TPTP Problem File: SLH0969^1.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_01755_066905__17550828_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1879 ( 533 unt; 591 typ;   0 def)
%            Number of atoms       : 5056 (2027 equ;   0 cnn)
%            Maximal formula atoms :   45 (   3 avg)
%            Number of connectives : 16971 ( 934   ~;  89   |; 499   &;12855   @)
%                                         (   0 <=>;2594  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   30 (   7 avg)
%            Number of types       :   75 (  74 usr)
%            Number of type conns  : 1758 (1758   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  520 ( 517 usr;  38 con; 0-4 aty)
%            Number of variables   : 4505 ( 439   ^;3824   !; 242   ?;4505   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:26:26.811
%------------------------------------------------------------------------------
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% Explicit typings (517)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_Itf__a_J_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_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_OFpow_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4300370950623927696at_nat: produc7299876168958306407at_nat > set_Pr1604112560894399303at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
    member6436561159828024080at_nat: produc1296870326321989863at_nat > set_Pr5540436994911067335at_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member3250002818903848048at_nat: set_Pr8693737435421807431at_nat > set_se6425085446898223655at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member6273874620674182256at_nat: set_Pr4329608150637261639at_nat > set_se5479201105039950375at_nat > $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_Q,type,
    q: relational_fmla_a_b ).

thf(sy_v_x,type,
    x: nat ).

thf(sy_v_y,type,
    y: nat ).

% Relevant facts (1274)
thf(fact_0_fmla_Oinject_I3_J,axiom,
    ! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
      ( ( ( relational_Eq_a_b @ X31 @ X32 )
        = ( relational_Eq_a_b @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% fmla.inject(3)
thf(fact_1_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_2_term_Oinject_I2_J,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( relational_Var_a @ X2 )
        = ( relational_Var_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% term.inject(2)
thf(fact_3_assms_I2_J,axiom,
    ~ ( finite_finite_list_a @ ( relational_eval_a_b @ q @ i ) ) ).

% assms(2)
thf(fact_4_assms_I1_J,axiom,
    ~ ( member_nat @ x @ ( relational_fv_a_b @ q ) ) ).

% assms(1)
thf(fact_5_fmla_Odistinct_I25_J,axiom,
    ! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(25)
thf(fact_6_cpropagated__simps_I3_J,axiom,
    ! [X: nat,T: relational_term_a] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X @ T ) )
      = ( T
       != ( relational_Var_a @ X ) ) ) ).

% cpropagated_simps(3)
thf(fact_7_genempty_Ointros_I8_J,axiom,
    ! [Q: relational_fmla_a_b,Y: nat,X: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) ) ) ) ).

% genempty.intros(8)
thf(fact_8_genempty_Ointros_I7_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ).

% genempty.intros(7)
thf(fact_9_qp__impl_Osimps_I5_J,axiom,
    ! [V: nat,Vb: nat] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ V @ ( relational_Var_a @ Vb ) ) ) ).

% qp_impl.simps(5)
thf(fact_10_finite__eval__Disj2D,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) )
     => ( finite_finite_list_a @ ( relational_eval_a_b @ Q2 @ I ) ) ) ).

% finite_eval_Disj2D
thf(fact_11_eqs__in,axiom,
    ! [Y: nat,X: nat,G: set_Re381260168593705685la_a_b] :
      ( ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) )
     => ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G ) ) ).

% eqs_in
thf(fact_12_notin__eqs,axiom,
    ! [X: nat,Y: nat,G: set_Re381260168593705685la_a_b] :
      ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
     => ~ ( member_nat @ Y @ ( relational_eqs_a_b @ X @ G ) ) ) ).

% notin_eqs
thf(fact_13_genempty_Ointros_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q1 )
     => ( ( relati5999705594545617851ty_a_b @ Q2 )
       => ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).

% genempty.intros(5)
thf(fact_14_finite__fv,axiom,
    ! [Phi: relational_fmla_a_b] : ( finite_finite_nat @ ( relational_fv_a_b @ Phi ) ) ).

% finite_fv
thf(fact_15_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_16_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_17_genempty_Ointros_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( ( relati5999705594545617851ty_a_b @ Q1 )
        | ( relati5999705594545617851ty_a_b @ Q2 ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ).

% genempty.intros(6)
thf(fact_18_qp__impl_Osimps_I7_J,axiom,
    ! [V: relational_fmla_a_b,Va: relational_fmla_a_b] :
      ~ ( relati3725921752842749053pl_a_b @ ( relational_Conj_a_b @ V @ Va ) ) ).

% qp_impl.simps(7)
thf(fact_19_sr__Conj__eq,axiom,
    ! [Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relational_sr_a_b @ Q )
     => ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
          | ( member_nat @ Y @ ( relational_fv_a_b @ Q ) ) )
       => ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ) ) ) ).

% sr_Conj_eq
thf(fact_20_Relational__Calculus_Oeval__def,axiom,
    ( relational_eval_a_b
    = ( ^ [Q3: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q3 ) @ Q3 ) ) ) ).

% Relational_Calculus.eval_def
thf(fact_21_genempty_Ointros_I4_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ) ) ).

% genempty.intros(4)
thf(fact_22_genempty_Ointros_I3_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) ) ) ).

% genempty.intros(3)
thf(fact_23_sr__Disj,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( ( relational_fv_a_b @ Q1 )
        = ( relational_fv_a_b @ Q2 ) )
     => ( ( relational_sr_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
        = ( ( relational_sr_a_b @ Q1 )
          & ( relational_sr_a_b @ Q2 ) ) ) ) ).

% sr_Disj
thf(fact_24_cpropagated__simps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q2 ) ) ) ).

% cpropagated_simps(5)
thf(fact_25_cpropagated__simps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q2 ) ) ) ).

% cpropagated_simps(6)
thf(fact_26_cpropagated__nocp,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_nocp
thf(fact_27_qp__eq,axiom,
    ! [X: nat,Y: nat] :
      ~ ( relational_qp_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) ) ).

% qp_eq
thf(fact_28_cpropagated__simps_I4_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Neg_a_b @ Q ) )
      = ( relational_nocp_a_b @ Q ) ) ).

% cpropagated_simps(4)
thf(fact_29_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_30_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_31_mem__Collect__eq,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( member8757157785044589968at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_32_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_33_Collect__mem__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_34_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_35_Collect__mem__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_36_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_37_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_38_fmla_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: relational_term_a,X42: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Neg_a_b @ X42 ) ) ).

% fmla.distinct(23)
thf(fact_39_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_40_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_41_nocp_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( ( relational_nocp_a_b @ Q1 )
        & ( relational_nocp_a_b @ Q2 ) ) ) ).

% nocp.simps(5)
thf(fact_42_genempty_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q ) ) ) ) ).

% genempty.intros(2)
thf(fact_43_qp__Conj,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ~ ( relational_qp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) ).

% qp_Conj
thf(fact_44_nocp__cpropagated,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ Q )
     => ( relati1591879772219623554ed_a_b @ Q ) ) ).

% nocp_cpropagated
thf(fact_45_nocp_Osimps_I3_J,axiom,
    ! [X: nat,T: relational_term_a] :
      ( ( relational_nocp_a_b @ ( relational_Eq_a_b @ X @ T ) )
      = ( T
       != ( relational_Var_a @ X ) ) ) ).

% nocp.simps(3)
thf(fact_46_cpropagated__simps_I7_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% cpropagated_simps(7)
thf(fact_47_qp__impl_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X )
     => ( ! [X4: nat,Q4: relational_fmla_a_b] :
            ( ( X
              = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
           => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
              & ( relational_qp_a_b @ Q4 ) ) )
       => ( ! [V2: $o] :
              ( X
             != ( relational_Bool_a_b @ V2 ) )
         => ( ! [V2: nat,Vb2: nat] :
                ( X
               != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
           => ( ! [V2: relational_fmla_a_b] :
                  ( X
                 != ( relational_Neg_a_b @ V2 ) )
             => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                    ( X
                   != ( relational_Conj_a_b @ V2 @ Va2 ) )
               => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X
                     != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ).

% qp_impl.elims(3)
thf(fact_48_qp__impl_Osimps_I3_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_qp_a_b @ Q ) ) ) ).

% qp_impl.simps(3)
thf(fact_49_nocp_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X )
     => ( ! [B: $o] :
            ( X
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X4: nat,T2: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X4 @ T2 ) )
             => ( T2
               != ( relational_Var_a @ X4 ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q4 ) )
               => ( relational_nocp_a_b @ Q4 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                 => ( ( relational_nocp_a_b @ Q12 )
                    & ( relational_nocp_a_b @ Q22 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                   => ( ( relational_nocp_a_b @ Q12 )
                      & ( relational_nocp_a_b @ Q22 ) ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(3)
thf(fact_50_nocp_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X )
     => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ P2 @ Ts ) )
       => ( ! [X4: nat,T2: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X4 @ T2 ) )
             => ( T2
                = ( relational_Var_a @ X4 ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q4 ) )
               => ~ ( relational_nocp_a_b @ Q4 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                 => ~ ( ( relational_nocp_a_b @ Q12 )
                      & ( relational_nocp_a_b @ Q22 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                   => ~ ( ( relational_nocp_a_b @ Q12 )
                        & ( relational_nocp_a_b @ Q22 ) ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                          & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ).

% nocp.elims(2)
thf(fact_51_gen_H_Ointros_I4_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).

% gen'.intros(4)
thf(fact_52_gen_H_Ointros_I5_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b2 @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).

% gen'.intros(5)
thf(fact_53_gen_Ointros_I4_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
     => ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).

% gen.intros(4)
thf(fact_54_gen_Ointros_I5_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ X @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q1 ) @ ( relational_Neg_a_b @ Q2 ) ) @ G )
     => ( relational_gen_a_b @ X @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) ) @ G ) ) ).

% gen.intros(5)
thf(fact_55_genempty_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ A )
     => ( ( A
         != ( relational_Bool_a_b @ $false ) )
       => ( ! [Q4: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q4 ) ) )
             => ~ ( relati5999705594545617851ty_a_b @ Q4 ) )
         => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                ( ( A
                  = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
               => ~ ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( A
                    = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
                 => ~ ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( A
                      = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                   => ( ( relati5999705594545617851ty_a_b @ Q12 )
                     => ~ ( relati5999705594545617851ty_a_b @ Q22 ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( A
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ~ ( ( relati5999705594545617851ty_a_b @ Q12 )
                          | ( relati5999705594545617851ty_a_b @ Q22 ) ) )
                 => ( ! [Q4: relational_fmla_a_b] :
                        ( ? [X4: nat,Y3: nat] :
                            ( A
                            = ( 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] :
                              ( A
                              = ( 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] :
                                ( A
                                = ( relati591517084277583526ts_a_b @ Y3 @ Q4 ) )
                           => ~ ( relati5999705594545617851ty_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% genempty.cases
thf(fact_56_cpropagated__simps_I1_J,axiom,
    ! [B2: $o] : ( relati1591879772219623554ed_a_b @ ( relational_Bool_a_b @ B2 ) ) ).

% cpropagated_simps(1)
thf(fact_57_cpropagated__simps_I2_J,axiom,
    ! [P3: b,Ts2: list_R6823256787227418703term_a] : ( relati1591879772219623554ed_a_b @ ( relational_Pred_b_a @ P3 @ Ts2 ) ) ).

% cpropagated_simps(2)
thf(fact_58_gen_Ointros_I7_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b @ X @ Q1 @ G )
        | ( relational_gen_a_b @ X @ Q2 @ G ) )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).

% gen.intros(7)
thf(fact_59_Gen__Conj_I1_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q1 @ X_1 )
     => ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_12 ) ) ).

% Gen_Conj(1)
thf(fact_60_Gen__Conj_I2_J,axiom,
    ! [X: nat,Q2: relational_fmla_a_b,Q1: relational_fmla_a_b] :
      ( ? [X_1: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q2 @ X_1 )
     => ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X_12 ) ) ).

% Gen_Conj(2)
thf(fact_61_fmla_Odistinct_I3_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X31: nat,X32: relational_term_a] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(3)
thf(fact_62_fmla_Odistinct_I7_J,axiom,
    ! [X11: b,X12: list_R6823256787227418703term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Pred_b_a @ X11 @ X12 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(7)
thf(fact_63_fmla_Odistinct_I29_J,axiom,
    ! [X31: nat,X32: relational_term_a,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(29)
thf(fact_64_fmla_Odistinct_I39_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X71: nat,X72: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relati591517084277583526ts_a_b @ X71 @ X72 ) ) ).

% fmla.distinct(39)
thf(fact_65_fmla_Odistinct_I13_J,axiom,
    ! [X2: $o,X31: nat,X32: relational_term_a] :
      ( ( relational_Bool_a_b @ X2 )
     != ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.distinct(13)
thf(fact_66_fmla_Odistinct_I17_J,axiom,
    ! [X2: $o,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Bool_a_b @ X2 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(17)
thf(fact_67_cp_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [X4: nat,T2: relational_term_a] :
          ( X
         != ( relational_Eq_a_b @ X4 @ T2 ) )
     => ( ! [Q4: relational_fmla_a_b] :
            ( X
           != ( relational_Neg_a_b @ Q4 ) )
       => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
              ( X
             != ( relational_Conj_a_b @ Q12 @ Q22 ) )
         => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                ( X
               != ( relational_Disj_a_b @ Q12 @ Q22 ) )
           => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                  ( X
                 != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
             => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                    ( X
                   != ( relational_Pred_b_a @ V2 @ Va2 ) )
               => ~ ! [V2: $o] :
                      ( X
                     != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ).

% cp.cases
thf(fact_68_fv_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
          ( X
         != ( relational_Pred_b_a @ Uu @ Ts ) )
     => ( ! [B: $o] :
            ( X
           != ( relational_Bool_a_b @ B ) )
       => ( ! [X4: nat,T3: relational_term_a] :
              ( X
             != ( relational_Eq_a_b @ X4 @ T3 ) )
         => ( ! [Phi2: relational_fmla_a_b] :
                ( X
               != ( relational_Neg_a_b @ Phi2 ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                  ( X
                 != ( relational_Conj_a_b @ Phi2 @ Psi ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( X
                   != ( relational_Disj_a_b @ Phi2 @ Psi ) )
               => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                      ( X
                     != ( relati591517084277583526ts_a_b @ Z @ Phi2 ) ) ) ) ) ) ) ) ).

% fv.cases
thf(fact_69_nocp_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [B: $o] :
          ( X
         != ( relational_Bool_a_b @ B ) )
     => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ P2 @ Ts ) )
       => ( ! [X4: nat,T2: relational_term_a] :
              ( X
             != ( relational_Eq_a_b @ X4 @ T2 ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( X
               != ( relational_Neg_a_b @ Q4 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( X
                 != ( relational_Conj_a_b @ Q12 @ Q22 ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( X
                   != ( relational_Disj_a_b @ Q12 @ Q22 ) )
               => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( X
                     != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) ) ) ) ) ) ).

% nocp.cases
thf(fact_70_fmla_Oexhaust,axiom,
    ! [Y: relational_fmla_a_b] :
      ( ! [X112: b,X122: list_R6823256787227418703term_a] :
          ( Y
         != ( relational_Pred_b_a @ X112 @ X122 ) )
     => ( ! [X22: $o] :
            ( Y
           != ( relational_Bool_a_b @ X22 ) )
       => ( ! [X312: nat,X322: relational_term_a] :
              ( Y
             != ( relational_Eq_a_b @ X312 @ X322 ) )
         => ( ! [X43: relational_fmla_a_b] :
                ( Y
               != ( relational_Neg_a_b @ X43 ) )
           => ( ! [X512: relational_fmla_a_b,X522: relational_fmla_a_b] :
                  ( Y
                 != ( relational_Conj_a_b @ X512 @ X522 ) )
             => ( ! [X612: relational_fmla_a_b,X622: relational_fmla_a_b] :
                    ( Y
                   != ( relational_Disj_a_b @ X612 @ X622 ) )
               => ~ ! [X712: nat,X722: relational_fmla_a_b] :
                      ( Y
                     != ( relati591517084277583526ts_a_b @ X712 @ X722 ) ) ) ) ) ) ) ) ).

