( ( zip_a_nat @ ( map_a_a @ F2 @ Xs ) @ Ys )
      = ( map_Pr6975357046622824128_a_nat
        @ ( produc8738682720600764961_a_nat
          @ ^ [X: a] : ( product_Pair_a_nat @ ( F2 @ X ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_671_zip__map1,axiom,
    ! [F2: nat > a,Xs: list_nat,Ys: list_a] :
      ( ( zip_a_a @ ( map_nat_a @ F2 @ Xs ) @ Ys )
      = ( map_Pr4215426277517063678od_a_a
        @ ( produc892607601009605285od_a_a
          @ ^ [X: nat] : ( product_Pair_a_a @ ( F2 @ X ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_672_zip__map1,axiom,
    ! [F2: a > a,Xs: list_a,Ys: list_a] :
      ( ( zip_a_a @ ( map_a_a @ F2 @ Xs ) @ Ys )
      = ( map_Pr7904243085458786820od_a_a
        @ ( produc408267641121961211od_a_a
          @ ^ [X: a] : ( product_Pair_a_a @ ( F2 @ X ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_673_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_674_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_a,Y2: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_675_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_nat,Y2: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_676_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_a,Y2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_677_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_b,Ys: list_nat,Y2: nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: b] :
              ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X4 @ Y2 ) @ ( set_Pr2160427703459429033_b_nat @ ( zip_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_678_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_set_nat,Y2: set_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3254054031482475050et_nat @ Ys ) )
     => ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member3782324328723991648et_nat @ ( produc4207506657711014383et_nat @ X4 @ Y2 ) @ ( set_Pr6187991034916610252et_nat @ ( zip_nat_set_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_679_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_list_a,Y2: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( member_list_a @ Y2 @ ( set_list_a2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X4 @ Y2 ) @ ( set_Pr8466731866354521812list_a @ ( zip_nat_list_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_680_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_list_nat @ Y2 @ ( set_list_nat2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X4 @ Y2 ) @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_681_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_R6823256787227418703term_a,Y2: relational_term_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) )
     => ( ( member5466445112152945266term_a @ Y2 @ ( set_Re3569617851344498910term_a @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member2722330264857718789term_a @ ( produc1842950163714375456term_a @ X4 @ Y2 ) @ ( set_Pr7861135356763781785term_a @ ( zip_na8968355248959840256term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_682_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_set_nat,Y2: set_nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3254054031482475050et_nat @ Ys ) )
     => ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member1257341626780437706et_nat @ ( produc845668862566212397et_nat @ X4 @ Y2 ) @ ( set_Pr2887221764955454302et_nat @ ( zip_a_set_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_683_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_684_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_a,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: a] :
              ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X2 @ Y3 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_685_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_nat,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X2 @ Y3 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_686_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_a,X2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_687_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_b,Ys: list_nat,X2: b] :
      ( ( ( size_size_list_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member6959632917342813205_b_nat @ ( product_Pair_b_nat @ X2 @ Y3 ) @ ( set_Pr2160427703459429033_b_nat @ ( zip_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_688_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_set_nat,Ys: list_nat,X2: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8873588116083876704at_nat @ ( produc641871753055645167at_nat @ X2 @ Y3 ) @ ( set_Pr2055882785421719500at_nat @ ( zip_set_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_689_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_set_nat,Ys: list_a,X2: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
       => ~ ! [Y3: a] :
              ~ ( member5257926396607048452_nat_a @ ( produc6206495014686954143_nat_a @ X2 @ Y3 ) @ ( set_Pr6887806534782065048_nat_a @ ( zip_set_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_690_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_list_a,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s349497388124573686list_a @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: list_a] :
              ~ ( member8189971380475638336list_a @ ( produc7903367357317368283list_a @ X2 @ Y3 ) @ ( set_Pr8466731866354521812list_a @ ( zip_nat_list_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_691_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_list_nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: list_nat] :
              ~ ( member1631564025489475386st_nat @ ( produc8282810413953273033st_nat @ X2 @ Y3 ) @ ( set_Pr1526418413117253030st_nat @ ( zip_nat_list_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_692_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_R6823256787227418703term_a,X2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s88622898042387131term_a @ Ys ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: relational_term_a] :
              ~ ( member2722330264857718789term_a @ ( produc1842950163714375456term_a @ X2 @ Y3 ) @ ( set_Pr7861135356763781785term_a @ ( zip_na8968355248959840256term_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_693_map__zip__map2,axiom,
    ! [F2: product_prod_nat_a > ( nat > a ) > nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( map_Pr2159937840280155150_nat_a @ F2 @ ( zip_nat_a @ Xs @ ( map_a_a @ G @ Ys ) ) )
      = ( map_Pr2159937840280155150_nat_a
        @ ( produc2909000522608705447_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_694_map__zip__map2,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a,Xs: list_a,G: a > nat,Ys: list_a] :
      ( ( map_Pr990148473491725286_nat_a @ F2 @ ( zip_a_nat @ Xs @ ( map_a_nat @ G @ Ys ) ) )
      = ( map_Pr5343040932936202630_nat_a
        @ ( produc5537466922496691919_nat_a
          @ ^ [X: a,Y: a] : ( F2 @ ( product_Pair_a_nat @ X @ ( G @ Y ) ) ) )
        @ ( zip_a_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_695_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > product_prod_a_nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( map_Pr3854537109475248058_a_nat @ F2 @ ( zip_nat_nat @ Xs @ ( map_a_nat @ G @ Ys ) ) )
      = ( map_Pr3415083616219229670_a_nat
        @ ( produc732906326552059263_a_nat
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_696_map__zip__map2,axiom,
    ! [F2: product_prod_nat_a > product_prod_a_nat,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( map_Pr3415083616219229670_a_nat @ F2 @ ( zip_nat_a @ Xs @ ( map_a_a @ G @ Ys ) ) )
      = ( map_Pr3415083616219229670_a_nat
        @ ( produc732906326552059263_a_nat
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_697_map__zip__map2,axiom,
    ! [F2: product_prod_nat_nat > nat > a,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( map_Pr329795578479598952_nat_a @ F2 @ ( zip_nat_nat @ Xs @ ( map_a_nat @ G @ Ys ) ) )
      = ( map_Pr7512474648352264252_nat_a
        @ ( produc4481717121449037155_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_nat @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_698_map__zip__map2,axiom,
    ! [F2: product_prod_nat_a > nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( map_Pr7512474648352264252_nat_a @ F2 @ ( zip_nat_a @ Xs @ ( map_a_a @ G @ Ys ) ) )
      = ( map_Pr7512474648352264252_nat_a
        @ ( produc4481717121449037155_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( product_Pair_nat_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_699_map__zip__map2,axiom,
    ! [F2: product_prod_a_nat > product_prod_nat_a,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( map_Pr990148473491725286_nat_a @ F2 @ ( zip_a_nat @ Xs @ ( map_nat_nat @ G @ Ys ) ) )
      = ( map_Pr990148473491725286_nat_a
        @ ( produc2753474147469666119_nat_a
          @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_a_nat @ X @ ( G @ Y ) ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_700_map__zip__map2,axiom,
    ! [F2: product_prod_a_a > product_prod_nat_a,Xs: list_a,G: nat > a,Ys: list_nat] :
      ( ( map_Pr5343040932936202630_nat_a @ F2 @ ( zip_a_a @ Xs @ ( map_nat_a @ G @ Ys ) ) )
      = ( map_Pr990148473491725286_nat_a
        @ ( produc2753474147469666119_nat_a
          @ ^ [X: a,Y: nat] : ( F2 @ ( product_Pair_a_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_701_map__zip__map2,axiom,
    ! [F2: produc6033039080626123092_nat_a > product_prod_a_nat,Xs: list_nat,G: a > product_prod_nat_a,Ys: list_a] :
      ( ( map_Pr2688339558924254733_a_nat @ F2 @ ( zip_na7849663319851687980_nat_a @ Xs @ ( map_a_8233067891256473437_nat_a @ G @ Ys ) ) )
      = ( map_Pr3415083616219229670_a_nat
        @ ( produc732906326552059263_a_nat
          @ ^ [X: nat,Y: a] : ( F2 @ ( produc3386236975552542988_nat_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_702_map__zip__map2,axiom,
    ! [F2: produc6033039080626123092_nat_a > nat > a,Xs: list_nat,G: a > product_prod_nat_a,Ys: list_a] :
      ( ( map_Pr5972415843224298325_nat_a @ F2 @ ( zip_na7849663319851687980_nat_a @ Xs @ ( map_a_8233067891256473437_nat_a @ G @ Ys ) ) )
      = ( map_Pr7512474648352264252_nat_a
        @ ( produc4481717121449037155_nat_a
          @ ^ [X: nat,Y: a] : ( F2 @ ( produc3386236975552542988_nat_a @ X @ ( G @ Y ) ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% map_zip_map2
thf(fact_703_in__fv__substs,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( member_nat @ ( relati8128731020529265620ar_nat @ Xs @ Ys @ X2 )
          @ ( relational_fv_a_b
            @ ( fold_P7970104616371074773la_a_b
              @ ( produc5586541307551673003la_a_b
                @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
              @ ( zip_nat_nat @ Xs @ Ys )
              @ Q ) ) ) ) ) ).

% in_fv_substs
thf(fact_704_subsetI,axiom,
    ! [A4: set_list_a,B4: set_list_a] :
      ( ! [X4: list_a] :
          ( ( member_list_a @ X4 @ A4 )
         => ( member_list_a @ X4 @ B4 ) )
     => ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_705_subsetI,axiom,
    ! [A4: set_nat_a_nat_a,B4: set_nat_a_nat_a] :
      ( ! [X4: ( nat > a ) > nat > a] :
          ( ( member_nat_a_nat_a @ X4 @ A4 )
         => ( member_nat_a_nat_a @ X4 @ B4 ) )
     => ( ord_le7151205609328483131_nat_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_706_subsetI,axiom,
    ! [A4: set_set_list_a,B4: set_set_list_a] :
      ( ! [X4: set_list_a] :
          ( ( member_set_list_a @ X4 @ A4 )
         => ( member_set_list_a @ X4 @ B4 ) )
     => ( ord_le8877086941679407844list_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_707_subsetI,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A4 )
         => ( member_set_nat @ X4 @ B4 ) )
     => ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_708_subsetI,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A4 )
         => ( member_a @ X4 @ B4 ) )
     => ( ord_less_eq_set_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_709_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( member_nat @ X4 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_710_subset__antisym,axiom,
    ! [A4: set_set_list_a,B4: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A4 @ B4 )
     => ( ( ord_le8877086941679407844list_a @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_711_subset__antisym,axiom,
    ! [A4: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
     => ( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_712_subset__antisym,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_713_subset__antisym,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( ord_less_eq_set_a @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_714_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_715_distinct__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( distinct_a @ ( union_a @ Xs @ Ys ) )
      = ( distinct_a @ Ys ) ) ).

% distinct_union
thf(fact_716_distinct__union,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( distinct_list_a @ ( union_list_a @ Xs @ Ys ) )
      = ( distinct_list_a @ Ys ) ) ).

% distinct_union
thf(fact_717_distinct__union,axiom,
    ! [Xs: list_P2851791750731487283_nat_a,Ys: list_P2851791750731487283_nat_a] :
      ( ( distin7399675782055410666_nat_a @ ( union_728571500015337257_nat_a @ Xs @ Ys ) )
      = ( distin7399675782055410666_nat_a @ Ys ) ) ).

% distinct_union
thf(fact_718_order__refl,axiom,
    ! [X2: set_set_list_a] : ( ord_le8877086941679407844list_a @ X2 @ X2 ) ).

% order_refl
thf(fact_719_order__refl,axiom,
    ! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_720_order__refl,axiom,
    ! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_721_order__refl,axiom,
    ! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).

% order_refl
thf(fact_722_order__refl,axiom,
    ! [X2: nat > $o] : ( ord_less_eq_nat_o @ X2 @ X2 ) ).

% order_refl
thf(fact_723_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_724_dual__order_Orefl,axiom,
    ! [A3: set_set_list_a] : ( ord_le8877086941679407844list_a @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_725_dual__order_Orefl,axiom,
    ! [A3: set_set_nat] : ( ord_le6893508408891458716et_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_726_dual__order_Orefl,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_727_dual__order_Orefl,axiom,
    ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_728_dual__order_Orefl,axiom,
    ! [A3: nat > $o] : ( ord_less_eq_nat_o @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_729_dual__order_Orefl,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_730_substs__Neg,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Neg_a_b @ Q ) )
        = ( relational_Neg_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% substs_Neg
thf(fact_731_substs__Conj,axiom,
    ! [Xs: list_nat,Ys: list_nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( fold_P7970104616371074773la_a_b
          @ ( produc5586541307551673003la_a_b
            @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
          @ ( zip_nat_nat @ Xs @ Ys )
          @ ( relational_Conj_a_b @ Q1 @ Q22 ) )
        = ( relational_Conj_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q1 )
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q22 ) ) ) ) ).

% substs_Conj
thf(fact_732_ap__substs,axiom,
    ! [Q: relational_fmla_a_b,Xs: list_nat,Ys: list_nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ( relational_ap_a_b
          @ ( fold_P7970104616371074773la_a_b
            @ ( produc5586541307551673003la_a_b
              @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
            @ ( zip_nat_nat @ Xs @ Ys )
            @ Q ) ) ) ) ).

% ap_substs
thf(fact_733_fmla_Oinject_I4_J,axiom,
    ! [X42: relational_fmla_a_b,Y4: relational_fmla_a_b] :
      ( ( ( relational_Neg_a_b @ X42 )
        = ( relational_Neg_a_b @ Y4 ) )
      = ( X42 = Y4 ) ) ).

% fmla.inject(4)
thf(fact_734_fmla_Oinject_I5_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
      ( ( ( relational_Conj_a_b @ X51 @ X52 )
        = ( relational_Conj_a_b @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% fmla.inject(5)
thf(fact_735_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_736_Collect__case__prod__mono,axiom,
    ! [A4: nat > nat > $o,B4: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A4 @ B4 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_737_fmla_Odistinct_I31_J,axiom,
    ! [X42: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X42 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(31)
thf(fact_738_less__eq__set__def,axiom,
    ( ord_le8861187494160871172list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( ord_less_eq_list_a_o
          @ ^ [X: list_a] : ( member_list_a @ X @ A6 )
          @ ^ [X: list_a] : ( member_list_a @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_739_less__eq__set__def,axiom,
    ( ord_le7151205609328483131_nat_a
    = ( ^ [A6: set_nat_a_nat_a,B6: set_nat_a_nat_a] :
          ( ord_le2385885742185689866at_a_o
          @ ^ [X: ( nat > a ) > nat > a] : ( member_nat_a_nat_a @ X @ A6 )
          @ ^ [X: ( nat > a ) > nat > a] : ( member_nat_a_nat_a @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_740_less__eq__set__def,axiom,
    ( ord_le8877086941679407844list_a
    = ( ^ [A6: set_set_list_a,B6: set_set_list_a] :
          ( ord_le897266612844759801st_a_o
          @ ^ [X: set_list_a] : ( member_set_list_a @ X @ A6 )
          @ ^ [X: set_list_a] : ( member_set_list_a @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_741_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_742_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X: a] : ( member_a @ X @ A6 )
          @ ^ [X: a] : ( member_a @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_743_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_744_fmla_Odistinct_I37_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(37)
thf(fact_745_fmla_Odistinct_I33_J,axiom,
    ! [X42: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X42 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(33)
thf(fact_746_fv_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Neg_a_b @ Phi ) )
      = ( relational_fv_a_b @ Phi ) ) ).

% fv.simps(4)
thf(fact_747_subst_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) @ X2 @ Y2 )
      = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X2 @ Y2 ) @ ( relational_subst_a_b @ Q22 @ X2 @ Y2 ) ) ) ).

% subst.simps(5)
thf(fact_748_subst_Osimps_I4_J,axiom,
    ! [Q: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( relational_subst_a_b @ ( relational_Neg_a_b @ Q ) @ X2 @ Y2 )
      = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q @ X2 @ Y2 ) ) ) ).

% subst.simps(4)
thf(fact_749_sat_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) @ I @ Sigma )
      = ( ( relational_sat_a_b @ Phi @ I @ Sigma )
        & ( relational_sat_a_b @ Psi @ I @ Sigma ) ) ) ).

% sat.simps(5)
thf(fact_750_sat_Osimps_I4_J,axiom,
    ! [Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Neg_a_b @ Phi ) @ I @ Sigma )
      = ( ~ ( relational_sat_a_b @ Phi @ I @ Sigma ) ) ) ).

% sat.simps(4)
thf(fact_751_ap__subst_H,axiom,
    ! [Q: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( relational_ap_a_b @ ( relational_subst_a_b @ Q @ X2 @ Y2 ) )
     => ( relational_ap_a_b @ Q ) ) ).

% ap_subst'
thf(fact_752_order__antisym__conv,axiom,
    ! [Y2: set_set_list_a,X2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ Y2 @ X2 )
     => ( ( ord_le8877086941679407844list_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_753_order__antisym__conv,axiom,
    ! [Y2: set_set_nat,X2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
     => ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_754_order__antisym__conv,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ X2 )
     => ( ( ord_less_eq_set_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_755_order__antisym__conv,axiom,
    ! [Y2: set_a,X2: set_a] :
      ( ( ord_less_eq_set_a @ Y2 @ X2 )
     => ( ( ord_less_eq_set_a @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_756_order__antisym__conv,axiom,
    ! [Y2: nat > $o,X2: nat > $o] :
      ( ( ord_less_eq_nat_o @ Y2 @ X2 )
     => ( ( ord_less_eq_nat_o @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_757_order__antisym__conv,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y2 )
        = ( X2 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_758_linorder__le__cases,axiom,
    ! [X2: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_le_cases
thf(fact_759_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_760_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_761_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_762_ord__le__eq__subst,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_763_ord__le__eq__subst,axiom,
    ! [A3: set_a,B2: set_a,F2: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_764_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le6893508408891458716et_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_765_ord__le__eq__subst,axiom,
    ! [A3: nat,B2: nat,F2: nat > nat > $o,C: nat > $o] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat_o @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat_o @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_766_ord__le__eq__subst,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,F2: set_set_nat > nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: set_set_nat,Y3: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_767_ord__le__eq__subst,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_768_ord__le__eq__subst,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > set_a,C: set_a] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ( F2 @ B2 )
          = C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_769_ord__eq__le__subst,axiom,
    ! [A3: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_770_ord__eq__le__subst,axiom,
    ! [A3: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_771_ord__eq__le__subst,axiom,
    ! [A3: set_a,F2: nat > set_a,B2: nat,C: nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_772_ord__eq__le__subst,axiom,
    ! [A3: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_773_ord__eq__le__subst,axiom,
    ! [A3: nat,F2: set_a > nat,B2: set_a,C: set_a] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X4: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_774_ord__eq__le__subst,axiom,
    ! [A3: set_set_nat,F2: nat > set_set_nat,B2: nat,C: nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le6893508408891458716et_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le6893508408891458716et_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_775_ord__eq__le__subst,axiom,
    ! [A3: nat > $o,F2: nat > nat > $o,B2: nat,C: nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat_o @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat_o @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_776_ord__eq__le__subst,axiom,
    ! [A3: nat,F2: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C )
       => ( ! [X4: set_set_nat,Y3: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_777_ord__eq__le__subst,axiom,
    ! [A3: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_778_ord__eq__le__subst,axiom,
    ! [A3: set_a,F2: set_nat > set_a,B2: set_nat,C: set_nat] :
      ( ( A3
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_779_linorder__linear,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X2 ) ) ).