% fmla.exhaust
thf(fact_71_gen_H_Ointros_I7_J,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
      ( ( ( relational_gen_a_b2 @ X @ Q1 @ G )
        | ( relational_gen_a_b2 @ X @ Q2 @ G ) )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ).

% gen'.intros(7)
thf(fact_72_genempty_Ointros_I9_J,axiom,
    ! [Q: relational_fmla_a_b,Y: nat] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( relati5999705594545617851ty_a_b @ ( relati591517084277583526ts_a_b @ Y @ Q ) ) ) ).

% genempty.intros(9)
thf(fact_73_genempty_Ointros_I1_J,axiom,
    relati5999705594545617851ty_a_b @ ( relational_Bool_a_b @ $false ) ).

% genempty.intros(1)
thf(fact_74_qp__Gen,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ? [X_12: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ X @ Q @ X_12 ) ) ) ).

% qp_Gen
thf(fact_75_nocp_Osimps_I7_J,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
      = ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
        & ( relational_nocp_a_b @ Q ) ) ) ).

% nocp.simps(7)
thf(fact_76_nocp_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X )
        = Y )
     => ( ( ? [B: $o] :
              ( X
              = ( relational_Bool_a_b @ B ) )
         => Y )
       => ( ( ? [P2: b,Ts: list_R6823256787227418703term_a] :
                ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
           => ~ Y )
         => ( ! [X4: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T2 ) )
               => ( Y
                  = ( T2
                    = ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y
                    = ( ~ ( relational_nocp_a_b @ Q4 ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( Y
                      = ( ~ ( ( relational_nocp_a_b @ Q12 )
                            & ( relational_nocp_a_b @ Q22 ) ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( Y
                        = ( ~ ( ( relational_nocp_a_b @ Q12 )
                              & ( relational_nocp_a_b @ Q22 ) ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( Y
                          = ( ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                                & ( relational_nocp_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% nocp.elims(1)
thf(fact_77_qp__Exists,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ) ).

% qp_Exists
thf(fact_78_qp__ExistsE,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_qp_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) )
     => ~ ( ( relational_qp_a_b @ Q )
         => ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ).

% qp_ExistsE
thf(fact_79_genempty_Osimps,axiom,
    ( relati5999705594545617851ty_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( ( A3
            = ( relational_Bool_a_b @ $false ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
              & ( relati5999705594545617851ty_a_b @ Q3 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
              & ( relati5999705594545617851ty_a_b @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Disj_a_b @ Q13 @ Q23 ) )
              & ( relati5999705594545617851ty_a_b @ Q13 )
              & ( relati5999705594545617851ty_a_b @ Q23 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A3
                = ( relational_Conj_a_b @ Q13 @ Q23 ) )
              & ( ( relati5999705594545617851ty_a_b @ Q13 )
                | ( relati5999705594545617851ty_a_b @ Q23 ) ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ? [X3: nat,Y4: nat] :
                  ( A3
                  = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y4 ) ) ) )
              & ( relati5999705594545617851ty_a_b @ Q3 ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ? [Y4: nat] :
                  ( A3
                  = ( relati591517084277583526ts_a_b @ Y4 @ Q3 ) )
              & ( relati5999705594545617851ty_a_b @ Q3 ) ) ) ) ) ).

% genempty.simps
thf(fact_80_qp__impl_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X )
        = Y )
     => ( ( ? [X4: nat,C: a] :
              ( X
              = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) )
         => ~ Y )
       => ( ( ? [X4: b,Ts: list_R6823256787227418703term_a] :
                ( X
                = ( relational_Pred_b_a @ X4 @ Ts ) )
           => ~ Y )
         => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                ( ( X
                  = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
               => ( Y
                  = ( ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                        & ( relational_qp_a_b @ Q4 ) ) ) ) )
           => ( ( ? [V2: $o] :
                    ( X
                    = ( relational_Bool_a_b @ V2 ) )
               => Y )
             => ( ( ? [V2: nat,Vb2: nat] :
                      ( X
                      = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
                 => Y )
               => ( ( ? [V2: relational_fmla_a_b] :
                        ( X
                        = ( relational_Neg_a_b @ V2 ) )
                   => Y )
                 => ( ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( X
                          = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => Y )
                   => ~ ( ? [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( X
                            = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => Y ) ) ) ) ) ) ) ) ) ).

% qp_impl.elims(1)
thf(fact_81_qp__impl_Ocases,axiom,
    ! [X: relational_fmla_a_b] :
      ( ! [X4: nat,C: a] :
          ( X
         != ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) )
     => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
            ( X
           != ( relational_Pred_b_a @ X4 @ Ts ) )
       => ( ! [X4: nat,Q4: relational_fmla_a_b] :
              ( X
             != ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
         => ( ! [V2: $o] :
                ( X
               != ( relational_Bool_a_b @ V2 ) )
           => ( ! [V2: nat,Vb2: nat] :
                  ( X
                 != ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( X
                   != ( relational_Neg_a_b @ V2 ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( X
                     != ( relational_Conj_a_b @ V2 @ Va2 ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( X
                       != ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ).

% qp_impl.cases
thf(fact_82_erase_Ocases,axiom,
    ! [X: produc7366699395886430672_b_nat] :
      ( ! [T2: $o,X4: nat] :
          ( X
         != ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T2 ) @ X4 ) )
     => ( ! [P2: b,Ts: list_R6823256787227418703term_a,X4: nat] :
            ( X
           != ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ X4 ) )
       => ( ! [Z: nat,T2: relational_term_a,X4: nat] :
              ( X
             != ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z @ T2 ) @ X4 ) )
         => ( ! [Q4: relational_fmla_a_b,X4: nat] :
                ( X
               != ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q4 ) @ X4 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat] :
                  ( X
                 != ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ X4 ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat] :
                    ( X
                   != ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ X4 ) )
               => ~ ! [Z: nat,Q4: relational_fmla_a_b,X4: nat] :
                      ( X
                     != ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ X4 ) ) ) ) ) ) ) ) ).

% erase.cases
thf(fact_83_qp__impl_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X )
     => ( ! [X4: nat,C: a] :
            ( X
           != ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) )
       => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
              ( X
             != ( relational_Pred_b_a @ X4 @ Ts ) )
         => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                ( ( X
                  = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
               => ~ ( ( member_nat @ X4 @ ( relational_fv_a_b @ Q4 ) )
                    & ( relational_qp_a_b @ Q4 ) ) ) ) ) ) ).

% qp_impl.elims(2)
thf(fact_84_qp__impl_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relati3725921752842749053pl_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X4: nat,Q4: relational_fmla_a_b] :
              ( ( X
                = ( 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 ) ) ) )
         => ( ! [V2: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ V2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) )
           => ( ! [V2: nat,Vb2: nat] :
                  ( ( X
                    = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) )
             => ( ! [V2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ V2 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) )
               => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ V2 @ Va2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) )
                 => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ V2 @ Va2 ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(3)
thf(fact_85_nocp_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relational_nocp_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ( ~ Y
               => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
         => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P2 @ Ts ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ) )
           => ( ! [X4: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T2 ) )
                 => ( ( Y
                      = ( T2
                       != ( relational_Var_a @ X4 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y
                        = ( relational_nocp_a_b @ Q4 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( Y
                          = ( ( relational_nocp_a_b @ Q12 )
                            & ( relational_nocp_a_b @ Q22 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( Y
                            = ( ( relational_nocp_a_b @ Q12 )
                              & ( relational_nocp_a_b @ Q22 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
                   => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ( ( Y
                              = ( ( 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_86_term_Odistinct_I1_J,axiom,
    ! [X1: a,X2: nat] :
      ( ( relational_Const_a @ X1 )
     != ( relational_Var_a @ X2 ) ) ).

% term.distinct(1)
thf(fact_87_Relational__Calculus_Oterm_Oexhaust,axiom,
    ! [Y: relational_term_a] :
      ( ! [X13: a] :
          ( Y
         != ( relational_Const_a @ X13 ) )
     => ~ ! [X22: nat] :
            ( Y
           != ( relational_Var_a @ X22 ) ) ) ).

% Relational_Calculus.term.exhaust
thf(fact_88_fv__fo__term__list_Ocases,axiom,
    ! [X: relational_term_a] :
      ( ! [N: nat] :
          ( X
         != ( relational_Var_a @ N ) )
     => ~ ! [V2: a] :
            ( X
           != ( relational_Const_a @ V2 ) ) ) ).

% fv_fo_term_list.cases
thf(fact_89_cpropagated__cp__triv,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = Q ) ) ).

% cpropagated_cp_triv
thf(fact_90_cpropagated__cp,axiom,
    ! [Q: relational_fmla_a_b] : ( relati1591879772219623554ed_a_b @ ( relational_cp_a_b @ Q ) ) ).

% cpropagated_cp
thf(fact_91_genempty__cp,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( relati5999705594545617851ty_a_b @ Q )
     => ( ( relational_cp_a_b @ Q )
        = ( relational_Bool_a_b @ $false ) ) ) ).

% genempty_cp
thf(fact_92_qp__impl_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relati3725921752842749053pl_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X4: nat,C: a] :
              ( ( X
                = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) ) )
         => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( 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] :
                  ( ( X
                    = ( 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_93_qp__impl_Osimps_I1_J,axiom,
    ! [X: nat,C2: a] : ( relati3725921752842749053pl_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C2 ) ) ) ).

% qp_impl.simps(1)
thf(fact_94_qp__impl_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Y: $o] :
      ( ( ( relati3725921752842749053pl_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ X )
       => ( ! [X4: nat,C: a] :
              ( ( X
                = ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) )
             => ( Y
               => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) ) ) )
         => ( ! [X4: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ X4 @ Ts ) )
               => ( Y
                 => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Pred_b_a @ X4 @ Ts ) ) ) )
           => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                 => ( ( Y
                      = ( ( 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 ) ) ) )
             => ( ! [V2: $o] :
                    ( ( X
                      = ( relational_Bool_a_b @ V2 ) )
                   => ( ~ Y
                     => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) )
               => ( ! [V2: nat,Vb2: nat] :
                      ( ( X
                        = ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) )
                     => ( ~ Y
                       => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Eq_a_b @ V2 @ ( relational_Var_a @ Vb2 ) ) ) ) )
                 => ( ! [V2: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Neg_a_b @ V2 ) )
                       => ( ~ Y
                         => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Neg_a_b @ V2 ) ) ) )
                   => ( ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                          ( ( X
                            = ( relational_Conj_a_b @ V2 @ Va2 ) )
                         => ( ~ Y
                           => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Conj_a_b @ V2 @ Va2 ) ) ) )
                     => ~ ! [V2: relational_fmla_a_b,Va2: relational_fmla_a_b] :
                            ( ( X
                              = ( relational_Disj_a_b @ V2 @ Va2 ) )
                           => ( ~ Y
                             => ~ ( accp_R989495437599811158la_a_b @ relati7364465619720499582el_a_b @ ( relational_Disj_a_b @ V2 @ Va2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% qp_impl.pelims(1)
thf(fact_95_nocp_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ~ ( relational_nocp_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Bool_a_b @ B ) ) )
         => ( ! [X4: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T2 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) )
                 => ( T2
                   != ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
                   => ( relational_nocp_a_b @ Q4 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( relational_nocp_a_b @ Q12 )
                        & ( relational_nocp_a_b @ Q22 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( relational_nocp_a_b @ Q12 )
                          & ( relational_nocp_a_b @ Q22 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( 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_96_nocp_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b] :
      ( ( relational_nocp_a_b @ X )
     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ X )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
             => ~ ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) )
         => ( ! [X4: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T2 ) )
               => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) )
                 => ( T2
                    = ( relational_Var_a @ X4 ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Neg_a_b @ Q4 ) )
                   => ~ ( relational_nocp_a_b @ Q4 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ~ ( ( relational_nocp_a_b @ Q12 )
                          & ( relational_nocp_a_b @ Q22 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( ( accp_R989495437599811158la_a_b @ relati3149960101488570543el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ~ ( ( relational_nocp_a_b @ Q12 )
                            & ( relational_nocp_a_b @ Q22 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( 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_97_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B2 )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_98_old_Oprod_Oinject,axiom,
    ! [A: nat,B2: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B2 )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_99_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B2 )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_100_old_Oprod_Oinject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B2 )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
      = ( ( A = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_101_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_102_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_103_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_104_prod_Oinject,axiom,
    ! [X1: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y1: set_Pr4329608150637261639at_nat,Y2: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ X1 @ X2 )
        = ( produc9060074326276436823at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_105_fmla_Oset__cases_I2_J,axiom,
    ! [E: b,A: relational_fmla_a_b] :
      ( ( member_b @ E @ ( relati8924981150291758614la_a_b @ A ) )
     => ( ! [Z2: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ E @ Z2 ) )
       => ( ! [Z: relational_fmla_a_b] :
              ( ( A
                = ( relational_Neg_a_b @ Z ) )
             => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z ) ) )
         => ( ! [Z1: relational_fmla_a_b] :
                ( ? [Z2: relational_fmla_a_b] :
                    ( A
                    = ( relational_Conj_a_b @ Z1 @ Z2 ) )
               => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
           => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                  ( ( A
                    = ( relational_Conj_a_b @ Z1 @ Z2 ) )
                 => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
             => ( ! [Z1: relational_fmla_a_b] :
                    ( ? [Z2: relational_fmla_a_b] :
                        ( A
                        = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                   => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z1 ) ) )
               => ( ! [Z1: relational_fmla_a_b,Z2: relational_fmla_a_b] :
                      ( ( A
                        = ( relational_Disj_a_b @ Z1 @ Z2 ) )
                     => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) )
                 => ~ ! [Z1: nat,Z2: relational_fmla_a_b] :
                        ( ( A
                          = ( relati591517084277583526ts_a_b @ Z1 @ Z2 ) )
                       => ~ ( member_b @ E @ ( relati8924981150291758614la_a_b @ Z2 ) ) ) ) ) ) ) ) ) ) ).

% fmla.set_cases(2)
thf(fact_106_qp__cp__erase,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
          = ( relational_Bool_a_b @ $false ) ) ) ) ).

% qp_cp_erase
thf(fact_107_eval__Bool__False,axiom,
    ! [I: product_prod_b_nat > set_list_a] :
      ( ( relational_eval_a_b @ ( relational_Bool_a_b @ $false ) @ I )
      = bot_bot_set_list_a ) ).

% eval_Bool_False
thf(fact_108_fmla_Osimps_I129_J,axiom,
    ! [X31: nat,X32: relational_term_a] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Eq_a_b @ X31 @ X32 ) )
      = bot_bot_set_b ) ).

% fmla.simps(129)
thf(fact_109_subst__term_Ocases,axiom,
    ! [X: produc6058688428250151583at_nat] :
      ( ! [Z: nat,X4: nat,Y3: nat] :
          ( X
         != ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
     => ~ ! [C: a,X4: nat,Y3: nat] :
            ( X
           != ( produc2180204704594896271at_nat @ ( relational_Const_a @ C ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ) ).

% subst_term.cases
thf(fact_110_erase_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat] :
      ( ( relational_erase_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X )
      = ( relational_Conj_a_b @ ( relational_erase_a_b @ Q1 @ X ) @ ( relational_erase_a_b @ Q2 @ X ) ) ) ).

% erase.simps(5)
thf(fact_111_fmla_Oset__intros_I11_J,axiom,
    ! [Yu: b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ X51 ) )
     => ( member_b @ Yu @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(11)
thf(fact_112_fmla_Oset__intros_I12_J,axiom,
    ! [Yw: b,X52: relational_fmla_a_b,X51: relational_fmla_a_b] :
      ( ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ X52 ) )
     => ( member_b @ Yw @ ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ) ) ).