% linorder_linear
thf(fact_780_order__eq__refl,axiom,
    ! [X2: set_set_list_a,Y2: set_set_list_a] :
      ( ( X2 = Y2 )
     => ( ord_le8877086941679407844list_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_781_order__eq__refl,axiom,
    ! [X2: set_set_nat,Y2: set_set_nat] :
      ( ( X2 = Y2 )
     => ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_782_order__eq__refl,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_783_order__eq__refl,axiom,
    ! [X2: set_a,Y2: set_a] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_784_order__eq__refl,axiom,
    ! [X2: nat > $o,Y2: nat > $o] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_nat_o @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_785_order__eq__refl,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 = Y2 )
     => ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% order_eq_refl
thf(fact_786_order__subst2,axiom,
    ! [A3: nat,B2: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_787_order__subst2,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_788_order__subst2,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_789_order__subst2,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_790_order__subst2,axiom,
    ! [A3: set_a,B2: set_a,F2: set_a > nat,C: nat] :
      ( ( ord_less_eq_set_a @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_791_order__subst2,axiom,
    ! [A3: nat,B2: nat,F2: nat > set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le6893508408891458716et_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le6893508408891458716et_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_792_order__subst2,axiom,
    ! [A3: nat,B2: nat,F2: nat > nat > $o,C: nat > $o] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat_o @ ( F2 @ B2 ) @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat_o @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat_o @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_793_order__subst2,axiom,
    ! [A3: set_set_nat,B2: set_set_nat,F2: set_set_nat > nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: set_set_nat,Y3: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_794_order__subst2,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_795_order__subst2,axiom,
    ! [A3: set_nat,B2: set_nat,F2: set_nat > set_a,C: set_a] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F2 @ A3 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_796_order__subst1,axiom,
    ! [A3: nat,F2: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_797_order__subst1,axiom,
    ! [A3: nat,F2: set_nat > nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_798_order__subst1,axiom,
    ! [A3: nat,F2: set_a > nat,B2: set_a,C: set_a] :
      ( ( ord_less_eq_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X4: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_799_order__subst1,axiom,
    ! [A3: set_nat,F2: nat > set_nat,B2: nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_800_order__subst1,axiom,
    ! [A3: set_a,F2: nat > set_a,B2: nat,C: nat] :
      ( ( ord_less_eq_set_a @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_801_order__subst1,axiom,
    ! [A3: nat,F2: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ C )
       => ( ! [X4: set_set_nat,Y3: set_set_nat] :
              ( ( ord_le6893508408891458716et_nat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_802_order__subst1,axiom,
    ! [A3: nat,F2: ( nat > $o ) > nat,B2: nat > $o,C: nat > $o] :
      ( ( ord_less_eq_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat_o @ B2 @ C )
       => ( ! [X4: nat > $o,Y3: nat > $o] :
              ( ( ord_less_eq_nat_o @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_803_order__subst1,axiom,
    ! [A3: set_set_nat,F2: nat > set_set_nat,B2: nat,C: nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_le6893508408891458716et_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_le6893508408891458716et_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_804_order__subst1,axiom,
    ! [A3: set_nat,F2: set_nat > set_nat,B2: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_nat @ B2 @ C )
       => ( ! [X4: set_nat,Y3: set_nat] :
              ( ( ord_less_eq_set_nat @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_805_order__subst1,axiom,
    ! [A3: set_nat,F2: set_a > set_nat,B2: set_a,C: set_a] :
      ( ( ord_less_eq_set_nat @ A3 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_set_a @ B2 @ C )
       => ( ! [X4: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X4 @ Y3 )
             => ( ord_less_eq_set_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_set_nat @ A3 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_806_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_set_list_a,Z4: set_set_list_a] : ( Y5 = Z4 ) )
    = ( ^ [A: set_set_list_a,B3: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ A @ B3 )
          & ( ord_le8877086941679407844list_a @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_807_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_set_nat,Z4: set_set_nat] : ( Y5 = Z4 ) )
    = ( ^ [A: set_set_nat,B3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ A @ B3 )
          & ( ord_le6893508408891458716et_nat @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_808_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : ( Y5 = Z4 ) )
    = ( ^ [A: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A @ B3 )
          & ( ord_less_eq_set_nat @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_809_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
    = ( ^ [A: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ A @ B3 )
          & ( ord_less_eq_set_a @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_810_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 ) )
    = ( ^ [A: nat > $o,B3: nat > $o] :
          ( ( ord_less_eq_nat_o @ A @ B3 )
          & ( ord_less_eq_nat_o @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_811_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [A: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A @ B3 )
          & ( ord_less_eq_nat @ B3 @ A ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_812_le__fun__def,axiom,
    ( ord_less_eq_nat_o
    = ( ^ [F: nat > $o,G2: nat > $o] :
        ! [X: nat] : ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).

% le_fun_def
thf(fact_813_le__funI,axiom,
    ! [F2: nat > $o,G: nat > $o] :
      ( ! [X4: nat] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G @ X4 ) )
     => ( ord_less_eq_nat_o @ F2 @ G ) ) ).

% le_funI
thf(fact_814_le__funE,axiom,
    ! [F2: nat > $o,G: nat > $o,X2: nat] :
      ( ( ord_less_eq_nat_o @ F2 @ G )
     => ( ord_less_eq_o @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funE
thf(fact_815_le__funD,axiom,
    ! [F2: nat > $o,G: nat > $o,X2: nat] :
      ( ( ord_less_eq_nat_o @ F2 @ G )
     => ( ord_less_eq_o @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ).

% le_funD
thf(fact_816_antisym,axiom,
    ! [A3: set_set_list_a,B2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A3 @ B2 )
     => ( ( ord_le8877086941679407844list_a @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_817_antisym,axiom,
    ! [A3: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
     => ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_818_antisym,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_819_antisym,axiom,
    ! [A3: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_820_antisym,axiom,
    ! [A3: nat > $o,B2: nat > $o] :
      ( ( ord_less_eq_nat_o @ A3 @ B2 )
     => ( ( ord_less_eq_nat_o @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_821_antisym,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A3 )
       => ( A3 = B2 ) ) ) ).

% antisym
thf(fact_822_dual__order_Otrans,axiom,
    ! [B2: set_set_list_a,A3: set_set_list_a,C: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B2 @ A3 )
     => ( ( ord_le8877086941679407844list_a @ C @ B2 )
       => ( ord_le8877086941679407844list_a @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_823_dual__order_Otrans,axiom,
    ! [B2: set_set_nat,A3: set_set_nat,C: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ C @ B2 )
       => ( ord_le6893508408891458716et_nat @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_824_dual__order_Otrans,axiom,
    ! [B2: set_nat,A3: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( ord_less_eq_set_nat @ C @ B2 )
       => ( ord_less_eq_set_nat @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_825_dual__order_Otrans,axiom,
    ! [B2: set_a,A3: set_a,C: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A3 )
     => ( ( ord_less_eq_set_a @ C @ B2 )
       => ( ord_less_eq_set_a @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_826_dual__order_Otrans,axiom,
    ! [B2: nat > $o,A3: nat > $o,C: nat > $o] :
      ( ( ord_less_eq_nat_o @ B2 @ A3 )
     => ( ( ord_less_eq_nat_o @ C @ B2 )
       => ( ord_less_eq_nat_o @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_827_dual__order_Otrans,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% dual_order.trans
thf(fact_828_dual__order_Oantisym,axiom,
    ! [B2: set_set_list_a,A3: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ B2 @ A3 )
     => ( ( ord_le8877086941679407844list_a @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_829_dual__order_Oantisym,axiom,
    ! [B2: set_set_nat,A3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
     => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_830_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_831_dual__order_Oantisym,axiom,
    ! [B2: set_a,A3: set_a] :
      ( ( ord_less_eq_set_a @ B2 @ A3 )
     => ( ( ord_less_eq_set_a @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_832_dual__order_Oantisym,axiom,
    ! [B2: nat > $o,A3: nat > $o] :
      ( ( ord_less_eq_nat_o @ B2 @ A3 )
     => ( ( ord_less_eq_nat_o @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_833_dual__order_Oantisym,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ B2 )
       => ( A3 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_834_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_set_list_a,Z4: set_set_list_a] : ( Y5 = Z4 ) )
    = ( ^ [A: set_set_list_a,B3: set_set_list_a] :
          ( ( ord_le8877086941679407844list_a @ B3 @ A )
          & ( ord_le8877086941679407844list_a @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_835_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_set_nat,Z4: set_set_nat] : ( Y5 = Z4 ) )
    = ( ^ [A: set_set_nat,B3: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ B3 @ A )
          & ( ord_le6893508408891458716et_nat @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_836_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : ( Y5 = Z4 ) )
    = ( ^ [A: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A )
          & ( ord_less_eq_set_nat @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_837_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
    = ( ^ [A: set_a,B3: set_a] :
          ( ( ord_less_eq_set_a @ B3 @ A )
          & ( ord_less_eq_set_a @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_838_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 ) )
    = ( ^ [A: nat > $o,B3: nat > $o] :
          ( ( ord_less_eq_nat_o @ B3 @ A )
          & ( ord_less_eq_nat_o @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_839_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [A: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A )
          & ( ord_less_eq_nat @ A @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_840_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A3: nat,B2: nat] :
      ( ! [A2: nat,B: nat] :
          ( ( ord_less_eq_nat @ A2 @ B )
         => ( P @ A2 @ B ) )
     => ( ! [A2: nat,B: nat] :
            ( ( P @ B @ A2 )
           => ( P @ A2 @ B ) )
       => ( P @ A3 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_841_order__trans,axiom,
    ! [X2: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
     => ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
       => ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_842_order__trans,axiom,
    ! [X2: set_nat,Y2: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_set_nat @ Y2 @ Z )
       => ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_843_order__trans,axiom,
    ! [X2: set_a,Y2: set_a,Z: set_a] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ord_less_eq_set_a @ Y2 @ Z )
       => ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_844_order__trans,axiom,
    ! [X2: nat > $o,Y2: nat > $o,Z: nat > $o] :
      ( ( ord_less_eq_nat_o @ X2 @ Y2 )
     => ( ( ord_less_eq_nat_o @ Y2 @ Z )
       => ( ord_less_eq_nat_o @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_845_order__trans,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).

% order_trans
thf(fact_846_order_Otrans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% order.trans
thf(fact_847_order__antisym,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X2 )
       => ( X2 = Y2 ) ) ) ).

% order_antisym
thf(fact_848_ord__le__eq__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_849_ord__eq__le__trans,axiom,
    ! [A3: nat,B2: nat,C: nat] :
      ( ( A3 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A3 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_850_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_851_le__cases3,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_852_nle__le,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A3 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A3 )
        & ( B2 != A3 ) ) ) ).

% nle_le
thf(fact_853_Collect__mono__iff,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
      = ( ! [X: list_a] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_854_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_855_Collect__mono,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).

% Collect_mono
thf(fact_856_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_857_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T3: nat] :
          ( ( member_nat @ T3 @ A6 )
         => ( member_nat @ T3 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_858_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_859_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_860_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ X2 @ A4 )
       => ( member_nat @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_861_Collect__subset,axiom,
    ! [A4: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_862_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A4 )
            & ( P @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_863_finite__lists__length__le,axiom,
    ! [A4: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_864_finite__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_865_finite__lists__length__le,axiom,
    ! [A4: set_a,N: nat] :
      ( ( finite_finite_a @ A4 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_a @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_866_split__cong,axiom,
    ! [Q3: product_prod_nat_a,F2: nat > a > ( nat > a ) > nat > a,G: nat > a > ( nat > a ) > nat > a,P2: product_prod_nat_a] :
      ( ! [X4: nat,Y3: a] :
          ( ( ( product_Pair_nat_a @ X4 @ Y3 )
            = Q3 )
         => ( ( F2 @ X4 @ Y3 )
            = ( G @ X4 @ Y3 ) ) )
     => ( ( P2 = Q3 )
       => ( ( produc2909000522608705447_nat_a @ F2 @ P2 )
          = ( produc2909000522608705447_nat_a @ G @ Q3 ) ) ) ) ).

% split_cong
thf(fact_867_set__n__lists,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
      = ( collect_list_nat
        @ ^ [Ys2: list_nat] :
            ( ( ( size_size_list_nat @ Ys2 )
              = N )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_868_set__n__lists,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( set_list_a2 @ ( n_lists_a @ N @ Xs ) )
      = ( collect_list_a
        @ ^ [Ys2: list_a] :
            ( ( ( size_size_list_a @ Ys2 )
              = N )
            & ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_869_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F2 @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_870_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_871_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_872_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_873_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_874_le__trans,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I3 @ K ) ) ) ).

% le_trans
thf(fact_875_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_876_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M2: nat] :
      ( ( P @ X2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M2 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_877_fmla_Odistinct_I19_J,axiom,
    ! [X23: $o,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X23 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(19)
thf(fact_878_fmla_Odistinct_I27_J,axiom,
    ! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(27)
thf(fact_879_sat_Osimps_I2_J,axiom,
    ! [B2: $o,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Bool_a_b @ B2 ) @ I @ Sigma )
      = B2 ) ).

% sat.simps(2)
thf(fact_880_genempty_Ointros_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q1 )
     => ( ( relati5999705594545617851ty_a_b @ Q22 )
       => ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(5)
thf(fact_881_qp__Disj,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ~ ( relational_qp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ).

% qp_Disj
thf(fact_882_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_883_length__n__lists__elem,axiom,
    ! [Ys: list_a,N: nat,Xs: list_a] :
      ( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
     => ( ( size_size_list_a @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_884_genempty_Ointros_I3_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(3)
thf(fact_885_genempty_Ointros_I4_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) ) ) ).

% genempty.intros(4)
thf(fact_886_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M4: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N4 )
         => ( ord_less_eq_nat @ X @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_887_Id__onI,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ ( id_on_nat @ A4 ) ) ) ).

% Id_onI
thf(fact_888_Id__on__def_H,axiom,
    ! [A4: list_a > $o] :
      ( ( id_on_list_a @ ( collect_list_a @ A4 ) )
      = ( collec943055143889122450list_a
        @ ( produc8172378796822260076st_a_o
          @ ^ [X: list_a,Y: list_a] :
              ( ( X = Y )
              & ( A4 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_889_Id__on__def_H,axiom,
    ! [A4: nat > $o] :
      ( ( id_on_nat @ ( collect_nat @ A4 ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( X = Y )
              & ( A4 @ X ) ) ) ) ) ).

% Id_on_def'
thf(fact_890_pred__subset__eq,axiom,
    ! [R2: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
      = ( ord_less_eq_set_nat @ R2 @ S3 ) ) ).

% pred_subset_eq
thf(fact_891_sr__Conj__eq,axiom,
    ! [Q: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( relational_sr_a_b @ Q )
     => ( ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
          | ( member_nat @ Y2 @ ( relational_fv_a_b @ Q ) ) )
       => ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X2 @ ( relational_Var_a @ Y2 ) ) ) ) ) ) ).

% sr_Conj_eq
thf(fact_892_Id__on__iff,axiom,
    ! [X2: nat,Y2: nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( id_on_nat @ A4 ) )
      = ( ( X2 = Y2 )
        & ( member_nat @ X2 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_893_Id__on__eqI,axiom,
    ! [A3: nat,B2: nat,A4: set_nat] :
      ( ( A3 = B2 )
     => ( ( member_nat @ A3 @ A4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B2 ) @ ( id_on_nat @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_894_Id__onE,axiom,
    ! [C: product_prod_nat_nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( id_on_nat @ A4 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( C
             != ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_895_genempty_Osimps,axiom,
    ( relati5999705594545617851ty_a_b
    = ( ^ [A: relational_fmla_a_b] :
          ( ( A
            = ( relational_Bool_a_b @ $false ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ( relati5999705594545617851ty_a_b @ Q12 )
              & ( relati5999705594545617851ty_a_b @ Q23 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relati5999705594545617851ty_a_b @ Q12 )
                | ( relati5999705594545617851ty_a_b @ Q23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ? [X: nat,Y: nat] :
                  ( A
                  = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ? [Y: nat] :
                  ( A
                  = ( relati591517084277583526ts_a_b @ Y @ Q2 ) )
              & ( relati5999705594545617851ty_a_b @ Q2 ) ) ) ) ) ).

% genempty.simps
thf(fact_896_genempty_Ocases,axiom,
    ! [A3: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ A3 )
     => ( ( A3
         != ( relational_Bool_a_b @ $false ) )
       => ( ! [Q4: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
             => ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( ( A3
                  = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
               => ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A3
                    = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A3
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( relati5999705594545617851ty_a_b @ Q13 )
                     => ~ ( relati5999705594545617851ty_a_b @ Q24 ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( A3
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ~ ( ( relati5999705594545617851ty_a_b @ Q13 )
                          | ( relati5999705594545617851ty_a_b @ Q24 ) ) )
                 => ( ! [Q4: relational_fmla_a_b] :
                        ( ? [X4: nat,Y3: nat] :
                            ( A3
                            = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) ) )
                       => ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
                   => ( ! [Q4: relational_fmla_a_b] :
                          ( ? [Y3: nat,X4: nat] :
                              ( A3
                              = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) ) )
                         => ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
                     => ~ ! [Q4: relational_fmla_a_b] :
                            ( ? [Y3: nat] :
                                ( A3
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q4 ) )
                           => ~ ( relati5999705594545617851ty_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% genempty.cases
thf(fact_897_eval__subst,axiom,
    ! [Sigma: nat > a,T: relational_term_a,X2: nat,Y2: nat] :
      ( ( relati1177013128715261720term_a @ Sigma @ ( relati7175845559408349773term_a @ T @ X2 @ Y2 ) )
      = ( relati1177013128715261720term_a @ ( fun_upd_nat_a @ Sigma @ X2 @ ( Sigma @ Y2 ) ) @ T ) ) ).

% eval_subst
thf(fact_898_fmla_Odistinct_I41_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Disj_a_b @ X61 @ X62 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(41)
thf(fact_899_exists__Exists,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
     => ( ( relati3989891337220013914ts_a_b @ X2 @ Q )
        = ( relati591517084277583526ts_a_b @ X2 @ Q ) ) ) ).

% exists_Exists
thf(fact_900_exists__def,axiom,
    ( relati3989891337220013914ts_a_b
    = ( ^ [X: nat,Q2: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) ) @ ( relati591517084277583526ts_a_b @ X @ Q2 ) @ Q2 ) ) ) ).

% exists_def
thf(fact_901_qp__Exists,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) ) ) ) ).

% qp_Exists
thf(fact_902_qp__ExistsE,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) )
     => ~ ( ( relational_qp_a_b @ Q )
         => ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) ) ) ) ).

% qp_ExistsE
thf(fact_903_sat_Osimps_I7_J,axiom,
    ! [Z: nat,Phi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi ) @ I @ Sigma )
      = ( ? [X: a] : ( relational_sat_a_b @ Phi @ I @ ( fun_upd_nat_a @ Sigma @ Z @ X ) ) ) ) ).

% sat.simps(7)
thf(fact_904_sat_Osimps_I3_J,axiom,
    ! [X2: nat,T4: relational_term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Eq_a_b @ X2 @ T4 ) @ I @ Sigma )
      = ( ( Sigma @ X2 )
        = ( relati1177013128715261720term_a @ Sigma @ T4 ) ) ) ).