% fmla.set_intros(12)
thf(fact_113_infinite__imp__nonempty,axiom,
    ! [S: set_list_a] :
      ( ~ ( finite_finite_list_a @ S )
     => ( S != bot_bot_set_list_a ) ) ).

% infinite_imp_nonempty
thf(fact_114_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_115_infinite__imp__nonempty,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_116_infinite__imp__nonempty,axiom,
    ! [S: set_Pr4329608150637261639at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ( S != bot_bo228742789529271731at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_117_finite_OemptyI,axiom,
    finite_finite_list_a @ bot_bot_set_list_a ).

% finite.emptyI
thf(fact_118_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_119_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_120_finite_OemptyI,axiom,
    finite4343798906461161616at_nat @ bot_bo228742789529271731at_nat ).

% finite.emptyI
thf(fact_121_cpropagated__sub,axiom,
    ! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b] :
      ( ( relati1591879772219623554ed_a_b @ Q )
     => ( ( member4680049679412964150la_a_b @ Q5 @ ( relational_sub_a_b @ Q ) )
       => ( relati1591879772219623554ed_a_b @ Q5 ) ) ) ).

% cpropagated_sub
thf(fact_122_eval__term_Ocases,axiom,
    ! [X: produc8608687409264118859term_a] :
      ( ! [Sigma: nat > a,C: a] :
          ( X
         != ( produc8917778089171359291term_a @ Sigma @ ( relational_Const_a @ C ) ) )
     => ~ ! [Sigma: nat > a,N: nat] :
            ( X
           != ( produc8917778089171359291term_a @ Sigma @ ( relational_Var_a @ N ) ) ) ) ).

% eval_term.cases
thf(fact_123_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_124_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_125_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1319942482725812455at_nat] :
      ~ ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
          ( Y
         != ( produc9060074326276436823at_nat @ A5 @ B ) ) ).

% old.prod.exhaust
thf(fact_127_surj__pair,axiom,
    ! [P3: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P3
      = ( produc6161850002892822231at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_128_surj__pair,axiom,
    ! [P3: product_prod_nat_nat] :
    ? [X4: nat,Y3: nat] :
      ( P3
      = ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_129_surj__pair,axiom,
    ! [P3: produc3843707927480180839at_nat] :
    ? [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P3
      = ( produc2922128104949294807at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_130_surj__pair,axiom,
    ! [P3: produc1319942482725812455at_nat] :
    ? [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
      ( P3
      = ( produc9060074326276436823at_nat @ X4 @ Y3 ) ) ).

% surj_pair
thf(fact_131_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P3: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_132_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P3: product_prod_nat_nat] :
      ( ! [A5: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_133_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P3: produc3843707927480180839at_nat] :
      ( ! [A5: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A5 @ B ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_134_prod__cases,axiom,
    ! [P: produc1319942482725812455at_nat > $o,P3: produc1319942482725812455at_nat] :
      ( ! [A5: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( P @ ( produc9060074326276436823at_nat @ A5 @ B ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_135_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B2 )
        = ( produc6161850002892822231at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_136_Pair__inject,axiom,
    ! [A: nat,B2: nat,A4: nat,B3: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B2 )
        = ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_137_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B2 )
        = ( produc2922128104949294807at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_138_Pair__inject,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
      ( ( ( produc9060074326276436823at_nat @ A @ B2 )
        = ( produc9060074326276436823at_nat @ A4 @ B3 ) )
     => ~ ( ( A = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_139_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B: nat,C: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_140_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B @ C ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_141_sat_Ocases,axiom,
    ! [X: produc1132964494702330949_nat_a] :
      ( ! [R: b,Ts: list_R6823256787227418703term_a,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
          ( X
         != ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
     => ( ! [B: $o,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
            ( X
           != ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
       => ( ! [X4: nat,T3: relational_term_a,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
              ( X
             != ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T3 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
         => ( ! [Phi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                ( X
               != ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
           => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                  ( X
                 != ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                    ( X
                   != ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) )
               => ~ ! [Z: nat,Phi2: relational_fmla_a_b,I2: product_prod_b_nat > set_list_a,Sigma: nat > a] :
                      ( X
                     != ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ I2 @ Sigma ) ) ) ) ) ) ) ) ) ).

% sat.cases
thf(fact_142_subst_Ocases,axiom,
    ! [X: produc8867654947514737559at_nat] :
      ( ! [T2: $o,X4: nat,Y3: nat] :
          ( X
         != ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T2 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
     => ( ! [P2: b,Ts: list_R6823256787227418703term_a,X4: nat,Y3: nat] :
            ( X
           != ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
       => ( ! [Z: nat,T2: relational_term_a,X4: nat,Y3: nat] :
              ( X
             != ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T2 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
         => ( ! [Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                ( X
               != ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat,Y3: nat] :
                  ( X
                 != ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat,Y3: nat] :
                    ( X
                   != ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
               => ~ ! [Z: nat,Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                      ( X
                     != ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) ) ) ) ) ) ) ) ).

% subst.cases
thf(fact_143_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_144_empty__iff,axiom,
    ! [C2: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ C2 @ bot_bo5327735625951526323at_nat ) ).

% empty_iff
thf(fact_145_empty__iff,axiom,
    ! [C2: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_146_empty__iff,axiom,
    ! [C2: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ C2 @ bot_bo228742789529271731at_nat ) ).

% empty_iff
thf(fact_147_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_148_all__not__in__conv,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( ! [X3: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ X3 @ A2 ) )
      = ( A2 = bot_bo5327735625951526323at_nat ) ) ).

% all_not_in_conv
thf(fact_149_all__not__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ! [X3: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X3 @ A2 ) )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_150_all__not__in__conv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ! [X3: produc3843707927480180839at_nat] :
            ~ ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% all_not_in_conv
thf(fact_151_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_152_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_153_Collect__empty__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( ( collec6321179662152712658at_nat @ P )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: produc3843707927480180839at_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_154_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_155_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X3: product_prod_nat_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_156_empty__Collect__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( bot_bo228742789529271731at_nat
        = ( collec6321179662152712658at_nat @ P ) )
      = ( ! [X3: produc3843707927480180839at_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_157_erase_Oelims,axiom,
    ! [X: relational_fmla_a_b,Xa: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_erase_a_b @ X @ Xa )
        = Y )
     => ( ! [T2: $o] :
            ( ( X
              = ( relational_Bool_a_b @ T2 ) )
           => ( Y
             != ( relational_Bool_a_b @ T2 ) ) )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
             => ~ ( ( ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts ) )
                   => ( Y
                      = ( relational_Bool_a_b @ $false ) ) )
                  & ( ~ ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts ) )
                   => ( Y
                      = ( relational_Pred_b_a @ P2 @ Ts ) ) ) ) )
         => ( ! [Z: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ Z @ T2 ) )
               => ~ ( ( ( T2
                        = ( relational_Var_a @ Z ) )
                     => ( Y
                        = ( relational_Bool_a_b @ $true ) ) )
                    & ( ( T2
                       != ( relational_Var_a @ Z ) )
                     => ( ( ( ( Xa = Z )
                            | ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T2 ) ) )
                         => ( Y
                            = ( relational_Bool_a_b @ $false ) ) )
                        & ( ~ ( ( Xa = Z )
                              | ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T2 ) ) )
                         => ( Y
                            = ( relational_Eq_a_b @ Z @ T2 ) ) ) ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y
                   != ( relational_Neg_a_b @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( Y
                     != ( relational_Conj_a_b @ ( relational_erase_a_b @ Q12 @ Xa ) @ ( relational_erase_a_b @ Q22 @ Xa ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( Y
                       != ( relational_Disj_a_b @ ( relational_erase_a_b @ Q12 @ Xa ) @ ( relational_erase_a_b @ Q22 @ Xa ) ) ) )
                 => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                       => ~ ( ( ( Xa = Z )
                             => ( Y
                                = ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
                            & ( ( Xa != Z )
                             => ( Y
                                = ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% erase.elims
thf(fact_158_erase_Osimps_I3_J,axiom,
    ! [T: relational_term_a,Z3: nat,X: nat] :
      ( ( ( T
          = ( relational_Var_a @ Z3 ) )
       => ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z3 @ T ) @ X )
          = ( relational_Bool_a_b @ $true ) ) )
      & ( ( T
         != ( relational_Var_a @ Z3 ) )
       => ( ( ( ( X = Z3 )
              | ( member_nat @ X @ ( relati6004689760767320788_set_a @ T ) ) )
           => ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z3 @ T ) @ X )
              = ( relational_Bool_a_b @ $false ) ) )
          & ( ~ ( ( X = Z3 )
                | ( member_nat @ X @ ( relati6004689760767320788_set_a @ T ) ) )
           => ( ( relational_erase_a_b @ ( relational_Eq_a_b @ Z3 @ T ) @ X )
              = ( relational_Eq_a_b @ Z3 @ T ) ) ) ) ) ) ).

% erase.simps(3)
thf(fact_159_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_160_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_161_bot__set__def,axiom,
    ( bot_bo228742789529271731at_nat
    = ( collec6321179662152712658at_nat @ bot_bo6829958733157050602_nat_o ) ) ).

% bot_set_def
thf(fact_162_fv__term__setD,axiom,
    ! [N2: nat,T: relational_term_a] :
      ( ( member_nat @ N2 @ ( relati6004689760767320788_set_a @ T ) )
     => ( T
        = ( relational_Var_a @ N2 ) ) ) ).

% fv_term_setD
thf(fact_163_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_164_gen__genempty,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ Z3 @ Q @ G )
     => ( ( G = bot_bo4495933725496725865la_a_b )
       => ( relati5999705594545617851ty_a_b @ Q ) ) ) ).

% gen_genempty
thf(fact_165_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_166_ex__in__conv,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( ? [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A2 ) )
      = ( A2 != bot_bo5327735625951526323at_nat ) ) ).

% ex_in_conv
thf(fact_167_ex__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ? [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 ) )
      = ( A2 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_168_ex__in__conv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ? [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 ) )
      = ( A2 != bot_bo228742789529271731at_nat ) ) ).

% ex_in_conv
thf(fact_169_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_170_equals0I,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ! [Y3: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ Y3 @ A2 )
     => ( A2 = bot_bo5327735625951526323at_nat ) ) ).

% equals0I
thf(fact_171_equals0I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y3 @ A2 )
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_172_equals0I,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ! [Y3: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ Y3 @ A2 )
     => ( A2 = bot_bo228742789529271731at_nat ) ) ).

% equals0I
thf(fact_173_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_174_equals0D,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( A2 = bot_bo5327735625951526323at_nat )
     => ~ ( member8206827879206165904at_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_175_equals0D,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_176_equals0D,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( A2 = bot_bo228742789529271731at_nat )
     => ~ ( member8757157785044589968at_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_177_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_178_emptyE,axiom,
    ! [A: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ A @ bot_bo5327735625951526323at_nat ) ).

% emptyE
thf(fact_179_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_180_emptyE,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ~ ( member8757157785044589968at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% emptyE
thf(fact_181_fv_Osimps_I1_J,axiom,
    ! [Uu2: b,Ts2: list_R6823256787227418703term_a] :
      ( ( relational_fv_a_b @ ( relational_Pred_b_a @ Uu2 @ Ts2 ) )
      = ( relati4569515538964159125_set_a @ Ts2 ) ) ).

% fv.simps(1)
thf(fact_182_erase_Opelims,axiom,
    ! [X: relational_fmla_a_b,Xa: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_erase_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ X @ Xa ) )
       => ( ! [T2: $o] :
              ( ( X
                = ( relational_Bool_a_b @ T2 ) )
             => ( ( Y
                  = ( relational_Bool_a_b @ T2 ) )
               => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Bool_a_b @ T2 ) @ Xa ) ) ) )
         => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P2 @ Ts ) )
               => ( ( ( ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts ) )
                     => ( Y
                        = ( relational_Bool_a_b @ $false ) ) )
                    & ( ~ ( member_nat @ Xa @ ( relati4569515538964159125_set_a @ Ts ) )
                     => ( Y
                        = ( relational_Pred_b_a @ P2 @ Ts ) ) ) )
                 => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ Xa ) ) ) )
           => ( ! [Z: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ Z @ T2 ) )
                 => ( ( ( ( T2
                          = ( relational_Var_a @ Z ) )
                       => ( Y
                          = ( relational_Bool_a_b @ $true ) ) )
                      & ( ( T2
                         != ( relational_Var_a @ Z ) )
                       => ( ( ( ( Xa = Z )
                              | ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T2 ) ) )
                           => ( Y
                              = ( relational_Bool_a_b @ $false ) ) )
                          & ( ~ ( ( Xa = Z )
                                | ( member_nat @ Xa @ ( relati6004689760767320788_set_a @ T2 ) ) )
                           => ( Y
                              = ( relational_Eq_a_b @ Z @ T2 ) ) ) ) ) )
                   => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Eq_a_b @ Z @ T2 ) @ Xa ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y
                        = ( relational_Neg_a_b @ ( relational_erase_a_b @ Q4 @ Xa ) ) )
                     => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Neg_a_b @ Q4 ) @ Xa ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( Y
                          = ( relational_Conj_a_b @ ( relational_erase_a_b @ Q12 @ Xa ) @ ( relational_erase_a_b @ Q22 @ Xa ) ) )
                       => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ Xa ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( Y
                            = ( relational_Disj_a_b @ ( relational_erase_a_b @ Q12 @ Xa ) @ ( relational_erase_a_b @ Q22 @ Xa ) ) )
                         => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ Xa ) ) ) )
                   => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                         => ( ( ( ( Xa = Z )
                               => ( Y
                                  = ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
                              & ( ( Xa != Z )
                               => ( Y
                                  = ( relati591517084277583526ts_a_b @ Z @ ( relational_erase_a_b @ Q4 @ Xa ) ) ) ) )
                           => ~ ( accp_P4351966040938400857_b_nat @ relati5987653437628155313el_a_b @ ( produc4282057684358614024_b_nat @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% erase.pelims
thf(fact_183_ap__cp__erase,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q @ X ) )
          = ( relational_Bool_a_b @ $false ) ) ) ) ).

% ap_cp_erase
thf(fact_184_Set_Ois__empty__def,axiom,
    ( is_emp1662574758705540307at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] : ( A6 = bot_bo2099793752762293965at_nat ) ) ) ).

% Set.is_empty_def
thf(fact_185_Set_Ois__empty__def,axiom,
    ( is_emp7572232645145209145at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] : ( A6 = bot_bo228742789529271731at_nat ) ) ) ).

% Set.is_empty_def
thf(fact_186_cov_H_OConjL,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_cov_a_b2 @ X @ Q1 @ G )
     => ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
          = ( relational_Bool_a_b @ $false ) )
       => ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).

% cov'.ConjL
thf(fact_187_cov_H_OConjR,axiom,
    ! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
      ( ( relational_cov_a_b2 @ X @ Q2 @ G )
     => ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
          = ( relational_Bool_a_b @ $false ) )
       => ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).

% cov'.ConjR
thf(fact_188_cov_H_Ononfree,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( relational_cov_a_b2 @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).

% cov'.nonfree
thf(fact_189_cov_H_OEq__self,axiom,
    ! [X: nat] : ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).

% cov'.Eq_self
thf(fact_190_Eqc,axiom,
    ! [X: nat,C2: a] : ( relational_ap_a_b @ ( relational_Eq_a_b @ X @ ( relational_Const_a @ C2 ) ) ) ).

% Eqc
thf(fact_191_ap_Osimps,axiom,
    ( relational_ap_a_b
    = ( ^ [A3: relational_fmla_a_b] :
          ( ? [P4: b,Ts3: list_R6823256787227418703term_a] :
              ( A3
              = ( relational_Pred_b_a @ P4 @ Ts3 ) )
          | ? [X3: nat,C3: a] :
              ( A3
              = ( relational_Eq_a_b @ X3 @ ( relational_Const_a @ C3 ) ) ) ) ) ) ).

% ap.simps
thf(fact_192_ap_Ocases,axiom,
    ! [A: relational_fmla_a_b] :
      ( ( relational_ap_a_b @ A )
     => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
            ( A
           != ( relational_Pred_b_a @ P2 @ Ts ) )
       => ~ ! [X4: nat,C: a] :
              ( A
             != ( relational_Eq_a_b @ X4 @ ( relational_Const_a @ C ) ) ) ) ) ).

% ap.cases
thf(fact_193_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_194_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_195_Collect__empty__eq__bot,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( ( ( collec6321179662152712658at_nat @ P )
        = bot_bo228742789529271731at_nat )
      = ( P = bot_bo6829958733157050602_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_196_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_197_bot__empty__eq,axiom,
    ( bot_bo7573314457883560170_nat_o
    = ( ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq
thf(fact_198_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_199_bot__empty__eq,axiom,
    ( bot_bo6829958733157050602_nat_o
    = ( ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq
thf(fact_200_gen__Eq,axiom,
    ! [Z3: nat,A: nat,T: relational_term_a,G: set_Re381260168593705685la_a_b] :
      ( ( relational_gen_a_b @ Z3 @ ( relational_Eq_a_b @ A @ T ) @ G )
      = ( ( Z3 = A )
        & ? [C3: a] :
            ( ( T
              = ( relational_Const_a @ C3 ) )
            & ( G
              = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ A @ T ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ).

% gen_Eq
thf(fact_201_sub_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X )
        = Y )
     => ( ! [T2: $o] :
            ( ( X
              = ( relational_Bool_a_b @ T2 ) )
           => ( Y
             != ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
             => ( Y
               != ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) @ bot_bo4495933725496725865la_a_b ) ) )
         => ( ! [X4: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T2 ) )
               => ( Y
                 != ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) @ bot_bo4495933725496725865la_a_b ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y
                   != ( insert7010464514620295119la_a_b @ ( relational_Neg_a_b @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( Y
                     != ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( Y
                       != ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) ) )
                 => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                       => ( Y
                         != ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ) ).

% sub.elims
thf(fact_202_fv__term__set_Oelims,axiom,
    ! [X: relational_term_a,Y: set_nat] :
      ( ( ( relati6004689760767320788_set_a @ X )
        = Y )
     => ( ! [N: nat] :
            ( ( X
              = ( relational_Var_a @ N ) )
           => ( Y
             != ( insert_nat @ N @ bot_bot_set_nat ) ) )
       => ~ ( ? [V2: a] :
                ( X
                = ( relational_Const_a @ V2 ) )
           => ( Y != bot_bot_set_nat ) ) ) ) ).

% fv_term_set.elims
thf(fact_203_gen_H_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen'.intros(2)
thf(fact_204_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_205_singletonI,axiom,
    ! [A: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ A @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) ).

% singletonI
thf(fact_206_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_207_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_208_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
      = ( finite4343798906461161616at_nat @ A2 ) ) ).

% finite_insert
thf(fact_209_finite__insert,axiom,
    ! [A: list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
      = ( finite_finite_list_a @ A2 ) ) ).

% finite_insert
thf(fact_210_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
      = ( finite_finite_nat @ A2 ) ) ).

% finite_insert
thf(fact_211_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( finite6177210948735845034at_nat @ A2 ) ) ).

% finite_insert
thf(fact_212_Un__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A2 = bot_bo2099793752762293965at_nat )
        & ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_213_Un__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ( A2 = bot_bo228742789529271731at_nat )
        & ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_214_finite__Un,axiom,
    ! [F: set_list_a,G: set_list_a] :
      ( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G ) )
      = ( ( finite_finite_list_a @ F )
        & ( finite_finite_list_a @ G ) ) ) ).

% finite_Un
thf(fact_215_finite__Un,axiom,
    ! [F: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
      = ( ( finite_finite_nat @ F )
        & ( finite_finite_nat @ G ) ) ) ).

% finite_Un
thf(fact_216_finite__Un,axiom,
    ! [F: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F @ G ) )
      = ( ( finite6177210948735845034at_nat @ F )
        & ( finite6177210948735845034at_nat @ G ) ) ) ).

% finite_Un
thf(fact_217_finite__Un,axiom,
    ! [F: set_Pr4329608150637261639at_nat,G: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F @ G ) )
      = ( ( finite4343798906461161616at_nat @ F )
        & ( finite4343798906461161616at_nat @ G ) ) ) ).

% finite_Un
thf(fact_218_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_219_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_220_Un__singleton__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A2 @ B4 )
        = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A2 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A2
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A2
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_221_Un__singleton__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A2 @ B4 )
        = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A2 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A2
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A2
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_222_singleton__Un__iff,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A2 @ B4 ) )
      = ( ( ( A2 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A2
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A2
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_223_singleton__Un__iff,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A2 @ B4 ) )
      = ( ( ( A2 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A2
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A2
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_224_sub_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_sub_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q1 ) @ ( relational_sub_a_b @ Q2 ) ) ) ) ).

% sub.simps(5)
thf(fact_225_Un__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_226_Un__empty__left,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_227_Un__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = A2 ) ).

% Un_empty_right
thf(fact_228_Un__empty__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = A2 ) ).

% Un_empty_right
thf(fact_229_finite__UnI,axiom,
    ! [F: set_list_a,G: set_list_a] :
      ( ( finite_finite_list_a @ F )
     => ( ( finite_finite_list_a @ G )
       => ( finite_finite_list_a @ ( sup_sup_set_list_a @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_230_finite__UnI,axiom,
    ! [F: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ F )
     => ( ( finite_finite_nat @ G )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_231_finite__UnI,axiom,
    ! [F: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( ( finite6177210948735845034at_nat @ G )
       => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_232_finite__UnI,axiom,
    ! [F: set_Pr4329608150637261639at_nat,G: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ F )
     => ( ( finite4343798906461161616at_nat @ G )
       => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_233_Un__infinite,axiom,
    ! [S: set_list_a,T4: set_list_a] :
      ( ~ ( finite_finite_list_a @ S )
     => ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T4 ) ) ) ).

% Un_infinite
thf(fact_234_Un__infinite,axiom,
    ! [S: set_nat,T4: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) ) ).

% Un_infinite
thf(fact_235_Un__infinite,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T4 ) ) ) ).

% Un_infinite
thf(fact_236_Un__infinite,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T4: set_Pr4329608150637261639at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T4 ) ) ) ).

% Un_infinite
thf(fact_237_infinite__Un,axiom,
    ! [S: set_list_a,T4: set_list_a] :
      ( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T4 ) ) )
      = ( ~ ( finite_finite_list_a @ S )
        | ~ ( finite_finite_list_a @ T4 ) ) ) ).

% infinite_Un
thf(fact_238_infinite__Un,axiom,
    ! [S: set_nat,T4: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T4 ) ) ) ).

% infinite_Un
thf(fact_239_infinite__Un,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T4 ) ) )
      = ( ~ ( finite6177210948735845034at_nat @ S )
        | ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ).

% infinite_Un
thf(fact_240_infinite__Un,axiom,
    ! [S: set_Pr4329608150637261639at_nat,T4: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T4 ) ) )
      = ( ~ ( finite4343798906461161616at_nat @ S )
        | ~ ( finite4343798906461161616at_nat @ T4 ) ) ) ).

% infinite_Un
thf(fact_241_singletonD,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_242_singletonD,axiom,
    ! [B2: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_243_singletonD,axiom,
    ! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_244_singletonD,axiom,
    ! [B2: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
     => ( B2 = A ) ) ).

% singletonD
thf(fact_245_singleton__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_246_singleton__iff,axiom,
    ! [B2: produc859450856879609959at_nat,A: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ B2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_247_singleton__iff,axiom,
    ! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_248_singleton__iff,axiom,
    ! [B2: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( B2 = A ) ) ).

% singleton_iff
thf(fact_249_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,C2: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C2 @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_250_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,C2: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C2 @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A = C2 )
          & ( B2 = D ) )
        | ( ( A = D )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_251_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ A2 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_252_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ A2 )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_253_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_254_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
     => ( A = B2 ) ) ).

% singleton_inject
thf(fact_255_finite_OinsertI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_256_finite_OinsertI,axiom,
    ! [A2: set_list_a,A: list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_257_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_258_finite_OinsertI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_259_eqs__insert_H,axiom,
    ! [Y: nat,X: nat,Qs: set_Re381260168593705685la_a_b] :
      ( ( Y != X )
     => ( ( relational_eqs_a_b @ X @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ Qs ) )
        = ( insert_nat @ Y @ ( relational_eqs_a_b @ X @ Qs ) ) ) ) ).

% eqs_insert'
thf(fact_260_finite_Ocases,axiom,
    ! [A: set_list_a] :
      ( ( finite_finite_list_a @ A )
     => ( ( A != bot_bot_set_list_a )
       => ~ ! [A7: set_list_a] :
              ( ? [A5: list_a] :
                  ( A
                  = ( insert_list_a @ A5 @ A7 ) )
             => ~ ( finite_finite_list_a @ A7 ) ) ) ) ).

% finite.cases
thf(fact_261_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A5: nat] :
                  ( A
                  = ( insert_nat @ A5 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_262_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A7: set_Pr1261947904930325089at_nat] :
              ( ? [A5: product_prod_nat_nat] :
                  ( A
                  = ( insert8211810215607154385at_nat @ A5 @ A7 ) )
             => ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_263_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A7: set_Pr4329608150637261639at_nat] :
              ( ? [A5: produc3843707927480180839at_nat] :
                  ( A
                  = ( insert9069300056098147895at_nat @ A5 @ A7 ) )
             => ~ ( finite4343798906461161616at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_264_finite_Osimps,axiom,
    ( finite_finite_list_a
    = ( ^ [A3: set_list_a] :
          ( ( A3 = bot_bot_set_list_a )
          | ? [A6: set_list_a,B5: list_a] :
              ( ( A3
                = ( insert_list_a @ B5 @ A6 ) )
              & ( finite_finite_list_a @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_265_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
          ( ( A3 = bot_bot_set_nat )
          | ? [A6: set_nat,B5: nat] :
              ( ( A3
                = ( insert_nat @ B5 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_266_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat] :
          ( ( A3 = bot_bo2099793752762293965at_nat )
          | ? [A6: set_Pr1261947904930325089at_nat,B5: product_prod_nat_nat] :
              ( ( A3
                = ( insert8211810215607154385at_nat @ B5 @ A6 ) )
              & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_267_finite_Osimps,axiom,
    ( finite4343798906461161616at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat] :
          ( ( A3 = bot_bo228742789529271731at_nat )
          | ? [A6: set_Pr4329608150637261639at_nat,B5: produc3843707927480180839at_nat] :
              ( ( A3
                = ( insert9069300056098147895at_nat @ B5 @ A6 ) )
              & ( finite4343798906461161616at_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_268_finite__induct,axiom,
    ! [F: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [X4: produc859450856879609959at_nat,F2: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ F2 )
             => ( ~ ( member8206827879206165904at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert5050368324300391991at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_269_finite__induct,axiom,
    ! [F: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,F2: set_list_a] :
              ( ( finite_finite_list_a @ F2 )
             => ( ~ ( member_list_a @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_270_finite__induct,axiom,
    ! [F: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F2: set_nat] :
              ( ( finite_finite_nat @ F2 )
             => ( ~ ( member_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_271_finite__induct,axiom,
    ! [F: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F2: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F2 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_272_finite__induct,axiom,
    ! [F: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat,F2: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F2 )
             => ( ~ ( member8757157785044589968at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert9069300056098147895at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_273_finite__ne__induct,axiom,
    ! [F: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F )
     => ( ( F != bot_bo5327735625951526323at_nat )
       => ( ! [X4: produc859450856879609959at_nat] : ( P @ ( insert5050368324300391991at_nat @ X4 @ bot_bo5327735625951526323at_nat ) )
         => ( ! [X4: produc859450856879609959at_nat,F2: set_Pr8693737435421807431at_nat] :
                ( ( finite4392333629123659920at_nat @ F2 )
               => ( ( F2 != bot_bo5327735625951526323at_nat )
                 => ( ~ ( member8206827879206165904at_nat @ X4 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert5050368324300391991at_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_274_finite__ne__induct,axiom,
    ! [F: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F )
     => ( ( F != bot_bot_set_list_a )
       => ( ! [X4: list_a] : ( P @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) )
         => ( ! [X4: list_a,F2: set_list_a] :
                ( ( finite_finite_list_a @ F2 )
               => ( ( F2 != bot_bot_set_list_a )
                 => ( ~ ( member_list_a @ X4 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_275_finite__ne__induct,axiom,
    ! [F: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( F != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( F2 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_276_finite__ne__induct,axiom,
    ! [F: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( ( F != bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X4: product_prod_nat_nat,F2: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F2 )
               => ( ( F2 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X4 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert8211810215607154385at_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_277_finite__ne__induct,axiom,
    ! [F: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F )
     => ( ( F != bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
         => ( ! [X4: produc3843707927480180839at_nat,F2: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F2 )
               => ( ( F2 != bot_bo228742789529271731at_nat )
                 => ( ~ ( member8757157785044589968at_nat @ X4 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert9069300056098147895at_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_278_infinite__finite__induct,axiom,
    ! [P: set_Pr8693737435421807431at_nat > $o,A2: set_Pr8693737435421807431at_nat] :
      ( ! [A7: set_Pr8693737435421807431at_nat] :
          ( ~ ( finite4392333629123659920at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [X4: produc859450856879609959at_nat,F2: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ F2 )
             => ( ~ ( member8206827879206165904at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert5050368324300391991at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_279_infinite__finite__induct,axiom,
    ! [P: set_list_a > $o,A2: set_list_a] :
      ( ! [A7: set_list_a] :
          ( ~ ( finite_finite_list_a @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,F2: set_list_a] :
              ( ( finite_finite_list_a @ F2 )
             => ( ~ ( member_list_a @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert_list_a @ X4 @ F2 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_280_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F2: set_nat] :
              ( ( finite_finite_nat @ F2 )
             => ( ~ ( member_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_281_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A7: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F2: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F2 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_282_infinite__finite__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
      ( ! [A7: set_Pr4329608150637261639at_nat] :
          ( ~ ( finite4343798906461161616at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat,F2: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F2 )
             => ( ~ ( member8757157785044589968at_nat @ X4 @ F2 )
               => ( ( P @ F2 )
                 => ( P @ ( insert9069300056098147895at_nat @ X4 @ F2 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_283_fmla_Osimps_I131_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relati8924981150291758614la_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) )
      = ( sup_sup_set_b @ ( relati8924981150291758614la_a_b @ X51 ) @ ( relati8924981150291758614la_a_b @ X52 ) ) ) ).

% fmla.simps(131)
thf(fact_284_cov_H_OConj,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
      ( ( relational_cov_a_b2 @ X @ Q1 @ G1 )
     => ( ( relational_cov_a_b2 @ X @ Q2 @ G2 )
       => ( relational_cov_a_b2 @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) ) ) ) ).

% cov'.Conj
thf(fact_285_sub_Osimps_I3_J,axiom,
    ! [X: nat,T: relational_term_a] :
      ( ( relational_sub_a_b @ ( relational_Eq_a_b @ X @ T ) )
      = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ T ) @ bot_bo4495933725496725865la_a_b ) ) ).

% sub.simps(3)
thf(fact_286_fv__term__set_Osimps_I1_J,axiom,
    ! [N2: nat] :
      ( ( relati6004689760767320788_set_a @ ( relational_Var_a @ N2 ) )
      = ( insert_nat @ N2 @ bot_bot_set_nat ) ) ).