% sat.simps(3)
thf(fact_905_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_906_in__set__product__lists__length,axiom,
    ! [Xs: list_a,Xss: list_list_a] :
      ( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_907_qp__impl_Oelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X2 )
     => ( ! [X4: nat,Q4: relational_fmla_a_b] :
            ( ( X2
              = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
           => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
              & ( relational_qp_a_b @ Q4 ) ) )
       => ( ! [V3: $o] :
              ( X2
             != ( relational_Bool_a_b @ V3 ) )
         => ( ! [V3: nat,Vb: nat] :
                ( X2
               != ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) )
           => ( ! [V3: relational_fmla_a_b] :
                  ( X2
                 != ( relational_Neg_a_b @ V3 ) )
             => ( ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                    ( X2
                   != ( relational_Conj_a_b @ V3 @ Va ) )
               => ~ ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                      ( X2
                     != ( relational_Disj_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ).

% qp_impl.elims(3)
thf(fact_908_subst_Oelims,axiom,
    ! [X2: relational_fmla_a_b,Xa2: nat,Xb: nat,Y2: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ! [T5: $o] :
            ( ( X2
              = ( relational_Bool_a_b @ T5 ) )
           => ( Y2
             != ( relational_Bool_a_b @ T5 ) ) )
       => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ P3 @ Ts ) )
             => ( Y2
               != ( relational_Pred_b_a @ P3
                  @ ( map_Re5736185711816362116term_a
                    @ ^ [T3: relational_term_a] : ( relati7175845559408349773term_a @ T3 @ Xa2 @ Xb )
                    @ Ts ) ) ) )
         => ( ! [Z3: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ Z3 @ T5 ) )
               => ( Y2
                 != ( relational_Eq_a_b @ ( if_nat @ ( Z3 = Xa2 ) @ Xb @ Z3 ) @ ( relati7175845559408349773term_a @ T5 @ Xa2 @ Xb ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y2
                   != ( relational_Neg_a_b @ ( relational_subst_a_b @ Q4 @ Xa2 @ Xb ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y2
                     != ( relational_Conj_a_b @ ( relational_subst_a_b @ Q13 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y2
                       != ( relational_Disj_a_b @ ( relational_subst_a_b @ Q13 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) ) )
                 => ~ ! [Z3: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) )
                       => ~ ( ( ( Xa2 = Z3 )
                             => ( Y2
                                = ( relati591517084277583526ts_a_b @ Xa2 @ Q4 ) ) )
                            & ( ( Xa2 != Z3 )
                             => ( ( ( Z3 = Xb )
                                 => ( Y2
                                    = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q4 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q4 @ Z3 @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q4 ) ) @ Xa2 @ Xb ) ) ) )
                                & ( ( Z3 != Xb )
                                 => ( Y2
                                    = ( relati591517084277583526ts_a_b @ Z3 @ ( relational_subst_a_b @ Q4 @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.elims
thf(fact_909_nocp_Oelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X2 )
     => ( ! [B: $o] :
            ( X2
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X4: nat,T5: relational_term_a] :
              ( ( X2
                = ( relational_Eq_a_b @ X4 @ T5 ) )
             => ( T5
               != ( relational_Var_a @ X4 ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X2
                  = ( relational_Neg_a_b @ Q4 ) )
               => ( relational_nocp_a_b @ Q4 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ( ( relational_nocp_a_b @ Q13 )
                    & ( relational_nocp_a_b @ Q24 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( relational_nocp_a_b @ Q13 )
                      & ( relational_nocp_a_b @ Q24 ) ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X2
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(3)
thf(fact_910_qp__impl_Ocases,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ! [X4: nat,C3: a] :
          ( X2
         != ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) )
     => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
            ( X2
           != ( relational_Pred_b_a @ X4 @ Ts ) )
       => ( ! [X4: nat,Q4: relational_fmla_a_b] :
              ( X2
             != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
         => ( ! [V3: $o] :
                ( X2
               != ( relational_Bool_a_b @ V3 ) )
           => ( ! [V3: nat,Vb: nat] :
                  ( X2
                 != ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) )
             => ( ! [V3: relational_fmla_a_b] :
                    ( X2
                   != ( relational_Neg_a_b @ V3 ) )
               => ( ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                      ( X2
                     != ( relational_Conj_a_b @ V3 @ Va ) )
                 => ~ ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                        ( X2
                       != ( relational_Disj_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ) ).

% qp_impl.cases
thf(fact_911_fmla_Odistinct_I9_J,axiom,
    ! [X11: b,X122: list_R6823256787227418703term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X122 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(9)
thf(fact_912_nocp_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% nocp.simps(6)
thf(fact_913_qp__impl_Osimps_I8_J,axiom,
    ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ).

% qp_impl.simps(8)
thf(fact_914_nocp_Osimps_I7_J,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) )
      = ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% nocp.simps(7)
thf(fact_915_qp__impl_Oelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X2 )
     => ( ! [X4: nat,C3: a] :
            ( X2
           != ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) )
       => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
              ( X2
             != ( relational_Pred_b_a @ X4 @ Ts ) )
         => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                ( ( X2
                  = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
               => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                    & ( relational_qp_a_b @ Q4 ) ) ) ) ) ) ).

% qp_impl.elims(2)
thf(fact_916_nocp_Oelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X2 )
     => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
            ( X2
           != ( relational_Pred_b_a @ P3 @ Ts ) )
       => ( ! [X4: nat,T5: relational_term_a] :
              ( ( X2
                = ( relational_Eq_a_b @ X4 @ T5 ) )
             => ( T5
                = ( relational_Var_a @ X4 ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X2
                  = ( relational_Neg_a_b @ Q4 ) )
               => ~ ( relational_nocp_a_b @ Q4 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ~ ( ( relational_nocp_a_b @ Q13 )
                      & ( relational_nocp_a_b @ Q24 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ~ ( ( relational_nocp_a_b @ Q13 )
                        & ( relational_nocp_a_b @ Q24 ) ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X2
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                          & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(2)
thf(fact_917_nocp_Oelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Y2: $o] :
      ( ( ( relational_nocp_a_b @ X2 )
        = Y2 )
     => ( ( ? [B: $o] :
              ( X2
              = ( relational_Bool_a_b @ B ) )
         => Y2 )
       => ( ( ? [P3: b,Ts: list_R6823256787227418703term_a] :
                ( X2
                = ( relational_Pred_b_a @ P3 @ Ts ) )
           => ~ Y2 )
         => ( ! [X4: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T5 ) )
               => ( Y2
                  = ( T5
                    = ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y2
                    = ( ~ ( relational_nocp_a_b @ Q4 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y2
                      = ( ~ ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y2
                        = ( ~ ( ( relational_nocp_a_b @ Q13 )
                              & ( relational_nocp_a_b @ Q24 ) ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( Y2
                          = ( ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                                & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.elims(1)
thf(fact_918_qp__impl_Osimps_I3_J,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) )
      = ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
        & ( relational_qp_a_b @ Q ) ) ) ).

% qp_impl.simps(3)
thf(fact_919_erase_Ocases,axiom,
    ! [X2: produc7366699395886430672_b_nat] :
      ( ! [T5: $o,X4: nat] :
          ( X2
         != ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T5 ) @ X4 ) )
     => ( ! [P3: b,Ts: list_R6823256787227418703term_a,X4: nat] :
            ( X2
           != ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ X4 ) )
       => ( ! [Z3: nat,T5: relational_term_a,X4: nat] :
              ( X2
             != ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z3 @ T5 ) @ X4 ) )
         => ( ! [Q4: relational_fmla_a_b,X4: nat] :
                ( X2
               != ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q4 ) @ X4 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat] :
                  ( X2
                 != ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ X4 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat] :
                    ( X2
                   != ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ X4 ) )
               => ~ ! [Z3: nat,Q4: relational_fmla_a_b,X4: nat] :
                      ( X2
                     != ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ X4 ) ) ) ) ) ) ) ) ).

% erase.cases
thf(fact_920_sat_Ocases,axiom,
    ! [X2: produc1132964494702330949_nat_a] :
      ( ! [R3: b,Ts: list_R6823256787227418703term_a,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
          ( X2
         != ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R3 @ Ts ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
     => ( ! [B: $o,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
            ( X2
           != ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
       => ( ! [X4: nat,T6: relational_term_a,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
              ( X2
             != ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T6 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
         => ( ! [Phi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
                ( X2
               != ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
                  ( X2
                 != ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
                    ( X2
                   != ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) )
               => ~ ! [Z3: nat,Phi2: relational_fmla_a_b,I4: product_prod_b_nat > set_list_a,Sigma4: nat > a] :
                      ( X2
                     != ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I4 @ Sigma4 ) ) ) ) ) ) ) ) ) ).

% sat.cases
thf(fact_921_cp_Ocases,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ! [X4: nat,T5: relational_term_a] :
          ( X2
         != ( relational_Eq_a_b @ X4 @ T5 ) )
     => ( ! [Q4: relational_fmla_a_b] :
            ( X2
           != ( relational_Neg_a_b @ Q4 ) )
       => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
              ( X2
             != ( relational_Conj_a_b @ Q13 @ Q24 ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( X2
               != ( relational_Disj_a_b @ Q13 @ Q24 ) )
           => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                  ( X2
                 != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
             => ( ! [V3: b,Va: list_R6823256787227418703term_a] :
                    ( X2
                   != ( relational_Pred_b_a @ V3 @ Va ) )
               => ~ ! [V3: $o] :
                      ( X2
                     != ( relational_Bool_a_b @ V3 ) ) ) ) ) ) ) ) ).

% cp.cases
thf(fact_922_fv_Ocases,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ! [Uu2: b,Ts: list_R6823256787227418703term_a] :
          ( X2
         != ( relational_Pred_b_a @ Uu2 @ Ts ) )
     => ( ! [B: $o] :
            ( X2
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X4: nat,T6: relational_term_a] :
              ( X2
             != ( relational_Eq_a_b @ X4 @ T6 ) )
         => ( ! [Phi2: relational_fmla_a_b] :
                ( X2
               != ( relational_Neg_a_b @ Phi2 ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                  ( X2
                 != ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                    ( X2
                   != ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
               => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                      ( X2
                     != ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) ) ) ) ) ) ) ) ).

% fv.cases
thf(fact_923_nocp_Ocases,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ! [B: $o] :
          ( X2
         != ( relational_Bool_a_b @ B ) )
     => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
            ( X2
           != ( relational_Pred_b_a @ P3 @ Ts ) )
       => ( ! [X4: nat,T5: relational_term_a] :
              ( X2
             != ( relational_Eq_a_b @ X4 @ T5 ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( X2
               != ( relational_Neg_a_b @ Q4 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( X2
                 != ( relational_Conj_a_b @ Q13 @ Q24 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( X2
                   != ( relational_Disj_a_b @ Q13 @ Q24 ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( X2
                     != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) ) ) ) ) ) ).

% nocp.cases
thf(fact_924_fmla_Oexhaust,axiom,
    ! [Y2: relational_fmla_a_b] :
      ( ! [X112: b,X123: list_R6823256787227418703term_a] :
          ( Y2
         != ( relational_Pred_b_a @ X112 @ X123 ) )
     => ( ! [X24: $o] :
            ( Y2
           != ( relational_Bool_a_b @ X24 ) )
       => ( ! [X312: nat,X322: relational_term_a] :
              ( Y2
             != ( relational_Eq_a_b @ X312 @ X322 ) )
         => ( ! [X43: relational_fmla_a_b] :
                ( Y2
               != ( relational_Neg_a_b @ X43 ) )
           => ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
                  ( Y2
                 != ( relational_Conj_a_b @ X512 @ X522 ) )
             => ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
                    ( Y2
                   != ( relational_Disj_a_b @ X612 @ X622 ) )
               => ~ ! [X712: nat,X722: relational_fmla_a_b] :
                      ( Y2
                     != ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).

% fmla.exhaust
thf(fact_925_qp__impl_Oelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Y2: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X2 )
        = Y2 )
     => ( ( ? [X4: nat,C3: a] :
              ( X2
              = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) )
         => ~ Y2 )
       => ( ( ? [X4: b,Ts: list_R6823256787227418703term_a] :
                ( X2
                = ( relational_Pred_b_a @ X4 @ Ts ) )
           => ~ Y2 )
         => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                ( ( X2
                  = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
               => ( Y2
                  = ( ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_qp_a_b @ Q4 ) ) ) ) )
           => ( ( ? [V3: $o] :
                    ( X2
                    = ( relational_Bool_a_b @ V3 ) )
               => Y2 )
             => ( ( ? [V3: nat,Vb: nat] :
                      ( X2
                      = ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) )
                 => Y2 )
               => ( ( ? [V3: relational_fmla_a_b] :
                        ( X2
                        = ( relational_Neg_a_b @ V3 ) )
                   => Y2 )
                 => ( ( ? [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                          ( X2
                          = ( relational_Conj_a_b @ V3 @ Va ) )
                     => Y2 )
                   => ~ ( ? [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                            ( X2
                            = ( relational_Disj_a_b @ V3 @ Va ) )
                       => Y2 ) ) ) ) ) ) ) ) ) ).

% qp_impl.elims(1)
thf(fact_926_subst_Ocases,axiom,
    ! [X2: produc8867654947514737559at_nat] :
      ( ! [T5: $o,X4: nat,Y3: nat] :
          ( X2
         != ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T5 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
     => ( ! [P3: b,Ts: list_R6823256787227418703term_a,X4: nat,Y3: nat] :
            ( X2
           != ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
       => ( ! [Z3: nat,T5: relational_term_a,X4: nat,Y3: nat] :
              ( X2
             != ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z3 @ T5 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
         => ( ! [Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                ( X2
               != ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat,Y3: nat] :
                  ( X2
                 != ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat,Y3: nat] :
                    ( X2
                   != ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
               => ~ ! [Z3: nat,Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                      ( X2
                     != ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ) ) ) ) ) ) ).

% subst.cases
thf(fact_927_sat_Oelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
     => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
            ( ( X2
              = ( relational_Pred_b_a @ R3 @ Ts ) )
           => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => B )
         => ( ! [X4: nat,T6: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T6 ) )
               => ( ( Xb @ X4 )
                  = ( relati1177013128715261720term_a @ Xb @ T6 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                   => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                      & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                     => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                        | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
                 => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                       => ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X5 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(3)
thf(fact_928_sat_Oelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
     => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
            ( ( X2
              = ( relational_Pred_b_a @ R3 @ Ts ) )
           => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => ~ B )
         => ( ! [X4: nat,T6: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T6 ) )
               => ( ( Xb @ X4 )
                 != ( relati1177013128715261720term_a @ Xb @ T6 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                   => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                        & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                     => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                          | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
                 => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                       => ~ ? [X4: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(2)
thf(fact_929_sat_Oelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a,Y2: $o] :
      ( ( ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
            ( ( X2
              = ( relational_Pred_b_a @ R3 @ Ts ) )
           => ( Y2
              = ( ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => ( Y2 = ~ B ) )
         => ( ! [X4: nat,T6: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T6 ) )
               => ( Y2
                  = ( ( Xb @ X4 )
                   != ( relati1177013128715261720term_a @ Xb @ T6 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y2
                    = ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                   => ( Y2
                      = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                     => ( Y2
                        = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) ) )
                 => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                       => ( Y2
                          = ( ~ ? [X: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(1)
thf(fact_930_subst_Opelims,axiom,
    ! [X2: relational_fmla_a_b,Xa2: nat,Xb: nat,Y2: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ X2 @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) )
       => ( ! [T5: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ T5 ) )
             => ( ( Y2
                  = ( relational_Bool_a_b @ T5 ) )
               => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T5 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
         => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ P3 @ Ts ) )
               => ( ( Y2
                    = ( relational_Pred_b_a @ P3
                      @ ( map_Re5736185711816362116term_a
                        @ ^ [T3: relational_term_a] : ( relati7175845559408349773term_a @ T3 @ Xa2 @ Xb )
                        @ Ts ) ) )
                 => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
           => ( ! [Z3: nat,T5: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ Z3 @ T5 ) )
                 => ( ( Y2
                      = ( relational_Eq_a_b @ ( if_nat @ ( Z3 = Xa2 ) @ Xb @ Z3 ) @ ( relati7175845559408349773term_a @ T5 @ Xa2 @ Xb ) ) )
                   => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z3 @ T5 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y2
                        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q4 @ Xa2 @ Xb ) ) )
                     => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y2
                          = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q13 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) )
                       => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y2
                            = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q13 @ Xa2 @ Xb ) @ ( relational_subst_a_b @ Q24 @ Xa2 @ Xb ) ) )
                         => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) )
                   => ~ ! [Z3: nat,Q4: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) )
                         => ( ( ( ( Xa2 = Z3 )
                               => ( Y2
                                  = ( relati591517084277583526ts_a_b @ Xa2 @ Q4 ) ) )
                              & ( ( Xa2 != Z3 )
                               => ( ( ( Z3 = Xb )
                                   => ( Y2
                                      = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q4 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q4 @ Z3 @ ( relati2677767559083392098h2_a_b @ Xa2 @ Xb @ Q4 ) ) @ Xa2 @ Xb ) ) ) )
                                  & ( ( Z3 != Xb )
                                   => ( Y2
                                      = ( relati591517084277583526ts_a_b @ Z3 @ ( relational_subst_a_b @ Q4 @ Xa2 @ Xb ) ) ) ) ) ) )
                           => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ ( product_Pair_nat_nat @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.pelims
thf(fact_931_sat__cp,axiom,
    ! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_cp_a_b @ Q ) @ I @ Sigma )
      = ( relational_sat_a_b @ Q @ I @ Sigma ) ) ).

% sat_cp
thf(fact_932_subst_Opsimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,X2: nat,Y2: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
     => ( ( relational_subst_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ X2 @ Y2 )
        = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q1 @ X2 @ Y2 ) @ ( relational_subst_a_b @ Q22 @ X2 @ Y2 ) ) ) ) ).

% subst.psimps(6)
thf(fact_933_fv__cp,axiom,
    ! [Q: relational_fmla_a_b] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_cp_a_b @ Q ) ) @ ( relational_fv_a_b @ Q ) ) ).

% fv_cp
thf(fact_934_sat_Osimps_I1_J,axiom,
    ! [R4: b,Ts2: list_R6823256787227418703term_a,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Pred_b_a @ R4 @ Ts2 ) @ I @ Sigma )
      = ( member_list_a @ ( relati4772805863405912879erms_a @ Sigma @ Ts2 ) @ ( I @ ( product_Pair_b_nat @ R4 @ ( size_s88622898042387131term_a @ Ts2 ) ) ) ) ) ).

% sat.simps(1)
thf(fact_935_subst_Opinduct,axiom,
    ! [A0: relational_fmla_a_b,A1: nat,A22: nat,P: relational_fmla_a_b > nat > nat > $o] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ A0 @ ( product_Pair_nat_nat @ A1 @ A22 ) ) )
     => ( ! [T5: $o,X4: nat,Y3: nat] :
            ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T5 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
           => ( P @ ( relational_Bool_a_b @ T5 ) @ X4 @ Y3 ) )
       => ( ! [P3: b,Ts: list_R6823256787227418703term_a,X4: nat,Y3: nat] :
              ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P3 @ Ts ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
             => ( P @ ( relational_Pred_b_a @ P3 @ Ts ) @ X4 @ Y3 ) )
         => ( ! [Z3: nat,T5: relational_term_a,X4: nat,Y3: nat] :
                ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z3 @ T5 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
               => ( P @ ( relational_Eq_a_b @ Z3 @ T5 ) @ X4 @ Y3 ) )
           => ( ! [Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                  ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                 => ( ( P @ Q4 @ X4 @ Y3 )
                   => ( P @ ( relational_Neg_a_b @ Q4 ) @ X4 @ Y3 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat,Y3: nat] :
                    ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                   => ( ( P @ Q13 @ X4 @ Y3 )
                     => ( ( P @ Q24 @ X4 @ Y3 )
                       => ( P @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ X4 @ Y3 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,X4: nat,Y3: nat] :
                      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                     => ( ( P @ Q13 @ X4 @ Y3 )
                       => ( ( P @ Q24 @ X4 @ Y3 )
                         => ( P @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ X4 @ Y3 ) ) ) )
                 => ( ! [Z3: nat,Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                        ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                       => ( ! [Xa: nat] :
                              ( ( X4 != Z3 )
                             => ( ( Z3 = Y3 )
                               => ( ( Xa
                                    = ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q4 ) )
                                 => ( P @ Q4 @ Z3 @ Xa ) ) ) )
                         => ( ! [Xa: nat] :
                                ( ( X4 != Z3 )
                               => ( ( Z3 = Y3 )
                                 => ( ( Xa
                                      = ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q4 ) )
                                   => ( P @ ( relational_subst_a_b @ Q4 @ Z3 @ Xa ) @ X4 @ Y3 ) ) ) )
                           => ( ( ( X4 != Z3 )
                               => ( ( Z3 != Y3 )
                                 => ( P @ Q4 @ X4 @ Y3 ) ) )
                             => ( P @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ X4 @ Y3 ) ) ) ) )
                   => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).