% fv_term_set.simps(1)
thf(fact_287_cov_H_OEqR,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).

% cov'.EqR
thf(fact_288_cov_H_OEqL,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( relational_cov_a_b2 @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).

% cov'.EqL
thf(fact_289_qp__gen,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_qp_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% qp_gen
thf(fact_290_gen_Ointros_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_gen_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% gen.intros(2)
thf(fact_291_cov_H_Oap,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_cov_a_b2 @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% cov'.ap
thf(fact_292_sub_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_Re381260168593705685la_a_b] :
      ( ( ( relational_sub_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ X )
       => ( ! [T2: $o] :
              ( ( X
                = ( relational_Bool_a_b @ T2 ) )
             => ( ( Y
                  = ( insert7010464514620295119la_a_b @ ( relational_Bool_a_b @ T2 ) @ bot_bo4495933725496725865la_a_b ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Bool_a_b @ T2 ) ) ) )
         => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P2 @ Ts ) )
               => ( ( Y
                    = ( insert7010464514620295119la_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) @ bot_bo4495933725496725865la_a_b ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ) )
           => ( ! [X4: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T2 ) )
                 => ( ( Y
                      = ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) @ bot_bo4495933725496725865la_a_b ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y
                        = ( 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 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( Y
                          = ( insert7010464514620295119la_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( Y
                            = ( insert7010464514620295119la_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ ( relational_sub_a_b @ Q12 ) @ ( relational_sub_a_b @ Q22 ) ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
                   => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                         => ( ( Y
                              = ( insert7010464514620295119la_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ ( relational_sub_a_b @ Q4 ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7309537865537208983el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sub.pelims
thf(fact_293_sup__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_294_sup__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_295_sup__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% sup_bot_right
thf(fact_296_sup__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% sup_bot_right
thf(fact_297_bot__eq__sup__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_298_bot__eq__sup__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ X @ Y ) )
      = ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_299_sup__eq__bot__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ( X = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_300_sup__eq__bot__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ X @ Y )
        = bot_bo228742789529271731at_nat )
      = ( ( X = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_301_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_302_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B2 )
        = bot_bo228742789529271731at_nat )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_303_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_304_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_305_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ A @ B2 ) )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B2 = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_306_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A @ B2 ) )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_307_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_308_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_309_fv_Osimps_I6_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Disj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(6)
thf(fact_310_fv_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) )
      = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi ) @ ( relational_fv_a_b @ Psi2 ) ) ) ).

% fv.simps(5)
thf(fact_311_fv_Osimps_I3_J,axiom,
    ! [X: nat,T5: relational_term_a] :
      ( ( relational_fv_a_b @ ( relational_Eq_a_b @ X @ T5 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T5 ) ) ) ).

% fv.simps(3)
thf(fact_312_fv_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X )
        = Y )
     => ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ Uu @ Ts ) )
           => ( Y
             != ( relati4569515538964159125_set_a @ Ts ) ) )
       => ( ( ? [B: $o] :
                ( X
                = ( relational_Bool_a_b @ B ) )
           => ( Y != bot_bot_set_nat ) )
         => ( ! [X4: nat,T3: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T3 ) )
               => ( Y
                 != ( sup_sup_set_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T3 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                   != ( relational_fv_a_b @ Phi2 ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                     != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                       != ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ( Y
                         != ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ) ).

% fv.elims
thf(fact_313_fv_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_nat] :
      ( ( ( relational_fv_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ X )
       => ( ! [Uu: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ Uu @ Ts ) )
             => ( ( Y
                  = ( relati4569515538964159125_set_a @ Ts ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Pred_b_a @ Uu @ Ts ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( Y = bot_bot_set_nat )
                 => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
           => ( ! [X4: nat,T3: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T3 ) )
                 => ( ( Y
                      = ( sup_sup_set_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) @ ( relati6004689760767320788_set_a @ T3 ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Eq_a_b @ X4 @ T3 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( relational_fv_a_b @ Phi2 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Neg_a_b @ Phi2 ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Conj_a_b @ Phi2 @ Psi ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( sup_sup_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( relational_fv_a_b @ Psi ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relational_Disj_a_b @ Phi2 @ Psi ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( Y
                              = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi2 ) @ ( insert_nat @ Z @ bot_bot_set_nat ) ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5703530512245835757el_a_b @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% fv.pelims
thf(fact_314_the__elem__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_315_the__elem__eq,axiom,
    ! [X: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_316_is__singletonI,axiom,
    ! [X: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_317_is__singletonI,axiom,
    ! [X: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_318_csts__term_Oelims,axiom,
    ! [X: relati8250976376238479564at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( relati7295153245664512040at_nat @ X )
        = Y )
     => ( ( ? [X4: nat] :
              ( X
              = ( relati1023757457726692818at_nat @ X4 ) )
         => ( Y != bot_bo2099793752762293965at_nat ) )
       => ~ ! [C: product_prod_nat_nat] :
              ( ( X
                = ( relati6667065761635354038at_nat @ C ) )
             => ( Y
               != ( insert8211810215607154385at_nat @ C @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% csts_term.elims
thf(fact_319_csts__term_Oelims,axiom,
    ! [X: relati2597207553810441522at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( relati8224766339378407694at_nat @ X )
        = Y )
     => ( ( ? [X4: nat] :
              ( X
              = ( relati2721891637006993336at_nat @ X4 ) )
         => ( Y != bot_bo228742789529271731at_nat ) )
       => ~ ! [C: produc3843707927480180839at_nat] :
              ( ( X
                = ( relati3582544344963639708at_nat @ C ) )
             => ( Y
               != ( insert9069300056098147895at_nat @ C @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% csts_term.elims
thf(fact_320_csts__term_Oelims,axiom,
    ! [X: relational_term_a,Y: set_a] :
      ( ( ( relati6638259395341848997term_a @ X )
        = Y )
     => ( ( ? [X4: nat] :
              ( X
              = ( relational_Var_a @ X4 ) )
         => ( Y != bot_bot_set_a ) )
       => ~ ! [C: a] :
              ( ( X
                = ( relational_Const_a @ C ) )
             => ( Y
               != ( insert_a @ C @ bot_bot_set_a ) ) ) ) ) ).

% csts_term.elims
thf(fact_321_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ bot_bo2099793752762293965at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_322_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X @ bot_bo228742789529271731at_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_323_Diff__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = A2 ) ).

% Diff_empty
thf(fact_324_Diff__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = A2 ) ).

% Diff_empty
thf(fact_325_empty__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_326_empty__Diff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ bot_bo228742789529271731at_nat @ A2 )
      = bot_bo228742789529271731at_nat ) ).

% empty_Diff
thf(fact_327_Diff__cancel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_328_Diff__cancel,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ A2 )
      = bot_bo228742789529271731at_nat ) ).

% Diff_cancel
thf(fact_329_finite__Diff,axiom,
    ! [A2: set_list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) ) ) ).

% finite_Diff
thf(fact_330_finite__Diff,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ).

% finite_Diff
thf(fact_331_finite__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B4 ) ) ) ).

% finite_Diff
thf(fact_332_finite__Diff2,axiom,
    ! [B4: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) )
        = ( finite_finite_list_a @ A2 ) ) ) ).

% finite_Diff2
thf(fact_333_finite__Diff2,axiom,
    ! [B4: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B4 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_334_finite__Diff2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B4 ) )
        = ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_335_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_336_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_337_finite__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B4 ) ) )
      = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_338_finite__Diff__insert,axiom,
    ! [A2: set_list_a,A: list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B4 ) ) )
      = ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_339_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B4: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B4 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_340_finite__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B4 ) ) )
      = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_341_Diff__infinite__finite,axiom,
    ! [T4: set_list_a,S: set_list_a] :
      ( ( finite_finite_list_a @ T4 )
     => ( ~ ( finite_finite_list_a @ S )
       => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_342_Diff__infinite__finite,axiom,
    ! [T4: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T4 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_343_Diff__infinite__finite,axiom,
    ! [T4: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ T4 )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_344_finite__csts__term,axiom,
    ! [T: relati383187284260386255list_a] : ( finite_finite_list_a @ ( relati1612993689924349611list_a @ T ) ) ).

% finite_csts_term
thf(fact_345_finite__csts__term,axiom,
    ! [T: relational_term_nat] : ( finite_finite_nat @ ( relati694035416245573993rm_nat @ T ) ) ).

% finite_csts_term
thf(fact_346_finite__csts__term,axiom,
    ! [T: relati8250976376238479564at_nat] : ( finite6177210948735845034at_nat @ ( relati7295153245664512040at_nat @ T ) ) ).

% finite_csts_term
thf(fact_347_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B4 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_348_Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B4 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_349_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_350_insert__Diff,axiom,
    ! [A: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A2 )
     => ( ( insert5050368324300391991at_nat @ A @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ A @ bot_bo5327735625951526323at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_351_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_352_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_353_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_354_Diff__insert2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_355_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_356_Diff__insert__absorb,axiom,
    ! [X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X @ A2 )
     => ( ( minus_8321449233255521966at_nat @ ( insert5050368324300391991at_nat @ X @ A2 ) @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_357_Diff__insert__absorb,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X @ A2 ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_358_Diff__insert__absorb,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X @ A2 ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_359_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_360_is__singleton__the__elem,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A6 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_361_is__singletonI_H,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ( member_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_362_is__singletonI_H,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ( A2 != bot_bo5327735625951526323at_nat )
     => ( ! [X4: produc859450856879609959at_nat,Y3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A2 )
           => ( ( member8206827879206165904at_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_sin3461952106595595099at_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_363_is__singletonI_H,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ( ( member8440522571783428010at_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_sin2850979758926227957at_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_364_is__singletonI_H,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( A2 != bot_bo228742789529271731at_nat )
     => ( ! [X4: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A2 )
           => ( ( member8757157785044589968at_nat @ Y3 @ A2 )
             => ( X4 = Y3 ) ) )
       => ( is_sin2937591304547752795at_nat @ A2 ) ) ) ).

% is_singletonI'
thf(fact_365_finite__empty__induct,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: produc859450856879609959at_nat,A7: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ A7 )
             => ( ( member8206827879206165904at_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_8321449233255521966at_nat @ A7 @ ( insert5050368324300391991at_nat @ A5 @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
         => ( P @ bot_bo5327735625951526323at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_366_finite__empty__induct,axiom,
    ! [A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: list_a,A7: set_list_a] :
              ( ( finite_finite_list_a @ A7 )
             => ( ( member_list_a @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ A5 @ bot_bot_set_list_a ) ) ) ) ) )
         => ( P @ bot_bot_set_list_a ) ) ) ) ).

% finite_empty_induct
thf(fact_367_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_368_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( member8440522571783428010at_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A5 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_369_finite__empty__induct,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( member8757157785044589968at_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ A5 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
         => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_370_infinite__coinduct,axiom,
    ! [X5: set_list_a > $o,A2: set_list_a] :
      ( ( X5 @ A2 )
     => ( ! [A7: set_list_a] :
            ( ( X5 @ A7 )
           => ? [X6: list_a] :
                ( ( member_list_a @ X6 @ A7 )
                & ( ( X5 @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X6 @ bot_bot_set_list_a ) ) )
                  | ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X6 @ bot_bot_set_list_a ) ) ) ) ) )
       => ~ ( finite_finite_list_a @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_371_infinite__coinduct,axiom,
    ! [X5: set_nat > $o,A2: set_nat] :
      ( ( X5 @ A2 )
     => ( ! [A7: set_nat] :
            ( ( X5 @ A7 )
           => ? [X6: nat] :
                ( ( member_nat @ X6 @ A7 )
                & ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_372_infinite__coinduct,axiom,
    ! [X5: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ( X5 @ A2 )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( X5 @ A7 )
           => ? [X6: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X6 @ A7 )
                & ( ( X5 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_373_infinite__coinduct,axiom,
    ! [X5: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
      ( ( X5 @ A2 )
     => ( ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( X5 @ A7 )
           => ? [X6: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X6 @ A7 )
                & ( ( X5 @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) )
                  | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
       => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_374_infinite__remove,axiom,
    ! [S: set_list_a,A: list_a] :
      ( ~ ( finite_finite_list_a @ S )
     => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).

% infinite_remove
thf(fact_375_infinite__remove,axiom,
    ! [S: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_376_infinite__remove,axiom,
    ! [S: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_377_infinite__remove,axiom,
    ! [S: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S )
     => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% infinite_remove
thf(fact_378_csts__term_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( relati7295153245664512040at_nat @ ( relati1023757457726692818at_nat @ X ) )
      = bot_bo2099793752762293965at_nat ) ).

% csts_term.simps(1)
thf(fact_379_csts__term_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( relati8224766339378407694at_nat @ ( relati2721891637006993336at_nat @ X ) )
      = bot_bo228742789529271731at_nat ) ).

% csts_term.simps(1)
thf(fact_380_csts__term_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( relati6638259395341848997term_a @ ( relational_Var_a @ X ) )
      = bot_bot_set_a ) ).

% csts_term.simps(1)
thf(fact_381_fv_Osimps_I7_J,axiom,
    ! [Z3: nat,Phi: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati591517084277583526ts_a_b @ Z3 @ Phi ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Phi ) @ ( insert_nat @ Z3 @ bot_bot_set_nat ) ) ) ).

% fv.simps(7)
thf(fact_382_is__singletonE,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A2 )
     => ~ ! [X4: product_prod_nat_nat] :
            ( A2
           != ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_383_is__singletonE,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A2 )
     => ~ ! [X4: produc3843707927480180839at_nat] :
            ( A2
           != ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_384_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [X3: product_prod_nat_nat] :
          ( A6
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_385_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
        ? [X3: produc3843707927480180839at_nat] :
          ( A6
          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_386_csts__term_Osimps_I2_J,axiom,
    ! [C2: product_prod_nat_nat] :
      ( ( relati7295153245664512040at_nat @ ( relati6667065761635354038at_nat @ C2 ) )
      = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ).

% csts_term.simps(2)
thf(fact_387_csts__term_Osimps_I2_J,axiom,
    ! [C2: produc3843707927480180839at_nat] :
      ( ( relati8224766339378407694at_nat @ ( relati3582544344963639708at_nat @ C2 ) )
      = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ).

% csts_term.simps(2)
thf(fact_388_eqs__minus,axiom,
    ! [X: nat,G: set_Re381260168593705685la_a_b,Y: nat] :
      ( ( relational_eqs_a_b @ X @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
      = ( minus_minus_set_nat @ ( relational_eqs_a_b @ X @ G ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ).

% eqs_minus
thf(fact_389_insert__remove__id,axiom,
    ! [X: nat,X5: set_nat] :
      ( ( member_nat @ X @ X5 )
     => ( X5
        = ( insert_nat @ X @ ( minus_minus_set_nat @ X5 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% insert_remove_id
thf(fact_390_insert__remove__id,axiom,
    ! [X: produc859450856879609959at_nat,X5: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X @ X5 )
     => ( X5
        = ( insert5050368324300391991at_nat @ X @ ( minus_8321449233255521966at_nat @ X5 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% insert_remove_id
thf(fact_391_insert__remove__id,axiom,
    ! [X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ X5 )
     => ( X5
        = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ X5 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% insert_remove_id
thf(fact_392_insert__remove__id,axiom,
    ! [X: produc3843707927480180839at_nat,X5: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X @ X5 )
     => ( X5
        = ( insert9069300056098147895at_nat @ X @ ( minus_3314409938677909166at_nat @ X5 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% insert_remove_id
thf(fact_393_remove__def,axiom,
    ( remove6466555014256735590at_nat
    = ( ^ [X3: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% remove_def
thf(fact_394_remove__def,axiom,
    ( remove5300788332298244428at_nat
    = ( ^ [X3: produc3843707927480180839at_nat,A6: set_Pr4329608150637261639at_nat] : ( minus_3314409938677909166at_nat @ A6 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% remove_def
thf(fact_395_fv__erase,axiom,
    ! [Q: relational_fmla_a_b,X: nat] : ( ord_less_eq_set_nat @ ( relational_fv_a_b @ ( relational_erase_a_b @ Q @ X ) ) @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% fv_erase
thf(fact_396_subset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_397_subset__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% subset_empty
thf(fact_398_empty__subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).

% empty_subsetI
thf(fact_399_empty__subsetI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A2 ) ).

% empty_subsetI
thf(fact_400_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A2 )
        = ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B2 )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_401_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ A2 )
        = ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) )
      = ( ( A = B2 )
        & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_402_singleton__insert__inj__eq,axiom,
    ! [B2: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_403_singleton__insert__inj__eq,axiom,
    ! [B2: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A @ A2 ) )
      = ( ( A = B2 )
        & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_404_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A2 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_405_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ord_le1268244103169919719at_nat @ A2 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_406_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_407_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
     => ( A = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_408_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_409_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_410_bot_Oextremum__unique,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ bot_bo228742789529271731at_nat )
      = ( A = bot_bo228742789529271731at_nat ) ) ).

% bot.extremum_unique
thf(fact_411_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_412_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_413_bot_Oextremum,axiom,
    ! [A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ bot_bo228742789529271731at_nat @ A ) ).

% bot.extremum
thf(fact_414_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_415_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_416_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_417_rev__finite__subset,axiom,
    ! [B4: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B4 )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_418_rev__finite__subset,axiom,
    ! [B4: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A2 @ B4 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_419_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B4 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_420_infinite__super,axiom,
    ! [S: set_list_a,T4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S @ T4 )
     => ( ~ ( finite_finite_list_a @ S )
       => ~ ( finite_finite_list_a @ T4 ) ) ) ).

% infinite_super
thf(fact_421_infinite__super,axiom,
    ! [S: set_nat,T4: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T4 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T4 ) ) ) ).

% infinite_super
thf(fact_422_infinite__super,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ T4 )
     => ( ~ ( finite6177210948735845034at_nat @ S )
       => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ).

% infinite_super
thf(fact_423_finite__subset,axiom,
    ! [A2: set_list_a,B4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B4 )
     => ( ( finite_finite_list_a @ B4 )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% finite_subset
thf(fact_424_finite__subset,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_425_finite__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_426_equiv__Conj__cong,axiom,
    ! [Q1: relational_fmla_a_b,Q14: relational_fmla_a_b,Q2: relational_fmla_a_b,Q24: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q1 @ Q14 )
     => ( ( relational_equiv_a_b @ Q2 @ Q24 )
       => ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( relational_Conj_a_b @ Q14 @ Q24 ) ) ) ) ).

% equiv_Conj_cong
thf(fact_427_Var__eq__subst__iff,axiom,
    ! [Z3: nat,T: relational_term_a,X: nat,Y: nat] :
      ( ( ( relational_Var_a @ Z3 )
        = ( relati7175845559408349773term_a @ T @ X @ Y ) )
      = ( ( ( Z3 = X )
         => ( ( X = Y )
            & ( T
              = ( relational_Var_a @ X ) ) ) )
        & ( ( Z3 != X )
         => ( ( ( Z3 = Y )
             => ( ( T
                  = ( relational_Var_a @ X ) )
                | ( T
                  = ( relational_Var_a @ Y ) ) ) )
            & ( ( Z3 != Y )
             => ( T
                = ( relational_Var_a @ Z3 ) ) ) ) ) ) ) ).

% Var_eq_subst_iff
thf(fact_428_subst__term_Osimps_I1_J,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( relati7175845559408349773term_a @ ( relational_Var_a @ Z3 ) @ X @ Y )
      = ( relational_Var_a @ ( if_nat @ ( X = Z3 ) @ Y @ Z3 ) ) ) ).

% subst_term.simps(1)
thf(fact_429_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_430_diff__shunt__var,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( minus_3314409938677909166at_nat @ X @ Y )
        = bot_bo228742789529271731at_nat )
      = ( ord_le1268244103169919719at_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_431_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_432_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_433_subset__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X5 = bot_bo2099793752762293965at_nat )
        | ( X5
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_434_subset__singleton__iff,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X5 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( ( X5 = bot_bo228742789529271731at_nat )
        | ( X5
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_435_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) )
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2
          = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_436_subset__singletonD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) )
     => ( ( A2 = bot_bo228742789529271731at_nat )
        | ( A2
          = ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_437_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_438_gen__fv,axiom,
    ! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_gen_a_b @ X @ Q @ G )
     => ( ( member4680049679412964150la_a_b @ Qqp @ G )
       => ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
          & ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).

% gen_fv
thf(fact_439_Exists__nonfree__equiv,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( relational_equiv_a_b @ ( relati591517084277583526ts_a_b @ X @ Q ) @ Q ) ) ).

% Exists_nonfree_equiv
thf(fact_440_subst__term_Oelims,axiom,
    ! [X: relational_term_a,Xa: nat,Xb: nat,Y: relational_term_a] :
      ( ( ( relati7175845559408349773term_a @ X @ Xa @ Xb )
        = Y )
     => ( ! [Z: nat] :
            ( ( X
              = ( relational_Var_a @ Z ) )
           => ( Y
             != ( relational_Var_a @ ( if_nat @ ( Xa = Z ) @ Xb @ Z ) ) ) )
       => ~ ! [C: a] :
              ( ( X
                = ( relational_Const_a @ C ) )
             => ( Y
               != ( relational_Const_a @ C ) ) ) ) ) ).

% subst_term.elims
thf(fact_441_finite__subset__induct,axiom,
    ! [F: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F )
     => ( ( ord_le3000389064537975527at_nat @ F @ A2 )
       => ( ( P @ bot_bo5327735625951526323at_nat )
         => ( ! [A5: produc859450856879609959at_nat,F2: set_Pr8693737435421807431at_nat] :
                ( ( finite4392333629123659920at_nat @ F2 )
               => ( ( member8206827879206165904at_nat @ A5 @ A2 )
                 => ( ~ ( member8206827879206165904at_nat @ A5 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert5050368324300391991at_nat @ A5 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_442_finite__subset__induct,axiom,
    ! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F )
     => ( ( ord_le8861187494160871172list_a @ F @ A2 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A5: list_a,F2: set_list_a] :
                ( ( finite_finite_list_a @ F2 )
               => ( ( member_list_a @ A5 @ A2 )
                 => ( ~ ( member_list_a @ A5 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert_list_a @ A5 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_443_finite__subset__induct,axiom,
    ! [F: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ~ ( member_nat @ A5 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert_nat @ A5 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_444_finite__subset__induct,axiom,
    ! [F: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( ( ord_le3146513528884898305at_nat @ F @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F2: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F2 )
               => ( ( member8440522571783428010at_nat @ A5 @ A2 )
                 => ( ~ ( member8440522571783428010at_nat @ A5 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert8211810215607154385at_nat @ A5 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_445_finite__subset__induct,axiom,
    ! [F: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F )
     => ( ( ord_le1268244103169919719at_nat @ F @ A2 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A5: produc3843707927480180839at_nat,F2: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F2 )
               => ( ( member8757157785044589968at_nat @ A5 @ A2 )
                 => ( ~ ( member8757157785044589968at_nat @ A5 @ F2 )
                   => ( ( P @ F2 )
                     => ( P @ ( insert9069300056098147895at_nat @ A5 @ F2 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_446_finite__subset__induct_H,axiom,
    ! [F: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ F )
     => ( ( ord_le3000389064537975527at_nat @ F @ A2 )
       => ( ( P @ bot_bo5327735625951526323at_nat )
         => ( ! [A5: produc859450856879609959at_nat,F2: set_Pr8693737435421807431at_nat] :
                ( ( finite4392333629123659920at_nat @ F2 )
               => ( ( member8206827879206165904at_nat @ A5 @ A2 )
                 => ( ( ord_le3000389064537975527at_nat @ F2 @ A2 )
                   => ( ~ ( member8206827879206165904at_nat @ A5 @ F2 )
                     => ( ( P @ F2 )
                       => ( P @ ( insert5050368324300391991at_nat @ A5 @ F2 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_447_finite__subset__induct_H,axiom,
    ! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ F )
     => ( ( ord_le8861187494160871172list_a @ F @ A2 )
       => ( ( P @ bot_bot_set_list_a )
         => ( ! [A5: list_a,F2: set_list_a] :
                ( ( finite_finite_list_a @ F2 )
               => ( ( member_list_a @ A5 @ A2 )
                 => ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
                   => ( ~ ( member_list_a @ A5 @ F2 )
                     => ( ( P @ F2 )
                       => ( P @ ( insert_list_a @ A5 @ F2 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_448_finite__subset__induct_H,axiom,
    ! [F: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F2 @ A2 )
                   => ( ~ ( member_nat @ A5 @ F2 )
                     => ( ( P @ F2 )
                       => ( P @ ( insert_nat @ A5 @ F2 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_449_finite__subset__induct_H,axiom,
    ! [F: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( ( ord_le3146513528884898305at_nat @ F @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A5: product_prod_nat_nat,F2: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F2 )
               => ( ( member8440522571783428010at_nat @ A5 @ A2 )
                 => ( ( ord_le3146513528884898305at_nat @ F2 @ A2 )
                   => ( ~ ( member8440522571783428010at_nat @ A5 @ F2 )
                     => ( ( P @ F2 )
                       => ( P @ ( insert8211810215607154385at_nat @ A5 @ F2 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_450_finite__subset__induct_H,axiom,
    ! [F: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F )
     => ( ( ord_le1268244103169919719at_nat @ F @ A2 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A5: produc3843707927480180839at_nat,F2: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F2 )
               => ( ( member8757157785044589968at_nat @ A5 @ A2 )
                 => ( ( ord_le1268244103169919719at_nat @ F2 @ A2 )
                   => ( ~ ( member8757157785044589968at_nat @ A5 @ F2 )
                     => ( ( P @ F2 )
                       => ( P @ ( insert9069300056098147895at_nat @ A5 @ F2 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_451_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_452_subset__insert__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ B4 ) )
      = ( ( ( member8206827879206165904at_nat @ X @ A2 )
         => ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) @ B4 ) )
        & ( ~ ( member8206827879206165904at_nat @ X @ A2 )
         => ( ord_le3000389064537975527at_nat @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_453_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B4 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ A2 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
         => ( ord_le3146513528884898305at_nat @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_454_subset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B4 ) )
      = ( ( ( member8757157785044589968at_nat @ X @ A2 )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B4 ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
         => ( ord_le1268244103169919719at_nat @ A2 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_455_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B4 )
     => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_456_Diff__single__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B4 )
     => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_457_finite__remove__induct,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ B4 )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [A7: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ A7 )
             => ( ( A7 != bot_bo5327735625951526323at_nat )
               => ( ( ord_le3000389064537975527at_nat @ A7 @ B4 )
                 => ( ! [X6: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ X6 @ A7 )
                       => ( P @ ( minus_8321449233255521966at_nat @ A7 @ ( insert5050368324300391991at_nat @ X6 @ bot_bo5327735625951526323at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_458_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 )
                 => ( ! [X6: list_a] :
                        ( ( member_list_a @ X6 @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X6 @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_459_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 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_460_finite__remove__induct,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B4 )
                 => ( ! [X6: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X6 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_461_finite__remove__induct,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B4 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B4 )
                 => ( ! [X6: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X6 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_462_remove__induct,axiom,
    ! [P: set_Pr8693737435421807431at_nat > $o,B4: set_Pr8693737435421807431at_nat] :
      ( ( P @ bot_bo5327735625951526323at_nat )
     => ( ( ~ ( finite4392333629123659920at_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ A7 )
             => ( ( A7 != bot_bo5327735625951526323at_nat )
               => ( ( ord_le3000389064537975527at_nat @ A7 @ B4 )
                 => ( ! [X6: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ X6 @ A7 )
                       => ( P @ ( minus_8321449233255521966at_nat @ A7 @ ( insert5050368324300391991at_nat @ X6 @ bot_bo5327735625951526323at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_463_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 )
                 => ( ! [X6: list_a] :
                        ( ( member_list_a @ X6 @ A7 )
                       => ( P @ ( minus_646659088055828811list_a @ A7 @ ( insert_list_a @ X6 @ bot_bot_set_list_a ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_464_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 )
                 => ( ! [X6: nat] :
                        ( ( member_nat @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_465_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B4: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B4 )
                 => ( ! [X6: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X6 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_466_remove__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,B4: set_Pr4329608150637261639at_nat] :
      ( ( P @ bot_bo228742789529271731at_nat )
     => ( ( ~ ( finite4343798906461161616at_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B4 )
                 => ( ! [X6: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X6 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X6 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_467_finite__ranking__induct,axiom,
    ! [S: set_Pr8693737435421807431at_nat,P: set_Pr8693737435421807431at_nat > $o,F3: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( P @ bot_bo5327735625951526323at_nat )
       => ( ! [X4: produc859450856879609959at_nat,S2: set_Pr8693737435421807431at_nat] :
              ( ( finite4392333629123659920at_nat @ S2 )
             => ( ! [Y5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y5 @ S2 )
                   => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X4 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert5050368324300391991at_nat @ X4 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_468_finite__ranking__induct,axiom,
    ! [S: set_list_a,P: set_list_a > $o,F3: list_a > nat] :
      ( ( finite_finite_list_a @ S )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [X4: list_a,S2: set_list_a] :
              ( ( finite_finite_list_a @ S2 )
             => ( ! [Y5: list_a] :
                    ( ( member_list_a @ Y5 @ S2 )
                   => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X4 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_list_a @ X4 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_469_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S2: set_nat] :
              ( ( finite_finite_nat @ S2 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S2 )
                   => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X4 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_nat @ X4 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_470_finite__ranking__induct,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,F3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,S2: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ S2 )
             => ( ! [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ S2 )
                   => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X4 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_471_finite__ranking__induct,axiom,
    ! [S: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o,F3: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat,S2: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ S2 )
             => ( ! [Y5: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Y5 @ S2 )
                   => ( ord_less_eq_nat @ ( F3 @ Y5 ) @ ( F3 @ X4 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert9069300056098147895at_nat @ X4 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_472_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_473_qps__minus,axiom,
    ! [G: set_Re381260168593705685la_a_b,X: nat,Y: nat] :
      ( ( relational_qps_a_b @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
      = ( relational_qps_a_b @ G ) ) ).

% qps_minus
thf(fact_474_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] :
          ~ ( member_nat @ X4 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_475_subset__emptyI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] :
          ~ ( member8206827879206165904at_nat @ X4 @ A2 )
     => ( ord_le3000389064537975527at_nat @ A2 @ bot_bo5327735625951526323at_nat ) ) ).

% subset_emptyI
thf(fact_476_subset__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X4 @ A2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_477_subset__emptyI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ~ ( member8757157785044589968at_nat @ X4 @ A2 )
     => ( ord_le1268244103169919719at_nat @ A2 @ bot_bo228742789529271731at_nat ) ) ).

% subset_emptyI
thf(fact_478_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_479_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ R2 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_480_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_481_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ! [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_482_subrelI,axiom,
    ! [R2: set_Pr7459493094073627847at_nat,S3: set_Pr7459493094073627847at_nat] :
      ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ R2 )
         => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Y3 ) @ S3 ) )
     => ( ord_le5997549366648089703at_nat @ R2 @ S3 ) ) ).