% subst.pinduct
thf(fact_936_gen__Gen__substs__Exists,axiom,
    ! [Xs: list_nat,Ys: list_nat,X2: nat,Y2: nat,Q: relational_fmla_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( X2 != Y2 )
       => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
         => ( ! [Xs2: list_nat,Ys3: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ? [X_12: set_Re381260168593705685la_a_b] :
                    ( relational_gen_a_b @ ( relati8128731020529265620ar_nat @ Xs2 @ Ys3 @ X2 )
                    @ ( relational_cp_a_b
                      @ ( fold_P7970104616371074773la_a_b
                        @ ( produc5586541307551673003la_a_b
                          @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
                        @ ( zip_nat_nat @ Xs2 @ Ys3 )
                        @ Q ) )
                    @ X_12 ) )
           => ? [X_1: set_Re381260168593705685la_a_b] :
                ( relational_gen_a_b @ ( relati8128731020529265620ar_nat @ Xs @ Ys @ X2 )
                @ ( relational_cp_a_b
                  @ ( fold_P7970104616371074773la_a_b
                    @ ( produc5586541307551673003la_a_b
                      @ ^ [X: nat,Y: nat,Q2: relational_fmla_a_b] : ( relational_subst_a_b @ Q2 @ X @ Y ) )
                    @ ( zip_nat_nat @ Xs @ Ys )
                    @ ( relati591517084277583526ts_a_b @ Y2 @ Q ) ) )
                @ X_1 ) ) ) ) ) ).

% gen_Gen_substs_Exists
thf(fact_937_fv__cp__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,A4: set_nat] :
      ( ( ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) )
       != ( relational_Bool_a_b @ $false ) )
     => ( ! [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
           => ( ( relati1591879772219623554ed_a_b @ X4 )
              & ( ( relational_fv_a_b @ X4 )
                = A4 ) ) )
       => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) ) )
          = A4 ) ) ) ).

% fv_cp_foldr1_Disj
thf(fact_938_sat_Opelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X2 @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ R3 @ Ts ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R3 @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
               => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X2
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                 => B ) )
           => ( ! [X4: nat,T6: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T6 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T6 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                   => ( ( Xb @ X4 )
                      = ( relati1177013128715261720term_a @ Xb @ T6 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                     => ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                       => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                          & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                         => ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) )
                   => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                           => ? [X5: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X5 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(3)
thf(fact_939_sat_Opelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X2 @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ R3 @ Ts ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R3 @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
               => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X2
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                 => ~ B ) )
           => ( ! [X4: nat,T6: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T6 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T6 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                   => ( ( Xb @ X4 )
                     != ( relati1177013128715261720term_a @ Xb @ T6 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                     => ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                       => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                         => ~ ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) ) )
                   => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
                           => ~ ? [X4: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(2)
thf(fact_940_cpropagated__simps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q22 ) ) ) ).

% cpropagated_simps(6)
thf(fact_941_cpropagated__simps_I7_J,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) )
      = ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_simps(7)
thf(fact_942_gen__sat,axiom,
    ! [X2: nat,Q: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_gen_a_b @ X2 @ Q @ G3 )
     => ( ( relational_sat_a_b @ Q @ I @ Sigma )
       => ? [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ G3 )
            & ( relational_sat_a_b @ X4 @ I @ Sigma ) ) ) ) ).

% gen_sat
thf(fact_943_qp__Gen,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X2 @ Q @ X_1 ) ) ) ).

% qp_Gen
thf(fact_944_gen__fv,axiom,
    ! [X2: nat,Q: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_gen_a_b @ X2 @ Q @ G3 )
     => ( ( member4680049679412964150la_a_b @ Qqp @ G3 )
       => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Qqp ) )
          & ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).

% gen_fv
thf(fact_945_gen_Ointros_I5_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X2 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G3 )
     => ( relational_gen_a_b @ X2 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G3 ) ) ).

% gen.intros(5)
thf(fact_946_gen_Ointros_I4_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X2 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G3 )
     => ( relational_gen_a_b @ X2 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G3 ) ) ).

% gen.intros(4)
thf(fact_947_cpropagated__nocp,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_nocp
thf(fact_948_sat_Opelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Xa2: product_prod_b_nat > set_list_a,Xb: nat > a,Y2: $o] :
      ( ( ( relational_sat_a_b @ X2 @ Xa2 @ Xb )
        = Y2 )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X2 @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) )
       => ( ! [R3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ R3 @ Ts ) )
             => ( ( Y2
                  = ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa2 @ ( product_Pair_b_nat @ R3 @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
               => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R3 @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X2
                  = ( relational_Bool_a_b @ B ) )
               => ( ( Y2 = B )
                 => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
           => ( ! [X4: nat,T6: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T6 ) )
                 => ( ( Y2
                      = ( ( Xb @ X4 )
                        = ( relati1177013128715261720term_a @ Xb @ T6 ) ) )
                   => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T6 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y2
                        = ( ~ ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb ) ) )
                     => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Phi2 @ Psi2 ) )
                     => ( ( Y2
                          = ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                            & ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
                       => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Phi2 @ Psi2 ) )
                       => ( ( Y2
                            = ( ( relational_sat_a_b @ Phi2 @ Xa2 @ Xb )
                              | ( relational_sat_a_b @ Psi2 @ Xa2 @ Xb ) ) )
                         => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) )
                   => ~ ! [Z3: nat,Phi2: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) )
                         => ( ( Y2
                              = ( ? [X: a] : ( relational_sat_a_b @ Phi2 @ Xa2 @ ( fun_upd_nat_a @ Xb @ Z3 @ X ) ) ) )
                           => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z3 @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa2 @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(1)
thf(fact_949_qp__impl_Opelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Y2: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X2 )
        = Y2 )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X2 )
       => ( ! [X4: nat,C3: a] :
              ( ( X2
                = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) )
             => ( Y2
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) ) ) )
         => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ X4 @ Ts ) )
               => ( Y2
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X4 @ Ts ) ) ) )
           => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                 => ( ( Y2
                      = ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_qp_a_b @ Q4 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) )
             => ( ! [V3: $o] :
                    ( ( X2
                      = ( relational_Bool_a_b @ V3 ) )
                   => ( ~ Y2
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V3 ) ) ) )
               => ( ! [V3: nat,Vb: nat] :
                      ( ( X2
                        = ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) )
                     => ( ~ Y2
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) ) ) )
                 => ( ! [V3: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Neg_a_b @ V3 ) )
                       => ( ~ Y2
                         => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V3 ) ) ) )
                   => ( ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                          ( ( X2
                            = ( relational_Conj_a_b @ V3 @ Va ) )
                         => ( ~ Y2
                           => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V3 @ Va ) ) ) )
                     => ~ ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                            ( ( X2
                              = ( relational_Disj_a_b @ V3 @ Va ) )
                           => ( ~ Y2
                             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(1)
thf(fact_950_qp__impl_Opelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X2 )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X2 )
       => ( ! [X4: nat,Q4: relational_fmla_a_b] :
              ( ( X2
                = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
             => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
               => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                  & ( relational_qp_a_b @ Q4 ) ) ) )
         => ( ! [V3: $o] :
                ( ( X2
                  = ( relational_Bool_a_b @ V3 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V3 ) ) )
           => ( ! [V3: nat,Vb: nat] :
                  ( ( X2
                    = ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V3 @ ( relational_Var_a @ Vb ) ) ) )
             => ( ! [V3: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ V3 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V3 ) ) )
               => ( ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ V3 @ Va ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V3 @ Va ) ) )
                 => ~ ! [V3: relational_fmla_a_b,Va: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ V3 @ Va ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V3 @ Va ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(3)
thf(fact_951_nocp_Opelims_I1_J,axiom,
    ! [X2: relational_fmla_a_b,Y2: $o] :
      ( ( ( relational_nocp_a_b @ X2 )
        = Y2 )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X2 )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => ( ~ Y2
               => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
         => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ P3 @ Ts ) )
               => ( Y2
                 => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
           => ( ! [X4: nat,T5: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T5 ) )
                 => ( ( Y2
                      = ( T5
                       != ( relational_Var_a @ X4 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y2
                        = ( relational_nocp_a_b @ Q4 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y2
                          = ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y2
                            = ( ( relational_nocp_a_b @ Q13 )
                              & ( relational_nocp_a_b @ Q24 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
                   => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ( ( Y2
                              = ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                                & ( relational_nocp_a_b @ Q4 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(1)
thf(fact_952_qp__impl_Opelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X2 )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X2 )
       => ( ! [X4: nat,C3: a] :
              ( ( X2
                = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C3 ) ) ) )
         => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ X4 @ Ts ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X4 @ Ts ) ) )
           => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                   => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_qp_a_b @ Q4 ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(2)
thf(fact_953_nocp_Opelims_I3_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X2 )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X2 )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) )
         => ( ! [X4: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T5 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) )
                 => ( T5
                   != ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
                   => ( relational_nocp_a_b @ Q4 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( relational_nocp_a_b @ Q13 )
                        & ( relational_nocp_a_b @ Q24 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( relational_nocp_a_b @ Q13 )
                          & ( relational_nocp_a_b @ Q24 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                            & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(3)
thf(fact_954_nocp_Opelims_I2_J,axiom,
    ! [X2: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X2 )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X2 )
       => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ P3 @ Ts ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) )
         => ( ! [X4: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T5 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) )
                 => ( T5
                    = ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
                   => ~ ( relational_nocp_a_b @ Q4 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ~ ( ( relational_nocp_a_b @ Q13 )
                          & ( relational_nocp_a_b @ Q24 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ~ ( ( relational_nocp_a_b @ Q13 )
                            & ( relational_nocp_a_b @ Q24 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                              & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.pelims(2)
thf(fact_955_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_956_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_957_Collect__empty__eq,axiom,
    ! [P: list_a > $o] :
      ( ( ( collect_list_a @ P )
        = bot_bot_set_list_a )
      = ( ! [X: list_a] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_958_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_959_empty__Collect__eq,axiom,
    ! [P: list_a > $o] :
      ( ( bot_bot_set_list_a
        = ( collect_list_a @ P ) )
      = ( ! [X: list_a] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_960_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_961_rrb__simps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_rrb_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( ( relational_rrb_a_b @ Q1 )
        & ( relational_rrb_a_b @ Q22 ) ) ) ).

% rrb_simps(5)
thf(fact_962_rrb__simps_I8_J,axiom,
    ! [Y2: nat,Qy: relational_fmla_a_b] :
      ( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y2 @ Qy ) )
      = ( ( ( member_nat @ Y2 @ ( relational_fv_a_b @ Qy ) )
         => ? [X6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y2 @ Qy @ X6 ) )
        & ( relational_rrb_a_b @ Qy ) ) ) ).

% rrb_simps(8)
thf(fact_963_infinite__imp__nonempty,axiom,
    ! [S3: set_list_a] :
      ( ~ ( finite_finite_list_a @ S3 )
     => ( S3 != bot_bot_set_list_a ) ) ).

% infinite_imp_nonempty
thf(fact_964_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_965_finite_OemptyI,axiom,
    finite_finite_list_a @ bot_bot_set_list_a ).

% finite.emptyI
thf(fact_966_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_967_bot_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A3 ) ).

% bot.extremum
thf(fact_968_bot_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
      = ( A3 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_969_bot_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ bot_bot_nat )
     => ( A3 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_970_emptyE,axiom,
    ! [A3: nat] :
      ~ ( member_nat @ A3 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_971_equals0D,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A3 @ A4 ) ) ).

% equals0D
thf(fact_972_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_973_empty__def,axiom,
    ( bot_bot_set_list_a
    = ( collect_list_a
      @ ^ [X: list_a] : $false ) ) ).

% empty_def
thf(fact_974_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_975_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_976_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_977_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_978_cp_Oelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X2 )
        = Y2 )
     => ( ! [X4: nat,T5: relational_term_a] :
            ( ( X2
              = ( relational_Eq_a_b @ X4 @ T5 ) )
           => ( Y2
             != ( relati582353067970734056la_a_b
                @ ^ [A: a] : ( relational_Eq_a_b @ X4 @ T5 )
                @ ^ [Y: nat] : ( if_Rel1279876242545935705la_a_b @ ( X4 = Y ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y ) ) )
                @ T5 ) ) )
       => ( ! [Q4: relational_fmla_a_b] :
              ( ( X2
                = ( relational_Neg_a_b @ Q4 ) )
             => ( Y2
               != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q4 ) )
                  @ ( relational_Bool_a_b
                    @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q4 ) ) )
                  @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q4 ) ) ) ) )
         => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                ( ( X2
                  = ( relational_Conj_a_b @ Q13 @ Q24 ) )
               => ( Y2
                 != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_cp_a_b @ Q24 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_cp_a_b @ Q13 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                 => ( Y2
                   != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) ) )
             => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                    ( ( X2
                      = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                   => ( Y2
                     != ( relati3989891337220013914ts_a_b @ X4 @ ( relational_cp_a_b @ Q4 ) ) ) )
               => ( ! [V3: b,Va: list_R6823256787227418703term_a] :
                      ( ( X2
                        = ( relational_Pred_b_a @ V3 @ Va ) )
                     => ( Y2
                       != ( relational_Pred_b_a @ V3 @ Va ) ) )
                 => ~ ! [V3: $o] :
                        ( ( X2
                          = ( relational_Bool_a_b @ V3 ) )
                       => ( Y2
                         != ( relational_Bool_a_b @ V3 ) ) ) ) ) ) ) ) ) ) ).

% cp.elims
thf(fact_979_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X4: nat] :
          ~ ( member_nat @ X4 @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_980_bot__set__def,axiom,
    ( bot_bot_set_list_a
    = ( collect_list_a @ bot_bot_list_a_o ) ) ).

% bot_set_def
thf(fact_981_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_982_fmla_Odisc_I13_J,axiom,
    ! [X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.disc(13)
thf(fact_983_cp_Osimps_I4_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q22 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q1 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) ).

% cp.simps(4)
thf(fact_984_fv_Osimps_I2_J,axiom,
    ! [B2: $o] :
      ( ( relational_fv_a_b @ ( relational_Bool_a_b @ B2 ) )
      = bot_bot_set_nat ) ).

% fv.simps(2)
thf(fact_985_prop__restrict,axiom,
    ! [X2: list_a,Z5: set_list_a,X3: set_list_a,P: list_a > $o] :
      ( ( member_list_a @ X2 @ Z5 )
     => ( ( ord_le8861187494160871172list_a @ Z5
          @ ( collect_list_a
            @ ^ [X: list_a] :
                ( ( member_list_a @ X @ X3 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_986_prop__restrict,axiom,
    ! [X2: nat,Z5: set_nat,X3: set_nat,P: nat > $o] :
      ( ( member_nat @ X2 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X3 )
                & ( P @ X ) ) ) )
       => ( P @ X2 ) ) ) ).

% prop_restrict
thf(fact_987_Collect__restrict,axiom,
    ! [X3: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ X3 )
            & ( P @ X ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_988_Collect__restrict,axiom,
    ! [X3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X3 )
            & ( P @ X ) ) )
      @ X3 ) ).

% Collect_restrict
thf(fact_989_Exists__in__sub__foldr__Disj,axiom,
    ! [X2: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
     => ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ Q ) )
        | ? [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ X4 ) ) ) ) ) ).

% Exists_in_sub_foldr_Disj
thf(fact_990_Exists__in__sub__cp__foldr__Disj,axiom,
    ! [X2: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
     => ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) )
        | ? [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X4 ) ) ) ) ) ) ).

% Exists_in_sub_cp_foldr_Disj
thf(fact_991_cp_Opelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X2 )
        = Y2 )
     => ( ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ X2 )
       => ( ! [X4: nat,T5: relational_term_a] :
              ( ( X2
                = ( relational_Eq_a_b @ X4 @ T5 ) )
             => ( ( Y2
                  = ( relati582353067970734056la_a_b
                    @ ^ [A: a] : ( relational_Eq_a_b @ X4 @ T5 )
                    @ ^ [Y: nat] : ( if_Rel1279876242545935705la_a_b @ ( X4 = Y ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y ) ) )
                    @ T5 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X2
                  = ( relational_Neg_a_b @ Q4 ) )
               => ( ( Y2
                    = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q4 ) )
                      @ ( relational_Bool_a_b
                        @ ~ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q4 ) ) )
                      @ ( relational_Neg_a_b @ ( relational_cp_a_b @ Q4 ) ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                 => ( ( Y2
                      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_cp_a_b @ Q24 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_cp_a_b @ Q13 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                   => ( ( Y2
                        = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q24 ) ) @ ( relational_Bool_a_b @ $true ) @ ( relational_cp_a_b @ Q13 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q13 ) @ ( relational_cp_a_b @ Q24 ) ) ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
               => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X2
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ( ( Y2
                          = ( relati3989891337220013914ts_a_b @ X4 @ ( relational_cp_a_b @ Q4 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) )
                 => ( ! [V3: b,Va: list_R6823256787227418703term_a] :
                        ( ( X2
                          = ( relational_Pred_b_a @ V3 @ Va ) )
                       => ( ( Y2
                            = ( relational_Pred_b_a @ V3 @ Va ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Pred_b_a @ V3 @ Va ) ) ) )
                   => ~ ! [V3: $o] :
                          ( ( X2
                            = ( relational_Bool_a_b @ V3 ) )
                         => ( ( Y2
                              = ( relational_Bool_a_b @ V3 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Bool_a_b @ V3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% cp.pelims
thf(fact_992_fv__close,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati7004044288120072026TS_a_b @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q ) ) @ Q ) )
      = bot_bot_set_nat ) ).

% fv_close
thf(fact_993_Exists__in__sub__cp__foldr1__Disj,axiom,
    ! [X2: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) ) )
     => ( ( ( Qs = nil_Re6358386334527539737la_a_b )
          & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ Q ) ) ) )
        | ? [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relational_cp_a_b @ X4 ) ) ) ) ) ) ).

% Exists_in_sub_cp_foldr1_Disj
thf(fact_994_image__eqI,axiom,
    ! [B2: nat,F2: nat > nat,X2: nat,A4: set_nat] :
      ( ( B2
        = ( F2 @ X2 ) )
     => ( ( member_nat @ X2 @ A4 )
       => ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A4 ) ) ) ) ).

% image_eqI
thf(fact_995_finite__imageI,axiom,
    ! [F4: set_list_a,H: list_a > list_a] :
      ( ( finite_finite_list_a @ F4 )
     => ( finite_finite_list_a @ ( image_list_a_list_a @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_996_finite__imageI,axiom,
    ! [F4: set_list_a,H: list_a > nat] :
      ( ( finite_finite_list_a @ F4 )
     => ( finite_finite_nat @ ( image_list_a_nat @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_997_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > list_a] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_list_a @ ( image_nat_list_a @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_998_finite__imageI,axiom,
    ! [F4: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F4 )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).

% finite_imageI
thf(fact_999_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( ( zip_nat_a @ Xs @ Ys )
        = nil_Pr1417316670369895453_nat_a )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_a ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1000_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( nil_Pr1417316670369895453_nat_a
        = ( zip_nat_a @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_a ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1001_zip__Nil,axiom,
    ! [Ys: list_a] :
      ( ( zip_nat_a @ nil_nat @ Ys )
      = nil_Pr1417316670369895453_nat_a ) ).

% zip_Nil
thf(fact_1002_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
    = nil_nat ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_1003_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( linord2614967742042102400et_nat @ A4 )
        = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1004_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( linord2614967742042102400et_nat @ A4 )
          = nil_nat )
        = ( A4 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1005_sat__foldr1__Disj,axiom,
    ! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Xs @ Q ) @ I @ Sigma )
      = ( ( ( Xs = nil_Re6358386334527539737la_a_b )
         => ( relational_sat_a_b @ Q @ I @ Sigma ) )
        & ( ( Xs != nil_Re6358386334527539737la_a_b )
         => ? [X: relational_fmla_a_b] :
              ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Xs ) )
              & ( relational_sat_a_b @ X @ I @ Sigma ) ) ) ) ) ).

% sat_foldr1_Disj
thf(fact_1006_image__Collect__subsetI,axiom,
    ! [P: list_a > $o,F2: list_a > nat,B4: set_nat] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
         => ( member_nat @ ( F2 @ X4 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_list_a_nat @ F2 @ ( collect_list_a @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_1007_image__Collect__subsetI,axiom,
    ! [P: nat > $o,F2: nat > nat,B4: set_nat] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( member_nat @ ( F2 @ X4 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_1008_pigeonhole__infinite,axiom,
    ! [A4: set_list_a,F2: list_a > list_a] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_list_a @ ( image_list_a_list_a @ F2 @ A4 ) )
       => ? [X4: list_a] :
            ( ( member_list_a @ X4 @ A4 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A: list_a] :
                      ( ( member_list_a @ A @ A4 )
                      & ( ( F2 @ A )
                        = ( F2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1009_pigeonhole__infinite,axiom,
    ! [A4: set_list_a,F2: list_a > nat] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_nat @ ( image_list_a_nat @ F2 @ A4 ) )
       => ? [X4: list_a] :
            ( ( member_list_a @ X4 @ A4 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A: list_a] :
                      ( ( member_list_a @ A @ A4 )
                      & ( ( F2 @ A )
                        = ( F2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1010_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F2: nat > list_a] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_list_a @ ( image_nat_list_a @ F2 @ A4 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A: nat] :
                      ( ( member_nat @ A @ A4 )
                      & ( ( F2 @ A )
                        = ( F2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1011_pigeonhole__infinite,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A4 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A: nat] :
                      ( ( member_nat @ A @ A4 )
                      & ( ( F2 @ A )
                        = ( F2 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_1012_setcompr__eq__image,axiom,
    ! [F2: list_a > list_a,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [Uu: list_a] :
          ? [X: list_a] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_list_a_list_a @ F2 @ ( collect_list_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1013_setcompr__eq__image,axiom,
    ! [F2: nat > list_a,P: nat > $o] :
      ( ( collect_list_a
        @ ^ [Uu: list_a] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_list_a @ F2 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1014_setcompr__eq__image,axiom,
    ! [F2: list_a > nat,P: list_a > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: list_a] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_list_a_nat @ F2 @ ( collect_list_a @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1015_setcompr__eq__image,axiom,
    ! [F2: nat > nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F2 @ ( collect_nat @ P ) ) ) ).

% setcompr_eq_image
thf(fact_1016_Setcompr__eq__image,axiom,
    ! [F2: nat > list_a,A4: set_nat] :
      ( ( collect_list_a
        @ ^ [Uu: list_a] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_nat @ X @ A4 ) ) )
      = ( image_nat_list_a @ F2 @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_1017_Setcompr__eq__image,axiom,
    ! [F2: nat > nat,A4: set_nat] :
      ( ( collect_nat
        @ ^ [Uu: nat] :
          ? [X: nat] :
            ( ( Uu
              = ( F2 @ X ) )
            & ( member_nat @ X @ A4 ) ) )
      = ( image_nat_nat @ F2 @ A4 ) ) ).

% Setcompr_eq_image
thf(fact_1018_image__subsetI,axiom,
    ! [A4: set_nat,F2: nat > nat,B4: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A4 )
         => ( member_nat @ ( F2 @ X4 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_1019_zip_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( zip_nat_a @ Xs @ nil_a )
      = nil_Pr1417316670369895453_nat_a ) ).

% zip.simps(1)
thf(fact_1020_imageE,axiom,
    ! [B2: nat,F2: nat > nat,A4: set_nat] :
      ( ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A4 ) )
     => ~ ! [X4: nat] :
            ( ( B2
              = ( F2 @ X4 ) )
           => ~ ( member_nat @ X4 @ A4 ) ) ) ).

% imageE
thf(fact_1021_imageI,axiom,
    ! [X2: nat,A4: set_nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A4 )
     => ( member_nat @ ( F2 @ X2 ) @ ( image_nat_nat @ F2 @ A4 ) ) ) ).

% imageI
thf(fact_1022_rev__image__eqI,axiom,
    ! [X2: nat,A4: set_nat,B2: nat,F2: nat > nat] :
      ( ( member_nat @ X2 @ A4 )
     => ( ( B2
          = ( F2 @ X2 ) )
       => ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_1023_Compr__image__eq,axiom,
    ! [F2: list_a > list_a,A4: set_list_a,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ ( image_list_a_list_a @ F2 @ A4 ) )
            & ( P @ X ) ) )
      = ( image_list_a_list_a @ F2
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( member_list_a @ X @ A4 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1024_Compr__image__eq,axiom,
    ! [F2: nat > list_a,A4: set_nat,P: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ ( image_nat_list_a @ F2 @ A4 ) )
            & ( P @ X ) ) )
      = ( image_nat_list_a @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1025_Compr__image__eq,axiom,
    ! [F2: list_a > nat,A4: set_list_a,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_list_a_nat @ F2 @ A4 ) )
            & ( P @ X ) ) )
      = ( image_list_a_nat @ F2
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( member_list_a @ X @ A4 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1026_Compr__image__eq,axiom,
    ! [F2: nat > nat,A4: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F2 @ A4 ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F2
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( P @ ( F2 @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_1027_fold__simps_I1_J,axiom,
    ! [F2: product_prod_nat_a > ( nat > a ) > nat > a,S: nat > a] :
      ( ( fold_P5280602285094830901_nat_a @ F2 @ nil_Pr1417316670369895453_nat_a @ S )
      = S ) ).

% fold_simps(1)
thf(fact_1028_infinite__surj,axiom,
    ! [A4: set_list_a,F2: list_a > list_a,B4: set_list_a] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( ord_le8861187494160871172list_a @ A4 @ ( image_list_a_list_a @ F2 @ B4 ) )
       => ~ ( finite_finite_list_a @ B4 ) ) ) ).

% infinite_surj
thf(fact_1029_infinite__surj,axiom,
    ! [A4: set_list_a,F2: nat > list_a,B4: set_nat] :
      ( ~ ( finite_finite_list_a @ A4 )
     => ( ( ord_le8861187494160871172list_a @ A4 @ ( image_nat_list_a @ F2 @ B4 ) )
       => ~ ( finite_finite_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_1030_infinite__surj,axiom,
    ! [A4: set_nat,F2: list_a > nat,B4: set_list_a] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( image_list_a_nat @ F2 @ B4 ) )
       => ~ ( finite_finite_list_a @ B4 ) ) ) ).

% infinite_surj
thf(fact_1031_infinite__surj,axiom,
    ! [A4: set_nat,F2: nat > nat,B4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( image_nat_nat @ F2 @ B4 ) )
       => ~ ( finite_finite_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_1032_all__finite__subset__image,axiom,
    ! [F2: list_a > list_a,A4: set_list_a,P: set_list_a > $o] :
      ( ( ! [B6: set_list_a] :
            ( ( ( finite_finite_list_a @ B6 )
              & ( ord_le8861187494160871172list_a @ B6 @ ( image_list_a_list_a @ F2 @ A4 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_list_a] :
            ( ( ( finite_finite_list_a @ B6 )
              & ( ord_le8861187494160871172list_a @ B6 @ A4 ) )
           => ( P @ ( image_list_a_list_a @ F2 @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1033_all__finite__subset__image,axiom,
    ! [F2: nat > list_a,A4: set_nat,P: set_list_a > $o] :
      ( ( ! [B6: set_list_a] :
            ( ( ( finite_finite_list_a @ B6 )
              & ( ord_le8861187494160871172list_a @ B6 @ ( image_nat_list_a @ F2 @ A4 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A4 ) )
           => ( P @ ( image_nat_list_a @ F2 @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1034_all__finite__subset__image,axiom,
    ! [F2: list_a > nat,A4: set_list_a,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_list_a_nat @ F2 @ A4 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_list_a] :
            ( ( ( finite_finite_list_a @ B6 )
              & ( ord_le8861187494160871172list_a @ B6 @ A4 ) )
           => ( P @ ( image_list_a_nat @ F2 @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1035_all__finite__subset__image,axiom,
    ! [F2: nat > nat,A4: set_nat,P: set_nat > $o] :
      ( ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A4 ) ) )
           => ( P @ B6 ) ) )
      = ( ! [B6: set_nat] :
            ( ( ( finite_finite_nat @ B6 )
              & ( ord_less_eq_set_nat @ B6 @ A4 ) )
           => ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_1036_ex__finite__subset__image,axiom,
    ! [F2: list_a > list_a,A4: set_list_a,P: set_list_a > $o] :
      ( ( ? [B6: set_list_a] :
            ( ( finite_finite_list_a @ B6 )
            & ( ord_le8861187494160871172list_a @ B6 @ ( image_list_a_list_a @ F2 @ A4 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_list_a] :
            ( ( finite_finite_list_a @ B6 )
            & ( ord_le8861187494160871172list_a @ B6 @ A4 )
            & ( P @ ( image_list_a_list_a @ F2 @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1037_ex__finite__subset__image,axiom,
    ! [F2: nat > list_a,A4: set_nat,P: set_list_a > $o] :
      ( ( ? [B6: set_list_a] :
            ( ( finite_finite_list_a @ B6 )
            & ( ord_le8861187494160871172list_a @ B6 @ ( image_nat_list_a @ F2 @ A4 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A4 )
            & ( P @ ( image_nat_list_a @ F2 @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1038_ex__finite__subset__image,axiom,
    ! [F2: list_a > nat,A4: set_list_a,P: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_list_a_nat @ F2 @ A4 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_list_a] :
            ( ( finite_finite_list_a @ B6 )
            & ( ord_le8861187494160871172list_a @ B6 @ A4 )
            & ( P @ ( image_list_a_nat @ F2 @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1039_ex__finite__subset__image,axiom,
    ! [F2: nat > nat,A4: set_nat,P: set_nat > $o] :
      ( ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ ( image_nat_nat @ F2 @ A4 ) )
            & ( P @ B6 ) ) )
      = ( ? [B6: set_nat] :
            ( ( finite_finite_nat @ B6 )
            & ( ord_less_eq_set_nat @ B6 @ A4 )
            & ( P @ ( image_nat_nat @ F2 @ B6 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_1040_finite__subset__image,axiom,
    ! [B4: set_list_a,F2: list_a > list_a,A4: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_list_a_list_a @ F2 @ A4 ) )
       => ? [C4: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C4 @ A4 )
            & ( finite_finite_list_a @ C4 )
            & ( B4
              = ( image_list_a_list_a @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1041_finite__subset__image,axiom,
    ! [B4: set_list_a,F2: nat > list_a,A4: set_nat] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_nat_list_a @ F2 @ A4 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A4 )
            & ( finite_finite_nat @ C4 )
            & ( B4
              = ( image_nat_list_a @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1042_finite__subset__image,axiom,
    ! [B4: set_nat,F2: list_a > nat,A4: set_list_a] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_list_a_nat @ F2 @ A4 ) )
       => ? [C4: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C4 @ A4 )
            & ( finite_finite_list_a @ C4 )
            & ( B4
              = ( image_list_a_nat @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1043_finite__subset__image,axiom,
    ! [B4: set_nat,F2: nat > nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A4 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A4 )
            & ( finite_finite_nat @ C4 )
            & ( B4
              = ( image_nat_nat @ F2 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_1044_finite__surj,axiom,
    ! [A4: set_list_a,B4: set_list_a,F2: list_a > list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_list_a_list_a @ F2 @ A4 ) )
       => ( finite_finite_list_a @ B4 ) ) ) ).

% finite_surj
thf(fact_1045_finite__surj,axiom,
    ! [A4: set_list_a,B4: set_nat,F2: list_a > nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_list_a_nat @ F2 @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_1046_finite__surj,axiom,
    ! [A4: set_nat,B4: set_list_a,F2: nat > list_a] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_nat_list_a @ F2 @ A4 ) )
       => ( finite_finite_list_a @ B4 ) ) ) ).

% finite_surj
thf(fact_1047_finite__surj,axiom,
    ! [A4: set_nat,B4: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F2 @ A4 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_1048_Exists__in__sub__foldr1__Disj,axiom,
    ! [X2: nat,Q5: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
     => ( ( ( Qs = nil_Re6358386334527539737la_a_b )
          & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ Q ) ) )
        | ? [X4: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
            & ( member4680049679412964150la_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q5 ) @ ( relational_sub_a_b @ X4 ) ) ) ) ) ).

% Exists_in_sub_foldr1_Disj
thf(fact_1049_sr__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,X3: set_nat,Q: relational_fmla_a_b] :
      ( ! [X4: relational_fmla_a_b] :
          ( ( member4680049679412964150la_a_b @ X4 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
         => ( ( relational_fv_a_b @ X4 )
            = X3 ) )
     => ( ( relational_sr_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
        = ( ( ( Qs = nil_Re6358386334527539737la_a_b )
           => ( relational_sr_a_b @ Q ) )
          & ( ( Qs != nil_Re6358386334527539737la_a_b )
           => ! [X: relational_fmla_a_b] :
                ( ( member4680049679412964150la_a_b @ X @ ( set_Re9104216502384355786la_a_b @ Qs ) )
               => ( relational_sr_a_b @ X ) ) ) ) ) ) ).

% sr_foldr1_Disj
thf(fact_1050_gen_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ A1 @ A22 @ A32 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A32 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A32
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
               => ~ ( relational_gen_a_b @ A1 @ Q4 @ A32 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                   => ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
               => ( ! [Q13: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ! [G22: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) )
                         => ( ( relational_gen_a_b @ A1 @ Q13 @ G1 )
                           => ~ ( relational_gen_a_b @ A1 @ Q24 @ G22 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                       => ( ( A32 = G4 )
                         => ~ ( ( relational_gen_a_b @ A1 @ Q13 @ G4 )
                              | ( relational_gen_a_b @ A1 @ Q24 @ G4 ) ) ) )
                   => ( ! [Y3: nat,Q4: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G4: set_Re381260168593705685la_a_b] :
                              ( ( A32
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                  @ G4 ) )
                             => ~ ( relational_gen_a_b @ Y3 @ Q4 @ G4 ) ) )
                     => ( ! [Y3: nat,Q4: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G4: set_Re381260168593705685la_a_b] :
                                ( ( A32
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y3 @ A1 ) )
                                    @ G4 ) )
                               => ~ ( relational_gen_a_b @ Y3 @ Q4 @ G4 ) ) )
                       => ~ ! [Y3: nat,Q4: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q4 ) )
                             => ! [G4: set_Re381260168593705685la_a_b] :
                                  ( ( A32
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b @ A1 @ Q4 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen.cases
thf(fact_1051_gen_Osimps,axiom,
    ( relational_gen_a_b
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A33 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A33
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relational_gen_a_b @ A12 @ Q2 @ A33 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ? [G23: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
                  & ( relational_gen_a_b @ A12 @ Q12 @ G12 )
                  & ( relational_gen_a_b @ A12 @ Q23 @ G23 ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relational_gen_a_b @ A12 @ Q12 @ A33 )
                | ( relational_gen_a_b @ A12 @ Q23 @ A33 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y ) ) ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A12 ) )
                      @ G5 ) )
                  & ( relational_gen_a_b @ Y @ Q2 @ G5 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y @ A12 ) )
                      @ G5 ) )
                  & ( relational_gen_a_b @ Y @ Q2 @ G5 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y @ Q2 ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G5 ) )
                  & ( A12 != Y )
                  & ( relational_gen_a_b @ A12 @ Q2 @ G5 ) ) ) ) ) ) ).

% gen.simps
thf(fact_1052_insert__iff,axiom,
    ! [A3: nat,B2: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ ( insert_nat @ B2 @ A4 ) )
      = ( ( A3 = B2 )
        | ( member_nat @ A3 @ A4 ) ) ) ).

% insert_iff
thf(fact_1053_insertCI,axiom,
    ! [A3: nat,B4: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A3 @ B4 )
       => ( A3 = B2 ) )
     => ( member_nat @ A3 @ ( insert_nat @ B2 @ B4 ) ) ) ).

% insertCI
thf(fact_1054_Un__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        | ( member_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1055_UnCI,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ A4 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1056_singletonI,axiom,
    ! [A3: nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_1057_finite__insert,axiom,
    ! [A3: list_a,A4: set_list_a] :
      ( ( finite_finite_list_a @ ( insert_list_a @ A3 @ A4 ) )
      = ( finite_finite_list_a @ A4 ) ) ).

% finite_insert
thf(fact_1058_finite__insert,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) )
      = ( finite_finite_nat @ A4 ) ) ).

% finite_insert
thf(fact_1059_insert__subset,axiom,
    ! [X2: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A4 ) @ B4 )
      = ( ( member_nat @ X2 @ B4 )
        & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_1060_finite__Un,axiom,
    ! [F4: set_list_a,G3: set_list_a] :
      ( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F4 @ G3 ) )
      = ( ( finite_finite_list_a @ F4 )
        & ( finite_finite_list_a @ G3 ) ) ) ).

% finite_Un
thf(fact_1061_finite__Un,axiom,
    ! [F4: set_nat,G3: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) )
      = ( ( finite_finite_nat @ F4 )
        & ( finite_finite_nat @ G3 ) ) ) ).

% finite_Un
thf(fact_1062_singleton__conv2,axiom,
    ! [A3: list_a] :
      ( ( collect_list_a
        @ ( ^ [Y5: list_a,Z4: list_a] : ( Y5 = Z4 )
          @ A3 ) )
      = ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ).

% singleton_conv2
thf(fact_1063_singleton__conv2,axiom,
    ! [A3: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
          @ A3 ) )
      = ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_1064_singleton__conv,axiom,
    ! [A3: list_a] :
      ( ( collect_list_a
        @ ^ [X: list_a] : ( X = A3 ) )
      = ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ).