% subrelI
thf(fact_483_ssubst__Pair__rhs,axiom,
    ! [R2: product_prod_nat_nat,S3: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S4: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R3 )
     => ( ( S4 = S3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_484_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S3: nat,R3: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 )
     => ( ( S4 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_485_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S3 ) @ R3 )
     => ( ( S4 = S3 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_486_ssubst__Pair__rhs,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat,S4: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S3 ) @ R3 )
     => ( ( S4 = S3 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ R2 @ S4 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_487_arg__min__least,axiom,
    ! [S: set_Pr8693737435421807431at_nat,Y: produc859450856879609959at_nat,F3: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( S != bot_bo5327735625951526323at_nat )
       => ( ( member8206827879206165904at_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic390166758595302878at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_488_arg__min__least,axiom,
    ! [S: set_list_a,Y: list_a,F3: list_a > nat] :
      ( ( finite_finite_list_a @ S )
     => ( ( S != bot_bot_set_list_a )
       => ( ( member_list_a @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic5043722365632780795_a_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_489_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic7446932960582359483at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_490_arg__min__least,axiom,
    ! [S: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,F3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( S != bot_bo2099793752762293965at_nat )
       => ( ( member8440522571783428010at_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic4984276347100956536at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_491_arg__min__least,axiom,
    ! [S: set_Pr4329608150637261639at_nat,Y: produc3843707927480180839at_nat,F3: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( S != bot_bo228742789529271731at_nat )
       => ( ( member8757157785044589968at_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F3 @ ( lattic3831113108725116382at_nat @ F3 @ S ) ) @ ( F3 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_492_rrb__simps_I3_J,axiom,
    ! [X: nat,T: relational_term_a] : ( relational_rrb_a_b @ ( relational_Eq_a_b @ X @ T ) ) ).

% rrb_simps(3)
thf(fact_493_rrb__simps_I6_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_rrb_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( ( relational_rrb_a_b @ Q1 )
        & ( relational_rrb_a_b @ Q2 ) ) ) ).

% rrb_simps(6)
thf(fact_494_sat__fv__cong,axiom,
    ! [Phi: relational_fmla_a_b,Sigma2: nat > a,Sigma3: nat > a,I: product_prod_b_nat > set_list_a] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ ( relational_fv_a_b @ Phi ) )
         => ( ( Sigma2 @ N )
            = ( Sigma3 @ N ) ) )
     => ( ( relational_sat_a_b @ Phi @ I @ Sigma2 )
        = ( relational_sat_a_b @ Phi @ I @ Sigma3 ) ) ) ).

% sat_fv_cong
thf(fact_495_sat_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi2 ) @ I @ Sigma2 )
      = ( ( relational_sat_a_b @ Phi @ I @ Sigma2 )
        & ( relational_sat_a_b @ Psi2 @ I @ Sigma2 ) ) ) ).

% sat.simps(5)
thf(fact_496_eval__cong,axiom,
    ! [Q: relational_fmla_a_b,Q5: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_fv_a_b @ Q )
        = ( relational_fv_a_b @ Q5 ) )
     => ( ! [Sigma: nat > a] :
            ( ( relational_sat_a_b @ Q @ I @ Sigma )
            = ( relational_sat_a_b @ Q5 @ I @ Sigma ) )
       => ( ( relational_eval_a_b @ Q @ I )
          = ( relational_eval_a_b @ Q5 @ I ) ) ) ) ).

% eval_cong
thf(fact_497_sat__fun__upd,axiom,
    ! [N2: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a,Z3: a] :
      ( ~ ( member_nat @ N2 @ ( relational_fv_a_b @ Q ) )
     => ( ( relational_sat_a_b @ Q @ I @ ( fun_upd_nat_a @ Sigma2 @ N2 @ Z3 ) )
        = ( relational_sat_a_b @ Q @ I @ Sigma2 ) ) ) ).

% sat_fun_upd
thf(fact_498_eval__empty__alt,axiom,
    ! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_eval_a_b @ Q @ I )
        = bot_bot_set_list_a )
      = ( ! [Sigma4: nat > a] :
            ~ ( relational_sat_a_b @ Q @ I @ Sigma4 ) ) ) ).

% eval_empty_alt
thf(fact_499_finite__psubset__induct,axiom,
    ! [A2: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ A2 )
     => ( ! [A7: set_list_a] :
            ( ( finite_finite_list_a @ A7 )
           => ( ! [B6: set_list_a] :
                  ( ( ord_less_set_list_a @ B6 @ A7 )
                 => ( P @ B6 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_500_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B6: set_nat] :
                  ( ( ord_less_set_nat @ B6 @ A7 )
                 => ( P @ B6 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_501_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A7 )
           => ( ! [B6: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B6 @ A7 )
                 => ( P @ B6 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_502_not__psubset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_503_not__psubset__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A2 @ bot_bo228742789529271731at_nat ) ).

% not_psubset_empty
thf(fact_504_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_505_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( A != bot_bo228742789529271731at_nat )
      = ( ord_le2604355607129572851at_nat @ bot_bo228742789529271731at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_506_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_507_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_508_bot_Oextremum__strict,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ~ ( ord_le2604355607129572851at_nat @ A @ bot_bo228742789529271731at_nat ) ).

% bot.extremum_strict
thf(fact_509_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_510_cov_Ononfree,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( relational_cov_a_b @ X @ Q @ bot_bo4495933725496725865la_a_b ) ) ).

% cov.nonfree
thf(fact_511_cov_OConj,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G1: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b,G2: set_Re381260168593705685la_a_b] :
      ( ( relational_cov_a_b @ X @ Q1 @ G1 )
     => ( ( relational_cov_a_b @ X @ Q2 @ G2 )
       => ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( sup_su5130108678486352897la_a_b @ G1 @ G2 ) ) ) ) ).

% cov.Conj
thf(fact_512_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_513_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X5 )
                & ( ord_less_nat @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_514_ex__cov,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_rrb_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ? [X_12: set_Re381260168593705685la_a_b] : ( relational_cov_a_b @ X @ Q @ X_12 ) ) ) ).

% ex_cov
thf(fact_515_cov__fv,axiom,
    ! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_cov_a_b @ X @ Q @ G )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( member4680049679412964150la_a_b @ Qqp @ G )
         => ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
            & ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).

% cov_fv
thf(fact_516_cov_OEq__self,axiom,
    ! [X: nat] : ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ).

% cov.Eq_self
thf(fact_517_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ X6 @ B ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_518_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat @ X6 @ A7 )
                   => ( ord_less_nat @ B @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_519_arg__min__if__finite_I2_J,axiom,
    ! [S: set_list_a,F3: list_a > nat] :
      ( ( finite_finite_list_a @ S )
     => ( ( S != bot_bot_set_list_a )
       => ~ ? [X6: list_a] :
              ( ( member_list_a @ X6 @ S )
              & ( ord_less_nat @ ( F3 @ X6 ) @ ( F3 @ ( lattic5043722365632780795_a_nat @ F3 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_520_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F3: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ S )
              & ( ord_less_nat @ ( F3 @ X6 ) @ ( F3 @ ( lattic7446932960582359483at_nat @ F3 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_521_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( S != bot_bo2099793752762293965at_nat )
       => ~ ? [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ S )
              & ( ord_less_nat @ ( F3 @ X6 ) @ ( F3 @ ( lattic4984276347100956536at_nat @ F3 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_522_arg__min__if__finite_I2_J,axiom,
    ! [S: set_Pr4329608150637261639at_nat,F3: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( S != bot_bo228742789529271731at_nat )
       => ~ ? [X6: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X6 @ S )
              & ( ord_less_nat @ ( F3 @ X6 ) @ ( F3 @ ( lattic3831113108725116382at_nat @ F3 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_523_cov_OConjL,axiom,
    ! [X: nat,Q1: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_cov_a_b @ X @ Q1 @ G )
     => ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q1 @ X ) )
          = ( relational_Bool_a_b @ $false ) )
       => ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).

% cov.ConjL
thf(fact_524_cov_OConjR,axiom,
    ! [X: nat,Q2: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Q1: relational_fmla_a_b] :
      ( ( relational_cov_a_b @ X @ Q2 @ G )
     => ( ( ( relational_cp_a_b @ ( relational_erase_a_b @ Q2 @ X ) )
          = ( relational_Bool_a_b @ $false ) )
       => ( relational_cov_a_b @ X @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ G ) ) ) ).

% cov.ConjR
thf(fact_525_cov_OEqL,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).

% cov.EqL
thf(fact_526_cov_OEqR,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( relational_cov_a_b @ X @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ).

% cov.EqR
thf(fact_527_finite__induct__select,axiom,
    ! [S: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ S )
     => ( ( P @ bot_bot_set_list_a )
       => ( ! [T6: set_list_a] :
              ( ( ord_less_set_list_a @ T6 @ S )
             => ( ( P @ T6 )
               => ? [X6: list_a] :
                    ( ( member_list_a @ X6 @ ( minus_646659088055828811list_a @ S @ T6 ) )
                    & ( P @ ( insert_list_a @ X6 @ T6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_528_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T6: set_nat] :
              ( ( ord_less_set_nat @ T6 @ S )
             => ( ( P @ T6 )
               => ? [X6: nat] :
                    ( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T6 ) )
                    & ( P @ ( insert_nat @ X6 @ T6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_529_finite__induct__select,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T6: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T6 @ S )
             => ( ( P @ T6 )
               => ? [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ ( minus_1356011639430497352at_nat @ S @ T6 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X6 @ T6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_530_finite__induct__select,axiom,
    ! [S: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ S )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [T6: set_Pr4329608150637261639at_nat] :
              ( ( ord_le2604355607129572851at_nat @ T6 @ S )
             => ( ( P @ T6 )
               => ? [X6: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X6 @ ( minus_3314409938677909166at_nat @ S @ T6 ) )
                    & ( P @ ( insert9069300056098147895at_nat @ X6 @ T6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_531_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
      = ( ( ( member_nat @ X @ B4 )
         => ( ord_less_set_nat @ A2 @ B4 ) )
        & ( ~ ( member_nat @ X @ B4 )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_532_psubset__insert__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ord_le6428140832669894131at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ B4 ) )
      = ( ( ( member8206827879206165904at_nat @ X @ B4 )
         => ( ord_le6428140832669894131at_nat @ A2 @ B4 ) )
        & ( ~ ( member8206827879206165904at_nat @ X @ B4 )
         => ( ( ( member8206827879206165904at_nat @ X @ A2 )
             => ( ord_le6428140832669894131at_nat @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) @ B4 ) )
            & ( ~ ( member8206827879206165904at_nat @ X @ A2 )
             => ( ord_le3000389064537975527at_nat @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_533_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ B4 ) )
      = ( ( ( member8440522571783428010at_nat @ X @ B4 )
         => ( ord_le7866589430770878221at_nat @ A2 @ B4 ) )
        & ( ~ ( member8440522571783428010at_nat @ X @ B4 )
         => ( ( ( member8440522571783428010at_nat @ X @ A2 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
            & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
             => ( ord_le3146513528884898305at_nat @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_534_psubset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ B4 ) )
      = ( ( ( member8757157785044589968at_nat @ X @ B4 )
         => ( ord_le2604355607129572851at_nat @ A2 @ B4 ) )
        & ( ~ ( member8757157785044589968at_nat @ X @ B4 )
         => ( ( ( member8757157785044589968at_nat @ X @ A2 )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B4 ) )
            & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
             => ( ord_le1268244103169919719at_nat @ A2 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_535_cov_Oap,axiom,
    ! [Q: relational_fmla_a_b,X: nat] :
      ( ( relational_ap_a_b @ Q )
     => ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( relational_cov_a_b @ X @ Q @ ( insert7010464514620295119la_a_b @ Q @ bot_bo4495933725496725865la_a_b ) ) ) ) ).

% cov.ap
thf(fact_536_cov__fv__aux,axiom,
    ! [X: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Qqp: relational_fmla_a_b] :
      ( ( relational_cov_a_b @ X @ Q @ G )
     => ( ( member4680049679412964150la_a_b @ Qqp @ G )
       => ( ( member_nat @ X @ ( relational_fv_a_b @ Qqp ) )
          & ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Qqp ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( relational_fv_a_b @ Q ) ) ) ) ) ).

% cov_fv_aux
thf(fact_537_Sup__fin_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A2 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A2 )
              = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_538_Sup__fin_Oremove,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( member6273874620674182256at_nat @ X @ A2 )
       => ( ( ( ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
              = bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ A2 )
              = X ) )
          & ( ( ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
             != bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ A2 )
              = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_539_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_540_Sup__fin_Oinsert__remove,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( ( ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
            = bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A2 ) )
            = X ) )
        & ( ( ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) )
           != bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A2 ) )
            = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ ( minus_5090130588299000206at_nat @ A2 @ ( insert2290795007606087959at_nat @ X @ bot_bo1534825418459847827at_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_541_fv__exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati3989891337220013914ts_a_b @ X @ Q ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% fv_exists
thf(fact_542_Sup__fin_Oinsert,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
          = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_543_Sup__fin_Oinsert,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( A2 != bot_bo1534825418459847827at_nat )
       => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A2 ) )
          = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A2 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_544_rrb__simps_I8_J,axiom,
    ! [Y: nat,Qy: relational_fmla_a_b] :
      ( ( relational_rrb_a_b @ ( relati3989891337220013914ts_a_b @ Y @ Qy ) )
      = ( ( ( member_nat @ Y @ ( relational_fv_a_b @ Qy ) )
         => ? [X7: set_Re381260168593705685la_a_b] : ( relational_gen_a_b @ Y @ Qy @ X7 ) )
        & ( relational_rrb_a_b @ Qy ) ) ) ).

% rrb_simps(8)
thf(fact_545_exists__def,axiom,
    ( relati3989891337220013914ts_a_b
    = ( ^ [X3: nat,Q3: relational_fmla_a_b] : ( if_Rel1279876242545935705la_a_b @ ( member_nat @ X3 @ ( relational_fv_a_b @ Q3 ) ) @ ( relati591517084277583526ts_a_b @ X3 @ Q3 ) @ Q3 ) ) ) ).

% exists_def
thf(fact_546_exists__Exists,axiom,
    ! [X: nat,Q: relational_fmla_a_b] :
      ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
     => ( ( relati3989891337220013914ts_a_b @ X @ Q )
        = ( relati591517084277583526ts_a_b @ X @ Q ) ) ) ).

% exists_Exists
thf(fact_547_Sup__fin_OcoboundedI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).

% Sup_fin.coboundedI
thf(fact_548_Sup__fin_Oin__idem,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
          = ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_549_Sup__fin_Oin__idem,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( member6273874620674182256at_nat @ X @ A2 )
       => ( ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A2 ) )
          = ( lattic8394993312974875902at_nat @ A2 ) ) ) ) ).

% Sup_fin.in_idem
thf(fact_550_Sup__fin_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A2 )
             => ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_551_Sup__fin_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ord_less_eq_nat @ A5 @ X ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_552_Sup__fin_Obounded__iff,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_553_Sup__fin_Osubset__imp,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B4 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_554_Sup__fin_Osubset,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( B4 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B4 @ A2 )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B4 ) @ ( lattic1093996805478795353in_nat @ A2 ) )
            = ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_555_Sup__fin_Osubset,axiom,
    ! [A2: set_se5479201105039950375at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( B4 != bot_bo1534825418459847827at_nat )
       => ( ( ord_le7966958417815640519at_nat @ B4 @ A2 )
         => ( ( sup_su5525570899277871387at_nat @ ( lattic8394993312974875902at_nat @ B4 ) @ ( lattic8394993312974875902at_nat @ A2 ) )
            = ( lattic8394993312974875902at_nat @ A2 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_556_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( A2 != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_557_Sup__fin_Oinsert__not__elem,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ~ ( member6273874620674182256at_nat @ X @ A2 )
       => ( ( A2 != bot_bo1534825418459847827at_nat )
         => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A2 ) )
            = ( sup_su5525570899277871387at_nat @ X @ ( lattic8394993312974875902at_nat @ A2 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_558_Sup__fin_Oclosed,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat,Y3: nat] : ( member_nat @ ( sup_sup_nat @ X4 @ Y3 ) @ ( insert_nat @ X4 @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) )
         => ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).