% singleton_conv
thf(fact_1065_singleton__conv,axiom,
    ! [A3: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( X = A3 ) )
      = ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_1066_pair__imageI,axiom,
    ! [A3: nat,B2: a,A4: set_Pr4193341848836149977_nat_a,F2: nat > a > ( nat > a ) > nat > a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A3 @ B2 ) @ A4 )
     => ( member_nat_a_nat_a @ ( F2 @ A3 @ B2 ) @ ( image_8641191218422233679_nat_a @ ( produc2909000522608705447_nat_a @ F2 ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_1067_sat__foldr__Disj,axiom,
    ! [Xs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma: nat > a] :
      ( ( relational_sat_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Xs @ Q ) @ I @ Sigma )
      = ( ? [X: relational_fmla_a_b] :
            ( ( member4680049679412964150la_a_b @ X @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Xs ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
            & ( relational_sat_a_b @ X @ I @ Sigma ) ) ) ) ).

% sat_foldr_Disj
thf(fact_1068_singletonD,axiom,
    ! [B2: nat,A3: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
     => ( B2 = A3 ) ) ).

% singletonD
thf(fact_1069_singleton__iff,axiom,
    ! [B2: nat,A3: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
      = ( B2 = A3 ) ) ).

% singleton_iff
thf(fact_1070_Collect__conv__if,axiom,
    ! [P: list_a > $o,A3: list_a] :
      ( ( ( P @ A3 )
       => ( ( collect_list_a
            @ ^ [X: list_a] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_list_a
            @ ^ [X: list_a] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_list_a ) ) ) ).

% Collect_conv_if
thf(fact_1071_Collect__conv__if,axiom,
    ! [P: nat > $o,A3: nat] :
      ( ( ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_1072_Collect__conv__if2,axiom,
    ! [P: list_a > $o,A3: list_a] :
      ( ( ( P @ A3 )
       => ( ( collect_list_a
            @ ^ [X: list_a] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_list_a
            @ ^ [X: list_a] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_list_a ) ) ) ).

% Collect_conv_if2
thf(fact_1073_Collect__conv__if2,axiom,
    ! [P: nat > $o,A3: nat] :
      ( ( ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_1074_subset__insert,axiom,
    ! [X2: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X2 @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ B4 ) )
        = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_1075_mk__disjoint__insert,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ? [B7: set_nat] :
          ( ( A4
            = ( insert_nat @ A3 @ B7 ) )
          & ~ ( member_nat @ A3 @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_1076_Collect__disj__eq,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( collect_list_a
        @ ^ [X: list_a] :
            ( ( P @ X )
            | ( Q @ X ) ) )
      = ( sup_sup_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_1077_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_1078_insert__Collect,axiom,
    ! [A3: list_a,P: list_a > $o] :
      ( ( insert_list_a @ A3 @ ( collect_list_a @ P ) )
      = ( collect_list_a
        @ ^ [U2: list_a] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1079_insert__Collect,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( insert_nat @ A3 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_1080_insert__eq__iff,axiom,
    ! [A3: nat,A4: set_nat,B2: nat,B4: set_nat] :
      ( ~ ( member_nat @ A3 @ A4 )
     => ( ~ ( member_nat @ B2 @ B4 )
       => ( ( ( insert_nat @ A3 @ A4 )
            = ( insert_nat @ B2 @ B4 ) )
          = ( ( ( A3 = B2 )
             => ( A4 = B4 ) )
            & ( ( A3 != B2 )
             => ? [C5: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B2 @ C5 ) )
                  & ~ ( member_nat @ B2 @ C5 )
                  & ( B4
                    = ( insert_nat @ A3 @ C5 ) )
                  & ~ ( member_nat @ A3 @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1081_insert__absorb,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ( ( insert_nat @ A3 @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_1082_insert__ident,axiom,
    ! [X2: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X2 @ A4 )
     => ( ~ ( member_nat @ X2 @ B4 )
       => ( ( ( insert_nat @ X2 @ A4 )
            = ( insert_nat @ X2 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_1083_insert__compr,axiom,
    ( insert_list_a
    = ( ^ [A: list_a,B6: set_list_a] :
          ( collect_list_a
          @ ^ [X: list_a] :
              ( ( X = A )
              | ( member_list_a @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_1084_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A: nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_1085_Set_Oset__insert,axiom,
    ! [X2: nat,A4: set_nat] :
      ( ( member_nat @ X2 @ A4 )
     => ~ ! [B7: set_nat] :
            ( ( A4
              = ( insert_nat @ X2 @ B7 ) )
           => ( member_nat @ X2 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_1086_insert__def,axiom,
    ( insert_list_a
    = ( ^ [A: list_a] :
          ( sup_sup_set_list_a
          @ ( collect_list_a
            @ ^ [X: list_a] : ( X = A ) ) ) ) ) ).

% insert_def
thf(fact_1087_insert__def,axiom,
    ( insert_nat
    = ( ^ [A: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( X = A ) ) ) ) ) ).

% insert_def
thf(fact_1088_insertI2,axiom,
    ! [A3: nat,B4: set_nat,B2: nat] :
      ( ( member_nat @ A3 @ B4 )
     => ( member_nat @ A3 @ ( insert_nat @ B2 @ B4 ) ) ) ).

% insertI2
thf(fact_1089_insertI1,axiom,
    ! [A3: nat,B4: set_nat] : ( member_nat @ A3 @ ( insert_nat @ A3 @ B4 ) ) ).

% insertI1
thf(fact_1090_insertE,axiom,
    ! [A3: nat,B2: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ ( insert_nat @ B2 @ A4 ) )
     => ( ( A3 != B2 )
       => ( member_nat @ A3 @ A4 ) ) ) ).

% insertE
thf(fact_1091_Un__def,axiom,
    ( sup_sup_set_list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( collect_list_a
          @ ^ [X: list_a] :
              ( ( member_list_a @ X @ A6 )
              | ( member_list_a @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1092_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              | ( member_nat @ X @ B6 ) ) ) ) ) ).

% Un_def
thf(fact_1093_UnI2,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ C @ B4 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1094_UnI1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1095_UnE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ~ ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% UnE
thf(fact_1096_infinite__Un,axiom,
    ! [S3: set_list_a,T2: set_list_a] :
      ( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S3 @ T2 ) ) )
      = ( ~ ( finite_finite_list_a @ S3 )
        | ~ ( finite_finite_list_a @ T2 ) ) ) ).

% infinite_Un
thf(fact_1097_infinite__Un,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T2 ) ) )
      = ( ~ ( finite_finite_nat @ S3 )
        | ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_1098_Un__infinite,axiom,
    ! [S3: set_list_a,T2: set_list_a] :
      ( ~ ( finite_finite_list_a @ S3 )
     => ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S3 @ T2 ) ) ) ).

% Un_infinite
thf(fact_1099_Un__infinite,axiom,
    ! [S3: set_nat,T2: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S3 @ T2 ) ) ) ).

% Un_infinite
thf(fact_1100_finite__UnI,axiom,
    ! [F4: set_list_a,G3: set_list_a] :
      ( ( finite_finite_list_a @ F4 )
     => ( ( finite_finite_list_a @ G3 )
       => ( finite_finite_list_a @ ( sup_sup_set_list_a @ F4 @ G3 ) ) ) ) ).

% finite_UnI
thf(fact_1101_finite__UnI,axiom,
    ! [F4: set_nat,G3: set_nat] :
      ( ( finite_finite_nat @ F4 )
     => ( ( finite_finite_nat @ G3 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) ) ) ) ).

% finite_UnI
thf(fact_1102_finite_OinsertI,axiom,
    ! [A4: set_list_a,A3: list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite_finite_list_a @ ( insert_list_a @ A3 @ A4 ) ) ) ).

% finite.insertI
thf(fact_1103_finite_OinsertI,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( insert_nat @ A3 @ A4 ) ) ) ).

% finite.insertI
thf(fact_1104_sub_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) ).

% sub.simps(6)
thf(fact_1105_insert__subsetI,axiom,
    ! [X2: nat,A4: set_nat,X3: set_nat] :
      ( ( member_nat @ X2 @ A4 )
     => ( ( ord_less_eq_set_nat @ X3 @ A4 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X3 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_1106_finite_Ocases,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( A3 != bot_bot_set_list_a )
       => ~ ! [A7: set_list_a] :
              ( ? [A2: list_a] :
                  ( A3
                  = ( insert_list_a @ A2 @ A7 ) )
             => ~ ( finite_finite_list_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1107_finite_Ocases,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A2: nat] :
                  ( A3
                  = ( insert_nat @ A2 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_1108_finite_Osimps,axiom,
    ( finite_finite_list_a
    = ( ^ [A: set_list_a] :
          ( ( A = bot_bot_set_list_a )
          | ? [A6: set_list_a,B3: list_a] :
              ( ( A
                = ( insert_list_a @ B3 @ A6 ) )
              & ( finite_finite_list_a @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1109_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A: set_nat] :
          ( ( A = bot_bot_set_nat )
          | ? [A6: set_nat,B3: nat] :
              ( ( A
                = ( insert_nat @ B3 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_1110_finite__induct,axiom,
    ! [F4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F4 )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,F5: set_list_a] :
              ( ( finite_finite_list_a @ F5 )
             => ( ~ ( member_list_a @ X4 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_list_a @ X4 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_1111_finite__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F5: set_nat] :
              ( ( finite_finite_nat @ F5 )
             => ( ~ ( member_nat @ X4 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_nat @ X4 @ F5 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_1112_finite__ne__induct,axiom,
    ! [F4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F4 )
     => ( ( F4 != bot_bot_set_list_a )
       => ( ! [X4: list_a] : ( P @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
         => ( ! [X4: list_a,F5: set_list_a] :
                ( ( finite_finite_list_a @ F5 )
               => ( ( F5 != bot_bot_set_list_a )
                 => ( ~ ( member_list_a @ X4 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_list_a @ X4 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1113_finite__ne__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( F5 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ X4 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_1114_infinite__finite__induct,axiom,
    ! [P: set_list_a > $o,A4: set_list_a] :
      ( ! [A7: set_list_a] :
          ( ~ ( finite_finite_list_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,F5: set_list_a] :
              ( ( finite_finite_list_a @ F5 )
             => ( ~ ( member_list_a @ X4 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_list_a @ X4 @ F5 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1115_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A4: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F5: set_nat] :
              ( ( finite_finite_nat @ F5 )
             => ( ~ ( member_nat @ X4 @ F5 )
               => ( ( P @ F5 )
                 => ( P @ ( insert_nat @ X4 @ F5 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_1116_gen_Ointros_I6_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G24: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X2 @ Q1 @ G13 )
     => ( ( relational_gen_a_b @ X2 @ Q22 @ G24 )
       => ( relational_gen_a_b @ X2 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) ) ) ) ).

% gen.intros(6)
thf(fact_1117_finite__subset__induct,axiom,
    ! [F4: set_list_a,A4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F4 )
     => ( ( ord_le8861187494160871172list_a @ F4 @ A4 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A2: list_a,F5: set_list_a] :
                ( ( finite_finite_list_a @ F5 )
               => ( ( member_list_a @ A2 @ A4 )
                 => ( ~ ( member_list_a @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_list_a @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1118_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A2: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A2 @ A4 )
                 => ( ~ ( member_nat @ A2 @ F5 )
                   => ( ( P @ F5 )
                     => ( P @ ( insert_nat @ A2 @ F5 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_1119_finite__subset__induct_H,axiom,
    ! [F4: set_list_a,A4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F4 )
     => ( ( ord_le8861187494160871172list_a @ F4 @ A4 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A2: list_a,F5: set_list_a] :
                ( ( finite_finite_list_a @ F5 )
               => ( ( member_list_a @ A2 @ A4 )
                 => ( ( ord_le8861187494160871172list_a @ F5 @ A4 )
                   => ( ~ ( member_list_a @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_list_a @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1120_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A2: nat,F5: set_nat] :
                ( ( finite_finite_nat @ F5 )
               => ( ( member_nat @ A2 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F5 @ A4 )
                   => ( ~ ( member_nat @ A2 @ F5 )
                     => ( ( P @ F5 )
                       => ( P @ ( insert_nat @ A2 @ F5 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_1121_fv__cp__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,A4: set_nat] :
      ( ! [X4: relational_fmla_a_b] :
          ( ( member4680049679412964150la_a_b @ X4 @ ( sup_su5130108678486352897la_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ( relati1591879772219623554ed_a_b @ X4 )
            & ( ( relational_fv_a_b @ X4 )
              = A4 ) ) )
     => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
        = A4 ) ) ).

% fv_cp_foldr_Disj
thf(fact_1122_qp__gen,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X2 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% qp_gen
thf(fact_1123_gen_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X2 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen.intros(2)
thf(fact_1124_sub_Oelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X2 )
        = Y2 )
     => ( ! [T5: $o] :
            ( ( X2
              = ( relational_Bool_a_b @ T5 ) )
           => ( Y2
             != ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T5 ) @ bot_bo4495933725496725865la_a_b ) ) )
       => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ P3 @ Ts ) )
             => ( Y2
               != ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ! [X4: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T5 ) )
               => ( Y2
                 != ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) @ bot_bo4495933725496725865la_a_b ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y2
                   != ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y2
                     != ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y2
                       != ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) ) )
                 => ~ ! [Z3: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) )
                       => ( Y2
                         != ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% sub.elims
thf(fact_1125_nongens__def,axiom,
    ( relati62690040636126068ns_a_b
    = ( ^ [Q2: relational_fmla_a_b] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ ( relational_fv_a_b @ Q2 ) )
              & ~ ? [X6: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X6 ) ) ) ) ) ).

% nongens_def
thf(fact_1126_gen__induct,axiom,
    ! [X12: nat,X23: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
      ( ( relational_gen_a_b @ X12 @ X23 @ X33 )
     => ( ! [X4: nat] : ( P @ X4 @ ( relational_Bool_a_b @ $false ) @ bot_bo4495933725496725865la_a_b )
       => ( ! [Q4: relational_fmla_a_b] :
              ( ( relational_ap_a_b @ Q4 )
             => ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                 => ( P @ X4 @ Q4 @ ( insert7010464514620295119la_a_b @ Q4 @ bot_bo4495933725496725865la_a_b ) ) ) )
         => ( ! [X4: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                ( ( relational_gen_a_b @ X4 @ Q4 @ G4 )
               => ( ( P @ X4 @ Q4 @ G4 )
                 => ( P @ X4 @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) @ G4 ) ) )
           => ( ! [X4: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                  ( ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                 => ( ( P @ X4 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                   => ( P @ X4 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
             => ( ! [X4: nat,Q13: relational_fmla_a_b,Q24: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                    ( ( relational_gen_a_b @ X4 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                   => ( ( P @ X4 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ G4 )
                     => ( P @ X4 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) @ G4 ) ) )
               => ( ! [X4: nat,Q13: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b] :
                      ( ( relational_gen_a_b @ X4 @ Q13 @ G1 )
                     => ( ( P @ X4 @ Q13 @ G1 )
                       => ! [Q24: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
                            ( ( relational_gen_a_b @ X4 @ Q24 @ G22 )
                           => ( ( P @ X4 @ Q24 @ G22 )
                             => ( P @ X4 @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) ) ) ) ) )
                 => ( ! [X4: nat,Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( ( ( relational_gen_a_b @ X4 @ Q13 @ G4 )
                            & ( P @ X4 @ Q13 @ G4 ) )
                          | ( ( relational_gen_a_b @ X4 @ Q24 @ G4 )
                            & ( P @ X4 @ Q24 @ G4 ) ) )
                       => ( P @ X4 @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ G4 ) )
                   => ( ! [Y3: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                          ( ( relational_gen_a_b @ Y3 @ Q4 @ G4 )
                         => ( ( P @ Y3 @ Q4 @ G4 )
                           => ! [X4: nat] :
                                ( P @ X4 @ ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y3 ) ) )
                                @ ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ X4 )
                                  @ G4 ) ) ) )
                     => ( ! [Y3: nat,Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                            ( ( relational_gen_a_b @ Y3 @ Q4 @ G4 )
                           => ( ( P @ Y3 @ Q4 @ G4 )
                             => ! [X4: nat] :
                                  ( P @ X4 @ ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ X4 ) ) )
                                  @ ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ X4 )
                                    @ G4 ) ) ) )
                       => ( ! [X4: nat,Y3: nat] :
                              ( ( X4 != Y3 )
                             => ! [Q4: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                                  ( ( relational_gen_a_b @ X4 @ Q4 @ G4 )
                                 => ( ( P @ X4 @ Q4 @ G4 )
                                   => ( P @ X4 @ ( relati591517084277583526ts_a_b @ Y3 @ Q4 ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) ) ) ) )
                         => ( P @ X12 @ X23 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen_induct
thf(fact_1127_sub_Opelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X2 )
        = Y2 )
     => ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X2 )
       => ( ! [T5: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ T5 ) )
             => ( ( Y2
                  = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T5 ) @ bot_bo4495933725496725865la_a_b ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T5 ) ) ) )
         => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ P3 @ Ts ) )
               => ( ( Y2
                    = ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) @ bot_bo4495933725496725865la_a_b ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
           => ( ! [X4: nat,T5: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T5 ) )
                 => ( ( Y2
                      = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) @ bot_bo4495933725496725865la_a_b ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y2
                        = ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y2
                          = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y2
                            = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q13 ) @ ( relational_sub_a_b @ Q24 ) ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
                   => ~ ! [Z3: nat,Q4: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) )
                         => ( ( Y2
                              = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sub.pelims
thf(fact_1128_sup_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
      = ( ( ord_less_eq_nat @ B2 @ A3 )
        & ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% sup.bounded_iff
thf(fact_1129_le__sup__iff,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_eq_nat @ X2 @ Z )
        & ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).

% le_sup_iff
thf(fact_1130_sup__Un__eq,axiom,
    ! [R2: set_nat,S3: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R2 )
        @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_1131_sup__set__def,axiom,
    ( sup_sup_set_list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( collect_list_a
          @ ( sup_sup_list_a_o
            @ ^ [X: list_a] : ( member_list_a @ X @ A6 )
            @ ^ [X: list_a] : ( member_list_a @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1132_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A6 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% sup_set_def
thf(fact_1133_fv_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).

% fv.simps(6)
thf(fact_1134_fv_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi ) ) ) ).

% fv.simps(5)
thf(fact_1135_inf__sup__ord_I4_J,axiom,
    ! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).

% inf_sup_ord(4)
thf(fact_1136_inf__sup__ord_I3_J,axiom,
    ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).

% inf_sup_ord(3)
thf(fact_1137_le__supE,axiom,
    ! [A3: nat,B2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X2 )
     => ~ ( ( ord_less_eq_nat @ A3 @ X2 )
         => ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).

% le_supE
thf(fact_1138_le__supI,axiom,
    ! [A3: nat,X2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ X2 )
     => ( ( ord_less_eq_nat @ B2 @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ X2 ) ) ) ).

% le_supI
thf(fact_1139_sup__ge1,axiom,
    ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).

% sup_ge1
thf(fact_1140_sup__ge2,axiom,
    ! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).

% sup_ge2
thf(fact_1141_le__supI1,axiom,
    ! [X2: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ X2 @ A3 )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% le_supI1
thf(fact_1142_le__supI2,axiom,
    ! [X2: nat,B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ X2 @ B2 )
     => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% le_supI2
thf(fact_1143_sup_Omono,axiom,
    ! [C: nat,A3: nat,D: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ( ord_less_eq_nat @ D @ B2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A3 @ B2 ) ) ) ) ).

% sup.mono
thf(fact_1144_sup__mono,axiom,
    ! [A3: nat,C: nat,B2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B2 @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A3 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_1145_sup__least,axiom,
    ! [Y2: nat,X2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ( ord_less_eq_nat @ Z @ X2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X2 ) ) ) ).

% sup_least
thf(fact_1146_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( sup_sup_nat @ X @ Y )
          = Y ) ) ) ).

% le_iff_sup
thf(fact_1147_sup_OorderE,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( A3
        = ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.orderE
thf(fact_1148_sup_OorderI,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3
        = ( sup_sup_nat @ A3 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A3 ) ) ).

% sup.orderI
thf(fact_1149_sup__unique,axiom,
    ! [F2: nat > nat > nat,X2: nat,Y2: nat] :
      ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ X4 @ ( F2 @ X4 @ Y3 ) )
     => ( ! [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X4 @ Y3 ) )
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X4 )
             => ( ( ord_less_eq_nat @ Z3 @ X4 )
               => ( ord_less_eq_nat @ ( F2 @ Y3 @ Z3 ) @ X4 ) ) )
         => ( ( sup_sup_nat @ X2 @ Y2 )
            = ( F2 @ X2 @ Y2 ) ) ) ) ) ).

% sup_unique
thf(fact_1150_sup_Oabsorb1,axiom,
    ! [B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( sup_sup_nat @ A3 @ B2 )
        = A3 ) ) ).

% sup.absorb1
thf(fact_1151_sup_Oabsorb2,axiom,
    ! [A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B2 )
     => ( ( sup_sup_nat @ A3 @ B2 )
        = B2 ) ) ).

% sup.absorb2
thf(fact_1152_sup__absorb1,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X2 )
     => ( ( sup_sup_nat @ X2 @ Y2 )
        = X2 ) ) ).

% sup_absorb1
thf(fact_1153_sup__absorb2,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y2 )
     => ( ( sup_sup_nat @ X2 @ Y2 )
        = Y2 ) ) ).

% sup_absorb2
thf(fact_1154_sup_OboundedE,axiom,
    ! [B2: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A3 )
         => ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% sup.boundedE
thf(fact_1155_sup_OboundedI,axiom,
    ! [B2: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A3 )
     => ( ( ord_less_eq_nat @ C @ A3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A3 ) ) ) ).

% sup.boundedI
thf(fact_1156_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A: nat] :
          ( A
          = ( sup_sup_nat @ A @ B3 ) ) ) ) ).

% sup.order_iff
thf(fact_1157_sup_Ocobounded1,axiom,
    ! [A3: nat,B2: nat] : ( ord_less_eq_nat @ A3 @ ( sup_sup_nat @ A3 @ B2 ) ) ).

% sup.cobounded1
thf(fact_1158_sup_Ocobounded2,axiom,
    ! [B2: nat,A3: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A3 @ B2 ) ) ).

% sup.cobounded2
thf(fact_1159_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A: nat] :
          ( ( sup_sup_nat @ A @ B3 )
          = A ) ) ) ).

% sup.absorb_iff1
thf(fact_1160_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A: nat,B3: nat] :
          ( ( sup_sup_nat @ A @ B3 )
          = B3 ) ) ) ).

% sup.absorb_iff2
thf(fact_1161_sup_OcoboundedI1,axiom,
    ! [C: nat,A3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.coboundedI1
thf(fact_1162_sup_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ C @ B2 )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A3 @ B2 ) ) ) ).

% sup.coboundedI2
thf(fact_1163_gen_H_Osimps,axiom,
    ( relational_gen_a_b2
    = ( ^ [A12: nat,A23: relational_fmla_a_b,A33: set_Re381260168593705685la_a_b] :
          ( ( ( A23
              = ( relational_Bool_a_b @ $false ) )
            & ( A33 = bot_bo4495933725496725865la_a_b ) )
          | ( ( A33
              = ( insert7010464514620295119la_a_b @ A23 @ bot_bo4495933725496725865la_a_b ) )
            & ( relational_ap_a_b @ A23 )
            & ( member_nat @ A12 @ ( relational_fv_a_b @ A23 ) ) )
          | ? [Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q2 ) ) )
              & ( relational_gen_a_b2 @ A12 @ Q2 @ A33 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q12 @ Q23 ) )
              & ? [G23: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G12 @ G23 ) )
                  & ( relational_gen_a_b2 @ A12 @ Q12 @ G12 )
                  & ( relational_gen_a_b2 @ A12 @ Q23 @ G23 ) ) )
          | ? [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q12 @ Q23 ) )
              & ( ( relational_gen_a_b2 @ A12 @ Q12 @ A33 )
                | ( relational_gen_a_b2 @ A12 @ Q23 @ A33 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y ) ) ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ A12 )
                      @ G5 ) )
                  & ( relational_gen_a_b2 @ Y @ Q2 @ G5 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q2 @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y @ A12 )
                      @ G5 ) )
                  & ( relational_gen_a_b2 @ Y @ Q2 @ G5 ) ) )
          | ? [Y: nat,Q2: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y @ Q2 ) )
              & ? [G5: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G5 ) )
                  & ( A12 != Y )
                  & ( relational_gen_a_b2 @ A12 @ Q2 @ G5 ) ) ) ) ) ) ).

% gen'.simps
thf(fact_1164_gen_H_Ocases,axiom,
    ! [A1: nat,A22: relational_fmla_a_b,A32: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ A1 @ A22 @ A32 )
     => ( ( ( A22
            = ( relational_Bool_a_b @ $false ) )
         => ( A32 != bot_bo4495933725496725865la_a_b ) )
       => ( ( ( A32
              = ( insert7010464514620295119la_a_b @ A22 @ bot_bo4495933725496725865la_a_b ) )
           => ( ( relational_ap_a_b @ A22 )
             => ~ ( member_nat @ A1 @ ( relational_fv_a_b @ A22 ) ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( A22
                  = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
               => ~ ( relational_gen_a_b2 @ A1 @ Q4 @ A32 ) )
           => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) )
                 => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) )
                   => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q24 ) ) @ A32 ) )
               => ( ! [Q13: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ! [G22: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G1 @ G22 ) )
                         => ( ( relational_gen_a_b2 @ A1 @ Q13 @ G1 )
                           => ~ ( relational_gen_a_b2 @ A1 @ Q24 @ G22 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q24: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                       => ( ( A32 = G4 )
                         => ~ ( ( relational_gen_a_b2 @ A1 @ Q13 @ G4 )
                              | ( relational_gen_a_b2 @ A1 @ Q24 @ G4 ) ) ) )
                   => ( ! [Y3: nat,Q4: relational_fmla_a_b] :
                          ( ( A22
                            = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ A1 @ ( relational_Var_a @ Y3 ) ) ) )
                         => ! [G4: set_Re381260168593705685la_a_b] :
                              ( ( A32
                                = ( image_6790371041703824709la_a_b
                                  @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                  @ G4 ) )
                             => ~ ( relational_gen_a_b2 @ Y3 @ Q4 @ G4 ) ) )
                     => ( ! [Y3: nat,Q4: relational_fmla_a_b] :
                            ( ( A22
                              = ( relational_Conj_a_b @ Q4 @ ( relational_Eq_a_b @ Y3 @ ( relational_Var_a @ A1 ) ) ) )
                           => ! [G4: set_Re381260168593705685la_a_b] :
                                ( ( A32
                                  = ( image_6790371041703824709la_a_b
                                    @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y3 @ A1 )
                                    @ G4 ) )
                               => ~ ( relational_gen_a_b2 @ Y3 @ Q4 @ G4 ) ) )
                       => ~ ! [Y3: nat,Q4: relational_fmla_a_b] :
                              ( ( A22
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q4 ) )
                             => ! [G4: set_Re381260168593705685la_a_b] :
                                  ( ( A32
                                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y3 ) @ G4 ) )
                                 => ( ( A1 != Y3 )
                                   => ~ ( relational_gen_a_b2 @ A1 @ Q4 @ G4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen'.cases
thf(fact_1165_gen_H_Ointros_I6_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b,G24: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X2 @ Q1 @ G13 )
     => ( ( relational_gen_a_b2 @ X2 @ Q22 @ G24 )
       => ( relational_gen_a_b2 @ X2 @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G13 @ G24 ) ) ) ) ).

% gen'.intros(6)
thf(fact_1166_gen_H_Ointros_I4_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X2 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G3 )
     => ( relational_gen_a_b2 @ X2 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ G3 ) ) ).

% gen'.intros(4)
thf(fact_1167_gen_H_Ointros_I5_J,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,G3: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X2 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G3 )
     => ( relational_gen_a_b2 @ X2 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q22 ) ) @ G3 ) ) ).

% gen'.intros(5)
thf(fact_1168_gen_H_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X2: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b2 @ X2 @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen'.intros(2)
thf(fact_1169_finite__ranking__induct,axiom,
    ! [S3: set_list_a,P: set_list_a > $o,F2: list_a > nat] :
      ( ( finite_finite_list_a @ S3 )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,S4: set_list_a] :
              ( ( finite_finite_list_a @ S4 )
             => ( ! [Y6: list_a] :
                    ( ( member_list_a @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_list_a @ X4 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1170_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X4 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_1171_fv_Osimps_I1_J,axiom,
    ! [Uu3: b,Ts2: list_R6823256787227418703term_a] :
      ( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu3 @ Ts2 ) )
      = ( relati4569515538964159125_set_a @ Ts2 ) ) ).

% fv.simps(1)
thf(fact_1172_fv__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( ( Qs = nil_Re6358386334527539737la_a_b )
       => ( ( relational_fv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
          = ( relational_fv_a_b @ Q ) ) )
      & ( ( Qs != nil_Re6358386334527539737la_a_b )
       => ( ( relational_fv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
          = ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ relational_fv_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) ) ) ) ) ) ).

% fv_foldr1_Disj
thf(fact_1173_finite__UN,axiom,
    ! [A4: set_list_a,B4: list_a > set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B4 @ A4 ) ) )
        = ( ! [X: list_a] :
              ( ( member_list_a @ X @ A4 )
             => ( finite_finite_list_a @ ( B4 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1174_finite__UN,axiom,
    ! [A4: set_list_a,B4: list_a > set_nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B4 @ A4 ) ) )
        = ( ! [X: list_a] :
              ( ( member_list_a @ X @ A4 )
             => ( finite_finite_nat @ ( B4 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1175_finite__UN,axiom,
    ! [A4: set_nat,B4: nat > set_list_a] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B4 @ A4 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A4 )
             => ( finite_finite_list_a @ ( B4 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1176_finite__UN,axiom,
    ! [A4: set_nat,B4: nat > set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A4 ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A4 )
             => ( finite_finite_nat @ ( B4 @ X ) ) ) ) ) ) ).

% finite_UN
thf(fact_1177_finite__Union,axiom,
    ! [A4: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A4 )
     => ( ! [M5: set_list_a] :
            ( ( member_set_list_a @ M5 @ A4 )
           => ( finite_finite_list_a @ M5 ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A4 ) ) ) ) ).

% finite_Union
thf(fact_1178_finite__Union,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ! [M5: set_nat] :
            ( ( member_set_nat @ M5 @ A4 )
           => ( finite_finite_nat @ M5 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A4 ) ) ) ) ).

% finite_Union
thf(fact_1179_finite__UN__I,axiom,
    ! [A4: set_list_a,B4: list_a > set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ! [A2: list_a] :
            ( ( member_list_a @ A2 @ A4 )
           => ( finite_finite_list_a @ ( B4 @ A2 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B4 @ A4 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1180_finite__UN__I,axiom,
    ! [A4: set_list_a,B4: list_a > set_nat] :
      ( ( finite_finite_list_a @ A4 )
     => ( ! [A2: list_a] :
            ( ( member_list_a @ A2 @ A4 )
           => ( finite_finite_nat @ ( B4 @ A2 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B4 @ A4 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1181_finite__UN__I,axiom,
    ! [A4: set_nat,B4: nat > set_list_a] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ( finite_finite_list_a @ ( B4 @ A2 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B4 @ A4 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1182_finite__UN__I,axiom,
    ! [A4: set_nat,B4: nat > set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A2: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ( finite_finite_nat @ ( B4 @ A2 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A4 ) ) ) ) ) ).

% finite_UN_I
thf(fact_1183_fv__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Q ) @ ( comple7399068483239264473et_nat @ ( image_8719518604786020652et_nat @ relational_fv_a_b @ ( set_Re9104216502384355786la_a_b @ Qs ) ) ) ) ) ).

% fv_foldr_Disj
thf(fact_1184_finite__UnionD,axiom,
    ! [A4: set_set_list_a] :
      ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A4 ) )
     => ( finite5282473924520328461list_a @ A4 ) ) ).

% finite_UnionD
thf(fact_1185_finite__UnionD,axiom,
    ! [A4: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A4 ) )
     => ( finite1152437895449049373et_nat @ A4 ) ) ).

% finite_UnionD
thf(fact_1186_UN__I,axiom,
    ! [A3: nat,A4: set_nat,B2: nat,B4: nat > set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ( ( member_nat @ B2 @ ( B4 @ A3 ) )
       => ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A4 ) ) ) ) ) ).

% UN_I
thf(fact_1187_Sup__set__def,axiom,
    ( comple6928918032620976721list_a
    = ( ^ [A6: set_set_list_a] :
          ( collect_list_a
          @ ^ [X: list_a] : ( complete_Sup_Sup_o @ ( image_set_list_a_o @ ( member_list_a @ X ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1188_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_nat
          @ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1189_SUP__Sup__eq,axiom,
    ! [S3: set_set_nat] :
      ( ( comple8317665133742190828_nat_o
        @ ( image_set_nat_nat_o
          @ ^ [I5: set_nat,X: nat] : ( member_nat @ X @ I5 )
          @ S3 ) )
      = ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S3 ) ) ) ) ).

% SUP_Sup_eq
thf(fact_1190_UN__E,axiom,
    ! [B2: nat,B4: nat > set_nat,A4: set_nat] :
      ( ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A4 ) ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ~ ( member_nat @ B2 @ ( B4 @ X4 ) ) ) ) ).

% UN_E
thf(fact_1191_finite__subset__Union,axiom,
    ! [A4: set_list_a,B8: set_set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( ord_le8861187494160871172list_a @ A4 @ ( comple6928918032620976721list_a @ B8 ) )
       => ~ ! [F6: set_set_list_a] :
              ( ( finite5282473924520328461list_a @ F6 )
             => ( ( ord_le8877086941679407844list_a @ F6 @ B8 )
               => ~ ( ord_le8861187494160871172list_a @ A4 @ ( comple6928918032620976721list_a @ F6 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1192_finite__subset__Union,axiom,
    ! [A4: set_nat,B8: set_set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( comple7399068483239264473et_nat @ B8 ) )
       => ~ ! [F6: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F6 )
             => ( ( ord_le6893508408891458716et_nat @ F6 @ B8 )
               => ~ ( ord_less_eq_set_nat @ A4 @ ( comple7399068483239264473et_nat @ F6 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_1193_cSup__eq__maximum,axiom,
    ! [Z: nat,X3: set_nat] :
      ( ( member_nat @ Z @ X3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X3 )
           => ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ( complete_Sup_Sup_nat @ X3 )
          = Z ) ) ) ).

% cSup_eq_maximum
thf(fact_1194_cSup__eq__non__empty,axiom,
    ! [X3: set_nat,A3: nat] :
      ( ( X3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X3 )
           => ( ord_less_eq_nat @ X4 @ A3 ) )
       => ( ! [Y3: nat] :
              ( ! [X5: nat] :
                  ( ( member_nat @ X5 @ X3 )
                 => ( ord_less_eq_nat @ X5 @ Y3 ) )
             => ( ord_less_eq_nat @ A3 @ Y3 ) )
         => ( ( complete_Sup_Sup_nat @ X3 )
            = A3 ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_1195_cSup__least,axiom,
    ! [X3: set_nat,Z: nat] :
      ( ( X3 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X3 )
           => ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X3 ) @ Z ) ) ) ).

% cSup_least
thf(fact_1196_le__cSup__finite,axiom,
    ! [X3: set_nat,X2: nat] :
      ( ( finite_finite_nat @ X3 )
     => ( ( member_nat @ X2 @ X3 )
       => ( ord_less_eq_nat @ X2 @ ( complete_Sup_Sup_nat @ X3 ) ) ) ) ).

% le_cSup_finite
thf(fact_1197_cSUP__least,axiom,
    ! [A4: set_nat,F2: nat > nat,M2: nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ord_less_eq_nat @ ( F2 @ X4 ) @ M2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A4 ) ) @ M2 ) ) ) ).

% cSUP_least
thf(fact_1198_conj__subset__def,axiom,
    ! [A4: set_list_a,P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ A4
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le8861187494160871172list_a @ A4 @ ( collect_list_a @ P ) )
        & ( ord_le8861187494160871172list_a @ A4 @ ( collect_list_a @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1199_conj__subset__def,axiom,
    ! [A4: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A4
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_1200_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_1201_Fpow__def,axiom,
    ( finite_Fpow_list_a
    = ( ^ [A6: set_list_a] :
          ( collect_set_list_a
          @ ^ [X6: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X6 @ A6 )
              & ( finite_finite_list_a @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_1202_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A6: set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( ord_less_eq_set_nat @ X6 @ A6 )
              & ( finite_finite_nat @ X6 ) ) ) ) ) ).

% Fpow_def
thf(fact_1203_csts_Oelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: set_a] :
      ( ( ( relational_csts_a_b @ X2 )
        = Y2 )
     => ( ( ? [B: $o] :
              ( X2
              = ( relational_Bool_a_b @ B ) )
         => ( Y2 != bot_bot_set_a ) )
       => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
              ( ( X2
                = ( relational_Pred_b_a @ P3 @ Ts ) )
             => ( Y2
               != ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) ) )
         => ( ! [X4: nat,T5: relational_term_a] :
                ( ( X2
                  = ( relational_Eq_a_b @ X4 @ T5 ) )
               => ( Y2
                 != ( relati6638259395341848997term_a @ T5 ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X2
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y2
                   != ( relational_csts_a_b @ Q4 ) ) )
             => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                   => ( Y2
                     != ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                     => ( Y2
                       != ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X2
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( Y2
                         != ( relational_csts_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ).

% csts.elims
thf(fact_1204_fold__closed__eq,axiom,
    ! [A4: set_nat,B4: set_nat,F2: nat > nat > nat,G: nat > nat > nat,Z: nat] :
      ( ! [A2: nat,B: nat] :
          ( ( member_nat @ A2 @ A4 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F2 @ A2 @ B )
              = ( G @ A2 @ B ) ) ) )
     => ( ! [A2: nat,B: nat] :
            ( ( member_nat @ A2 @ A4 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G @ A2 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z @ B4 )
         => ( ( finite_fold_nat_nat @ F2 @ Z @ A4 )
            = ( finite_fold_nat_nat @ G @ Z @ A4 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_1205_finite__csts__term,axiom,
    ! [T: relati383187284260386255list_a] : ( finite_finite_list_a @ ( relati1612993689924349611list_a @ T ) ) ).

% finite_csts_term
thf(fact_1206_finite__csts__term,axiom,
    ! [T: relational_term_nat] : ( finite_finite_nat @ ( relati694035416245573993rm_nat @ T ) ) ).

% finite_csts_term
thf(fact_1207_csts_Osimps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_csts_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) )
      = ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q22 ) ) ) ).

% csts.simps(6)
thf(fact_1208_union__fold__insert,axiom,
    ! [A4: set_list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( sup_sup_set_list_a @ A4 @ B4 )
        = ( finite9084501323850038056list_a @ insert_list_a @ B4 @ A4 ) ) ) ).

% union_fold_insert
thf(fact_1209_union__fold__insert,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( sup_sup_set_nat @ A4 @ B4 )
        = ( finite5529483035118572448et_nat @ insert_nat @ B4 @ A4 ) ) ) ).

% union_fold_insert
thf(fact_1210_Id__on__fold,axiom,
    ! [A4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( id_on_list_a @ A4 )
        = ( finite5942777621280601035list_a
          @ ^ [X: list_a] : ( insert1856800524785285367list_a @ ( produc6837034575241423639list_a @ X @ X ) )
          @ bot_bo2955605580254355571list_a
          @ A4 ) ) ) ).

% Id_on_fold
thf(fact_1211_Id__on__fold,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( id_on_nat @ A4 )
        = ( finite3745491028973389255at_nat
          @ ^ [X: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) )
          @ bot_bo2099793752762293965at_nat
          @ A4 ) ) ) ).

% Id_on_fold
thf(fact_1212_csts_Opelims,axiom,
    ! [X2: relational_fmla_a_b,Y2: set_a] :
      ( ( ( relational_csts_a_b @ X2 )
        = Y2 )
     => ( ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ X2 )
       => ( ! [B: $o] :
              ( ( X2
                = ( relational_Bool_a_b @ B ) )
             => ( ( Y2 = bot_bot_set_a )
               => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
         => ( ! [P3: b,Ts: list_R6823256787227418703term_a] :
                ( ( X2
                  = ( relational_Pred_b_a @ P3 @ Ts ) )
               => ( ( Y2
                    = ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Pred_b_a @ P3 @ Ts ) ) ) )
           => ( ! [X4: nat,T5: relational_term_a] :
                  ( ( X2
                    = ( relational_Eq_a_b @ X4 @ T5 ) )
                 => ( ( Y2
                      = ( relati6638259395341848997term_a @ T5 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Eq_a_b @ X4 @ T5 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X2
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y2
                        = ( relational_csts_a_b @ Q4 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
               => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                      ( ( X2
                        = ( relational_Conj_a_b @ Q13 @ Q24 ) )
                     => ( ( Y2
                          = ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Conj_a_b @ Q13 @ Q24 ) ) ) )
                 => ( ! [Q13: relational_fmla_a_b,Q24: relational_fmla_a_b] :
                        ( ( X2
                          = ( relational_Disj_a_b @ Q13 @ Q24 ) )
                       => ( ( Y2
                            = ( sup_sup_set_a @ ( relational_csts_a_b @ Q13 ) @ ( relational_csts_a_b @ Q24 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Disj_a_b @ Q13 @ Q24 ) ) ) )
                   => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                          ( ( X2
                            = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ( ( Y2
                              = ( relational_csts_a_b @ Q4 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% csts.pelims
thf(fact_1213_Set__filter__fold,axiom,
    ! [A4: set_list_a,P: list_a > $o] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( filter_list_a @ P @ A4 )
        = ( finite9084501323850038056list_a
          @ ^ [X: list_a,A8: set_list_a] : ( if_set_list_a @ ( P @ X ) @ ( insert_list_a @ X @ A8 ) @ A8 )
          @ bot_bot_set_list_a
          @ A4 ) ) ) ).

% Set_filter_fold
thf(fact_1214_Set__filter__fold,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( filter_nat @ P @ A4 )
        = ( finite5529483035118572448et_nat
          @ ^ [X: nat,A8: set_nat] : ( if_set_nat @ ( P @ X ) @ ( insert_nat @ X @ A8 ) @ A8 )
          @ bot_bot_set_nat
          @ A4 ) ) ) ).

% Set_filter_fold
thf(fact_1215_member__filter,axiom,
    ! [X2: nat,P: nat > $o,A4: set_nat] :
      ( ( member_nat @ X2 @ ( filter_nat @ P @ A4 ) )
      = ( ( member_nat @ X2 @ A4 )
        & ( P @ X2 ) ) ) ).

% member_filter
thf(fact_1216_finite__filter,axiom,
    ! [S3: set_list_a,P: list_a > $o] :
      ( ( finite_finite_list_a @ S3 )
     => ( finite_finite_list_a @ ( filter_list_a @ P @ S3 ) ) ) ).

% finite_filter
thf(fact_1217_finite__filter,axiom,
    ! [S3: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( finite_finite_nat @ ( filter_nat @ P @ S3 ) ) ) ).

% finite_filter
thf(fact_1218_Set_Ofilter__def,axiom,
    ( filter_list_a
    = ( ^ [P4: list_a > $o,A6: set_list_a] :
          ( collect_list_a
          @ ^ [A: list_a] :
              ( ( member_list_a @ A @ A6 )
              & ( P4 @ A ) ) ) ) ) ).

% Set.filter_def
thf(fact_1219_Set_Ofilter__def,axiom,
    ( filter_nat
    = ( ^ [P4: nat > $o,A6: set_nat] :
          ( collect_nat
          @ ^ [A: nat] :
              ( ( member_nat @ A @ A6 )
              & ( P4 @ A ) ) ) ) ) ).

% Set.filter_def
thf(fact_1220_arg__min__least,axiom,
    ! [S3: set_list_a,Y2: list_a,F2: list_a > nat] :
      ( ( finite_finite_list_a @ S3 )
     => ( ( S3 != bot_bot_set_list_a )
       => ( ( member_list_a @ Y2 @ S3 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic5043722365632780795_a_nat @ F2 @ S3 ) ) @ ( F2 @ Y2 ) ) ) ) ) ).

% arg_min_least
thf(fact_1221_arg__min__least,axiom,
    ! [S3: set_nat,Y2: nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y2 @ S3 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S3 ) ) @ ( F2 @ Y2 ) ) ) ) ) ).

% arg_min_least
thf(fact_1222_equiv__Exists__Disj,axiom,
    ! [X2: nat,Q1: relational_fmla_a_b,Q22: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X2 @ ( relational_Disj_a_b @ Q1 @ Q22 ) ) @ ( relational_Disj_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q1 ) @ ( relati591517084277583526ts_a_b @ X2 @ Q22 ) ) ) ).

% equiv_Exists_Disj
thf(fact_1223_Exists__nonfree__equiv,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) )
     => ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X2 @ Q ) @ Q ) ) ).

% Exists_nonfree_equiv
thf(fact_1224_equiv__Disj__cong,axiom,
    ! [Q1: relational_fmla_a_b,Q14: relational_fmla_a_b,Q22: relational_fmla_a_b,Q25: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q1 @ Q14 )
     => ( ( relational_equiv_a_b @ Q22 @ Q25 )
       => ( relational_equiv_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ ( relational_Disj_a_b @ Q14 @ Q25 ) ) ) ) ).

% equiv_Disj_cong
thf(fact_1225_equiv__Disj__Assoc,axiom,
    ! [Q1: relational_fmla_a_b,Q22: relational_fmla_a_b,Q32: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relational_Disj_a_b @ ( relational_Disj_a_b @ Q1 @ Q22 ) @ Q32 ) @ ( relational_Disj_a_b @ Q1 @ ( relational_Disj_a_b @ Q22 @ Q32 ) ) ) ).

% equiv_Disj_Assoc
thf(fact_1226_Exists__foldr1__Disj,axiom,
    ! [X2: nat,Xs: list_R8263082107343818799la_a_b,B2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X2 @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Xs @ B2 ) ) @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ ( map_Re5057026939262599940la_a_b @ ( relati3989891337220013914ts_a_b @ X2 ) @ Xs ) @ ( relati3989891337220013914ts_a_b @ X2 @ B2 ) ) ) ).

% Exists_foldr1_Disj
thf(fact_1227_Exists__foldr__Disj,axiom,
    ! [X2: nat,Xs: list_R8263082107343818799la_a_b,B2: relational_fmla_a_b] : ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X2 @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Xs @ B2 ) ) @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ ( map_Re5057026939262599940la_a_b @ ( relati3989891337220013914ts_a_b @ X2 ) @ Xs ) @ ( relati3989891337220013914ts_a_b @ X2 @ B2 ) ) ) ).

% Exists_foldr_Disj
thf(fact_1228_foldr1__Disj__equiv,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Qs2: list_R8263082107343818799la_a_b] :
      ( ( ( set_Re9104216502384355786la_a_b @ Qs )
        = ( set_Re9104216502384355786la_a_b @ Qs2 ) )
     => ( relational_equiv_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs2 @ ( relational_Bool_a_b @ $false ) ) ) ) ).

% foldr1_Disj_equiv
thf(fact_1229_foldr__Disj__equiv,axiom,
    ! [Q: relational_fmla_a_b,Qs: list_R8263082107343818799la_a_b,Q5: relational_fmla_a_b,Qs2: list_R8263082107343818799la_a_b] :
      ( ( ( insert7010464514620295119la_a_b @ Q @ ( set_Re9104216502384355786la_a_b @ Qs ) )
        = ( insert7010464514620295119la_a_b @ Q5 @ ( set_Re9104216502384355786la_a_b @ Qs2 ) ) )
     => ( relational_equiv_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs2 @ Q5 ) ) ) ).

% foldr_Disj_equiv
thf(fact_1230_equiv__eval__on__eval__eqI,axiom,
    ! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
      ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
     => ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q5 ) )
       => ( ( relational_equiv_a_b @ Q @ Q5 )
         => ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q5 ) @ Q @ I )
            = ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).

% equiv_eval_on_eval_eqI
thf(fact_1231_DiffI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1232_Diff__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        & ~ ( member_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1233_finite__Diff,axiom,
    ! [A4: set_list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ A4 )
     => ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1234_finite__Diff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff
thf(fact_1235_finite__Diff2,axiom,
    ! [B4: set_list_a,A4: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ B4 ) )
        = ( finite_finite_list_a @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1236_finite__Diff2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
        = ( finite_finite_nat @ A4 ) ) ) ).

% finite_Diff2
thf(fact_1237_insert__Diff1,axiom,
    ! [X2: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ X2 @ B4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B4 )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1238_Diff__insert0,axiom,
    ! [X2: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X2 @ A4 )
     => ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ B4 ) )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1239_finite__Diff__insert,axiom,
    ! [A4: set_list_a,A3: list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ A3 @ B4 ) ) )
      = ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1240_finite__Diff__insert,axiom,
    ! [A4: set_nat,A3: nat,B4: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ B4 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_1241_Diff__infinite__finite,axiom,
    ! [T2: set_list_a,S3: set_list_a] :
      ( ( finite_finite_list_a @ T2 )
     => ( ~ ( finite_finite_list_a @ S3 )
       => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S3 @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1242_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ T2 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_1243_infinite__remove,axiom,
    ! [S3: set_list_a,A3: list_a] :
      ( ~ ( finite_finite_list_a @ S3 )
     => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S3 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ).

% infinite_remove
thf(fact_1244_infinite__remove,axiom,
    ! [S3: set_nat,A3: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_1245_infinite__coinduct,axiom,
    ! [X3: set_list_a > $o,A4: set_list_a] :
      ( ( X3 @ A4 )
     => ( ! [A7: set_list_a] :
            ( ( X3 @ A7 )
           => ? [X5: list_a] :
                ( ( member_list_a @ X5 @ A7 )
                & ( ( X3 @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
                  | ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
       => ~ ( finite_finite_list_a @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1246_infinite__coinduct,axiom,
    ! [X3: set_nat > $o,A4: set_nat] :
      ( ( X3 @ A4 )
     => ( ! [A7: set_nat] :
            ( ( X3 @ A7 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A7 )
                & ( ( X3 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_1247_finite__empty__induct,axiom,
    ! [A4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ A4 )
     => ( ( P @ A4 )
       => ( ! [A2: list_a,A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( member_list_a @ A2 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ) ) ) )
         => ( P @ bot_bot_set_list_a ) ) ) ) ).

% finite_empty_induct
thf(fact_1248_finite__empty__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ A4 )
       => ( ! [A2: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A2 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1249_subset__insert__iff,axiom,
    ! [A4: set_nat,X2: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X2 @ B4 ) )
      = ( ( ( member_nat @ X2 @ A4 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X2 @ A4 )
         => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1250_insert__Diff__if,axiom,
    ! [X2: nat,B4: set_nat,A4: set_nat] :
      ( ( ( member_nat @ X2 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B4 )
          = ( minus_minus_set_nat @ A4 @ B4 ) ) )
      & ( ~ ( member_nat @ X2 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ B4 )
          = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1251_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X2: nat,C6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X2 @ C6 ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C6 ) )
        & ~ ( member_nat @ X2 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1252_DiffE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ C @ A4 )
         => ( member_nat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1253_DiffD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C @ A4 ) ) ).

% DiffD1
thf(fact_1254_DiffD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( member_nat @ C @ B4 ) ) ).

% DiffD2
thf(fact_1255_set__diff__eq,axiom,
    ( minus_646659088055828811list_a
    = ( ^ [A6: set_list_a,B6: set_list_a] :
          ( collect_list_a
          @ ^ [X: list_a] :
              ( ( member_list_a @ X @ A6 )
              & ~ ( member_list_a @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1256_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A6 )
              & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1257_insert__remove__id,axiom,
    ! [X2: nat,X3: set_nat] :
      ( ( member_nat @ X2 @ X3 )
     => ( X3
        = ( insert_nat @ X2 @ ( minus_minus_set_nat @ X3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).

% insert_remove_id
thf(fact_1258_insert__Diff,axiom,
    ! [A3: nat,A4: set_nat] :
      ( ( member_nat @ A3 @ A4 )
     => ( ( insert_nat @ A3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1259_Diff__insert__absorb,axiom,
    ! [X2: nat,A4: set_nat] :
      ( ~ ( member_nat @ X2 @ A4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A4 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1260_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,X2: nat,A4: set_nat] :
      ( ! [C4: set_nat] :
          ( ( member_set_nat @ C4 @ B4 )
         => ~ ( member_nat @ X2 @ C4 ) )
     => ( ( member_set_nat @ A4 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B4 ) )
        = ( ( member_nat @ X2 @ A4 )
          & ( member_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_1261_finite__remove__induct,axiom,
    ! [B4: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( A7 != bot_bot_set_list_a )
               => ( ( ord_le8861187494160871172list_a @ A7 @ B4 )
                 => ( ! [X5: list_a] :
                        ( ( member_list_a @ X5 @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1262_finite__remove__induct,axiom,
    ! [B4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B4 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_1263_remove__induct,axiom,
    ! [P: set_list_a > $o,B4: set_list_a] :
      ( ( P @ bot_bot_set_list_a )
     => ( ( ~ ( finite_finite_list_a @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( A7 != bot_bot_set_list_a )
               => ( ( ord_le8861187494160871172list_a @ A7 @ B4 )
                 => ( ! [X5: list_a] :
                        ( ( member_list_a @ X5 @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1264_remove__induct,axiom,
    ! [P: set_nat > $o,B4: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B4 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_1265_fun__upd__image,axiom,
    ! [X2: nat,A4: set_nat,F2: nat > a,Y2: a] :
      ( ( ( member_nat @ X2 @ A4 )
       => ( ( image_nat_a @ ( fun_upd_nat_a @ F2 @ X2 @ Y2 ) @ A4 )
          = ( insert_a @ Y2 @ ( image_nat_a @ F2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X2 @ A4 )
       => ( ( image_nat_a @ ( fun_upd_nat_a @ F2 @ X2 @ Y2 ) @ A4 )
          = ( image_nat_a @ F2 @ A4 ) ) ) ) ).

% fun_upd_image
thf(fact_1266_Relational__Calculus_Oequiv__def,axiom,
    ( relational_equiv_a_b
    = ( ^ [Q12: relational_fmla_a_b,Q23: relational_fmla_a_b] :
        ! [I2: product_prod_b_nat > set_list_a,Sigma3: nat > a] :
          ( ( finite_finite_a @ ( relational_adom_b_a @ I2 ) )
         => ( ( relational_sat_a_b @ Q12 @ I2 @ Sigma3 )
            = ( relational_sat_a_b @ Q23 @ I2 @ Sigma3 ) ) ) ) ) ).

% Relational_Calculus.equiv_def
thf(fact_1267_equiv__eval__on__eqI,axiom,
    ! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b,X3: set_nat] :
      ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
     => ( ( relational_equiv_a_b @ Q @ Q5 )
       => ( ( relati8814510239606734169on_a_b @ X3 @ Q @ I )
          = ( relati8814510239606734169on_a_b @ X3 @ Q5 @ I ) ) ) ) ).

% equiv_eval_on_eqI
thf(fact_1268_qp__fresh__val,axiom,
    ! [Q: relational_fmla_a_b,Sigma: nat > a,X2: nat,I: product_prod_b_nat > set_list_a] :
      ( ( relational_qp_a_b @ Q )
     => ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_adom_b_a @ I ) )
       => ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_csts_a_b @ Q ) )
         => ( ( relational_sat_a_b @ Q @ I @ Sigma )
           => ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).

% qp_fresh_val
thf(fact_1269_ap__fresh__val,axiom,
    ! [Q: relational_fmla_a_b,Sigma: nat > a,X2: nat,I: product_prod_b_nat > set_list_a] :
      ( ( relational_ap_a_b @ Q )
     => ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_adom_b_a @ I ) )
       => ( ~ ( member_a @ ( Sigma @ X2 ) @ ( relational_csts_a_b @ Q ) )
         => ( ( relational_sat_a_b @ Q @ I @ Sigma )
           => ~ ( member_nat @ X2 @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).

% ap_fresh_val
thf(fact_1270_equiv__eval__eqI,axiom,
    ! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
      ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
     => ( ( ( relational_fv_a_b @ Q )
          = ( relational_fv_a_b @ Q5 ) )
       => ( ( relational_equiv_a_b @ Q @ Q5 )
         => ( ( relational_eval_a_b @ Q @ I )
            = ( relational_eval_a_b @ Q5 @ I ) ) ) ) ) ).

% equiv_eval_eqI
thf(fact_1271_diff__diff__cancel,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
        = I3 ) ) ).

% diff_diff_cancel
thf(fact_1272_fv__exists,axiom,
    ! [X2: nat,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X2 @ Q ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% fv_exists
thf(fact_1273_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1274_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff

% Helper facts (11)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X2: a,Y2: a] :
      ( ( if_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X2: a,Y2: a] :
      ( ( if_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: set_list_a,Y2: set_list_a] :
      ( ( if_set_list_a @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X2: set_list_a,Y2: set_list_a] :
      ( ( if_set_list_a @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X2: relational_fmla_a_b,Y2: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ( finite_finite_list_a
    @ ( collect_list_a
      @ ^ [Ds: list_a] :
          ( ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ ( relational_Disj_a_b @ q1 @ q2 ) ) ) )
            = ( size_size_list_a @ Ds ) )
          & ? [Sigma3: nat > a] :
              ( relational_sat_a_b @ ( relational_Disj_a_b @ q1 @ q2 ) @ i
              @ ( fold_P5280602285094830901_nat_a
                @ ( produc2909000522608705447_nat_a
                  @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
                @ ( zip_nat_a @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ ( relational_Disj_a_b @ q1 @ q2 ) ) ) @ Ds )
                @ Sigma3 ) ) ) ) ) ).

thf(conj_1,conjecture,
    ( finite_finite_list_a
    @ ( collect_list_a
      @ ^ [Ds: list_a] :
          ( ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ q2 ) ) )
            = ( size_size_list_a @ Ds ) )
          & ? [Sigma3: nat > a] :
              ( relational_sat_a_b @ q2 @ i
              @ ( fold_P5280602285094830901_nat_a
                @ ( produc2909000522608705447_nat_a
                  @ ^ [X: nat,Y: a,F: nat > a] : ( fun_upd_nat_a @ F @ X @ Y ) )
                @ ( zip_nat_a @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ q2 ) ) @ Ds )
                @ Sigma3 ) ) ) ) ) ).

%------------------------------------------------------------------------------