% Sup_fin.closed
thf(fact_559_Sup__fin_Oclosed,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( A2 != bot_bo1534825418459847827at_nat )
       => ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ ( sup_su5525570899277871387at_nat @ X4 @ Y3 ) @ ( insert2290795007606087959at_nat @ X4 @ ( insert2290795007606087959at_nat @ Y3 @ bot_bo1534825418459847827at_nat ) ) )
         => ( member6273874620674182256at_nat @ ( lattic8394993312974875902at_nat @ A2 ) @ A2 ) ) ) ) ).

% Sup_fin.closed
thf(fact_560_Sup__fin_Ounion,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B4 )
         => ( ( B4 != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B4 ) )
              = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_561_Sup__fin_Ounion,axiom,
    ! [A2: set_se5479201105039950375at_nat,B4: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( A2 != bot_bo1534825418459847827at_nat )
       => ( ( finite4768004253697144688at_nat @ B4 )
         => ( ( B4 != bot_bo1534825418459847827at_nat )
           => ( ( lattic8394993312974875902at_nat @ ( sup_su474284620922681339at_nat @ A2 @ B4 ) )
              = ( sup_su5525570899277871387at_nat @ ( lattic8394993312974875902at_nat @ A2 ) @ ( lattic8394993312974875902at_nat @ B4 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_562_image__is__empty,axiom,
    ! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( image_5168914502847457605at_nat @ F3 @ A2 )
        = bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% image_is_empty
thf(fact_563_image__is__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > product_prod_nat_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( image_7325330684212377951at_nat @ F3 @ A2 )
        = bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% image_is_empty
thf(fact_564_image__is__empty,axiom,
    ! [F3: product_prod_nat_nat > produc3843707927480180839at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( image_1672857615394513579at_nat @ F3 @ A2 )
        = bot_bo228742789529271731at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% image_is_empty
thf(fact_565_image__is__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( image_960689587819713477at_nat @ F3 @ A2 )
        = bot_bo228742789529271731at_nat )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% image_is_empty
thf(fact_566_empty__is__image,axiom,
    ! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( image_5168914502847457605at_nat @ F3 @ A2 ) )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% empty_is_image
thf(fact_567_empty__is__image,axiom,
    ! [F3: produc3843707927480180839at_nat > product_prod_nat_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( image_7325330684212377951at_nat @ F3 @ A2 ) )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% empty_is_image
thf(fact_568_empty__is__image,axiom,
    ! [F3: product_prod_nat_nat > produc3843707927480180839at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( image_1672857615394513579at_nat @ F3 @ A2 ) )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% empty_is_image
thf(fact_569_empty__is__image,axiom,
    ! [F3: produc3843707927480180839at_nat > produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( image_960689587819713477at_nat @ F3 @ A2 ) )
      = ( A2 = bot_bo228742789529271731at_nat ) ) ).

% empty_is_image
thf(fact_570_image__empty,axiom,
    ! [F3: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( image_5168914502847457605at_nat @ F3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% image_empty
thf(fact_571_image__empty,axiom,
    ! [F3: product_prod_nat_nat > produc3843707927480180839at_nat] :
      ( ( image_1672857615394513579at_nat @ F3 @ bot_bo2099793752762293965at_nat )
      = bot_bo228742789529271731at_nat ) ).

% image_empty
thf(fact_572_image__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > product_prod_nat_nat] :
      ( ( image_7325330684212377951at_nat @ F3 @ bot_bo228742789529271731at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% image_empty
thf(fact_573_image__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > produc3843707927480180839at_nat] :
      ( ( image_960689587819713477at_nat @ F3 @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% image_empty
thf(fact_574_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > list_a] :
      ( ( finite_finite_list_a @ F )
     => ( finite_finite_list_a @ ( image_list_a_list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_575_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > nat] :
      ( ( finite_finite_list_a @ F )
     => ( finite_finite_nat @ ( image_list_a_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_576_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > product_prod_nat_nat] :
      ( ( finite_finite_list_a @ F )
     => ( finite6177210948735845034at_nat @ ( image_3288686587095672578at_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_577_finite__imageI,axiom,
    ! [F: set_nat,H: nat > list_a] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_list_a @ ( image_nat_list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_578_finite__imageI,axiom,
    ! [F: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_579_finite__imageI,axiom,
    ! [F: set_nat,H: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ F )
     => ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_580_finite__imageI,axiom,
    ! [F: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > list_a] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( finite_finite_list_a @ ( image_3765944151573559240list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_581_finite__imageI,axiom,
    ! [F: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( finite_finite_nat @ ( image_2486076414777270412at_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_582_finite__imageI,axiom,
    ! [F: set_Pr1261947904930325089at_nat,H: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ F )
     => ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_583_finite__surj,axiom,
    ! [A2: set_list_a,B4: set_list_a,F3: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_list_a_list_a @ F3 @ A2 ) )
       => ( finite_finite_list_a @ B4 ) ) ) ).

% finite_surj
thf(fact_584_finite__surj,axiom,
    ! [A2: set_list_a,B4: set_nat,F3: list_a > nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_list_a_nat @ F3 @ A2 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_585_finite__surj,axiom,
    ! [A2: set_list_a,B4: set_Pr1261947904930325089at_nat,F3: list_a > product_prod_nat_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_3288686587095672578at_nat @ F3 @ A2 ) )
       => ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_586_finite__surj,axiom,
    ! [A2: set_nat,B4: set_list_a,F3: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_nat_list_a @ F3 @ A2 ) )
       => ( finite_finite_list_a @ B4 ) ) ) ).

% finite_surj
thf(fact_587_finite__surj,axiom,
    ! [A2: set_nat,B4: set_nat,F3: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A2 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_588_finite__surj,axiom,
    ! [A2: set_nat,B4: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_5846123807819985514at_nat @ F3 @ A2 ) )
       => ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_589_finite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_list_a,F3: product_prod_nat_nat > list_a] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_3765944151573559240list_a @ F3 @ A2 ) )
       => ( finite_finite_list_a @ B4 ) ) ) ).

% finite_surj
thf(fact_590_finite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_nat,F3: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_2486076414777270412at_nat @ F3 @ A2 ) )
       => ( finite_finite_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_591_finite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_5168914502847457605at_nat @ F3 @ A2 ) )
       => ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% finite_surj
thf(fact_592_finite__subset__image,axiom,
    ! [B4: set_list_a,F3: list_a > list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_list_a_list_a @ F3 @ A2 ) )
       => ? [C4: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C4 @ A2 )
            & ( finite_finite_list_a @ C4 )
            & ( B4
              = ( image_list_a_list_a @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_593_finite__subset__image,axiom,
    ! [B4: set_list_a,F3: nat > list_a,A2: set_nat] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_nat_list_a @ F3 @ A2 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A2 )
            & ( finite_finite_nat @ C4 )
            & ( B4
              = ( image_nat_list_a @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_594_finite__subset__image,axiom,
    ! [B4: set_list_a,F3: product_prod_nat_nat > list_a,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( ord_le8861187494160871172list_a @ B4 @ ( image_3765944151573559240list_a @ F3 @ A2 ) )
       => ? [C4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ C4 @ A2 )
            & ( finite6177210948735845034at_nat @ C4 )
            & ( B4
              = ( image_3765944151573559240list_a @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_595_finite__subset__image,axiom,
    ! [B4: set_nat,F3: list_a > nat,A2: set_list_a] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_list_a_nat @ F3 @ A2 ) )
       => ? [C4: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C4 @ A2 )
            & ( finite_finite_list_a @ C4 )
            & ( B4
              = ( image_list_a_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_596_finite__subset__image,axiom,
    ! [B4: set_nat,F3: nat > nat,A2: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F3 @ A2 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A2 )
            & ( finite_finite_nat @ C4 )
            & ( B4
              = ( image_nat_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_597_finite__subset__image,axiom,
    ! [B4: set_nat,F3: product_prod_nat_nat > nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ ( image_2486076414777270412at_nat @ F3 @ A2 ) )
       => ? [C4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ C4 @ A2 )
            & ( finite6177210948735845034at_nat @ C4 )
            & ( B4
              = ( image_2486076414777270412at_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_598_finite__subset__image,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,F3: list_a > product_prod_nat_nat,A2: set_list_a] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_3288686587095672578at_nat @ F3 @ A2 ) )
       => ? [C4: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C4 @ A2 )
            & ( finite_finite_list_a @ C4 )
            & ( B4
              = ( image_3288686587095672578at_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_599_finite__subset__image,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat,A2: set_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_5846123807819985514at_nat @ F3 @ A2 ) )
       => ? [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A2 )
            & ( finite_finite_nat @ C4 )
            & ( B4
              = ( image_5846123807819985514at_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_600_finite__subset__image,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ ( image_5168914502847457605at_nat @ F3 @ A2 ) )
       => ? [C4: set_Pr1261947904930325089at_nat] :
            ( ( ord_le3146513528884898305at_nat @ C4 @ A2 )
            & ( finite6177210948735845034at_nat @ C4 )
            & ( B4
              = ( image_5168914502847457605at_nat @ F3 @ C4 ) ) ) ) ) ).

% finite_subset_image
thf(fact_601_ex__finite__subset__image,axiom,
    ! [F3: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ ( image_list_a_list_a @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ A2 )
            & ( P @ ( image_list_a_list_a @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_602_ex__finite__subset__image,axiom,
    ! [F3: nat > list_a,A2: set_nat,P: set_list_a > $o] :
      ( ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ ( image_nat_list_a @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ A2 )
            & ( P @ ( image_nat_list_a @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_603_ex__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > list_a,A2: set_Pr1261947904930325089at_nat,P: set_list_a > $o] :
      ( ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ ( image_3765944151573559240list_a @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ A2 )
            & ( P @ ( image_3765944151573559240list_a @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_604_ex__finite__subset__image,axiom,
    ! [F3: list_a > nat,A2: set_list_a,P: set_nat > $o] :
      ( ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ ( image_list_a_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ A2 )
            & ( P @ ( image_list_a_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_605_ex__finite__subset__image,axiom,
    ! [F3: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ A2 )
            & ( P @ ( image_nat_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_606_ex__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > nat,A2: set_Pr1261947904930325089at_nat,P: set_nat > $o] :
      ( ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ ( image_2486076414777270412at_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ A2 )
            & ( P @ ( image_2486076414777270412at_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_607_ex__finite__subset__image,axiom,
    ! [F3: list_a > product_prod_nat_nat,A2: set_list_a,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ ( image_3288686587095672578at_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_list_a] :
            ( ( finite_finite_list_a @ B7 )
            & ( ord_le8861187494160871172list_a @ B7 @ A2 )
            & ( P @ ( image_3288686587095672578at_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_608_ex__finite__subset__image,axiom,
    ! [F3: nat > product_prod_nat_nat,A2: set_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ ( image_5846123807819985514at_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_nat] :
            ( ( finite_finite_nat @ B7 )
            & ( ord_less_eq_set_nat @ B7 @ A2 )
            & ( P @ ( image_5846123807819985514at_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_609_ex__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ ( image_5168914502847457605at_nat @ F3 @ A2 ) )
            & ( P @ B7 ) ) )
      = ( ? [B7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ B7 )
            & ( ord_le3146513528884898305at_nat @ B7 @ A2 )
            & ( P @ ( image_5168914502847457605at_nat @ F3 @ B7 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_610_all__finite__subset__image,axiom,
    ! [F3: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
      ( ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ ( image_list_a_list_a @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ A2 ) )
           => ( P @ ( image_list_a_list_a @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_611_all__finite__subset__image,axiom,
    ! [F3: nat > list_a,A2: set_nat,P: set_list_a > $o] :
      ( ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ ( image_nat_list_a @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ A2 ) )
           => ( P @ ( image_nat_list_a @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_612_all__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > list_a,A2: set_Pr1261947904930325089at_nat,P: set_list_a > $o] :
      ( ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ ( image_3765944151573559240list_a @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ A2 ) )
           => ( P @ ( image_3765944151573559240list_a @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_613_all__finite__subset__image,axiom,
    ! [F3: list_a > nat,A2: set_list_a,P: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ ( image_list_a_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ A2 ) )
           => ( P @ ( image_list_a_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_614_all__finite__subset__image,axiom,
    ! [F3: nat > nat,A2: set_nat,P: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ A2 ) )
           => ( P @ ( image_nat_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_615_all__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > nat,A2: set_Pr1261947904930325089at_nat,P: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ ( image_2486076414777270412at_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ A2 ) )
           => ( P @ ( image_2486076414777270412at_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_616_all__finite__subset__image,axiom,
    ! [F3: list_a > product_prod_nat_nat,A2: set_list_a,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ ( image_3288686587095672578at_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_list_a] :
            ( ( ( finite_finite_list_a @ B7 )
              & ( ord_le8861187494160871172list_a @ B7 @ A2 ) )
           => ( P @ ( image_3288686587095672578at_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_617_all__finite__subset__image,axiom,
    ! [F3: nat > product_prod_nat_nat,A2: set_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ ( image_5846123807819985514at_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ( finite_finite_nat @ B7 )
              & ( ord_less_eq_set_nat @ B7 @ A2 ) )
           => ( P @ ( image_5846123807819985514at_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_618_all__finite__subset__image,axiom,
    ! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ ( image_5168914502847457605at_nat @ F3 @ A2 ) ) )
           => ( P @ B7 ) ) )
      = ( ! [B7: set_Pr1261947904930325089at_nat] :
            ( ( ( finite6177210948735845034at_nat @ B7 )
              & ( ord_le3146513528884898305at_nat @ B7 @ A2 ) )
           => ( P @ ( image_5168914502847457605at_nat @ F3 @ B7 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_619_infinite__surj,axiom,
    ! [A2: set_list_a,F3: list_a > list_a,B4: set_list_a] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( image_list_a_list_a @ F3 @ B4 ) )
       => ~ ( finite_finite_list_a @ B4 ) ) ) ).

% infinite_surj
thf(fact_620_infinite__surj,axiom,
    ! [A2: set_list_a,F3: nat > list_a,B4: set_nat] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( image_nat_list_a @ F3 @ B4 ) )
       => ~ ( finite_finite_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_621_infinite__surj,axiom,
    ! [A2: set_list_a,F3: product_prod_nat_nat > list_a,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( image_3765944151573559240list_a @ F3 @ B4 ) )
       => ~ ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_622_infinite__surj,axiom,
    ! [A2: set_nat,F3: list_a > nat,B4: set_list_a] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_list_a_nat @ F3 @ B4 ) )
       => ~ ( finite_finite_list_a @ B4 ) ) ) ).

% infinite_surj
thf(fact_623_infinite__surj,axiom,
    ! [A2: set_nat,F3: nat > nat,B4: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F3 @ B4 ) )
       => ~ ( finite_finite_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_624_infinite__surj,axiom,
    ! [A2: set_nat,F3: product_prod_nat_nat > nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( image_2486076414777270412at_nat @ F3 @ B4 ) )
       => ~ ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_625_infinite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: list_a > product_prod_nat_nat,B4: set_list_a] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( image_3288686587095672578at_nat @ F3 @ B4 ) )
       => ~ ( finite_finite_list_a @ B4 ) ) ) ).

% infinite_surj
thf(fact_626_infinite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat,B4: set_nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( image_5846123807819985514at_nat @ F3 @ B4 ) )
       => ~ ( finite_finite_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_627_infinite__surj,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( image_5168914502847457605at_nat @ F3 @ B4 ) )
       => ~ ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% infinite_surj
thf(fact_628_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F3: nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_5846123807819985514at_nat @ ( fun_up4795876855168711666at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_5846123807819985514at_nat @ F3 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_5846123807819985514at_nat @ ( fun_up4795876855168711666at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_5846123807819985514at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_629_fun__upd__image,axiom,
    ! [X: nat,A2: set_nat,F3: nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member_nat @ X @ A2 )
       => ( ( image_2499013659892586576at_nat @ ( fun_up8749971096457065944at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_2499013659892586576at_nat @ F3 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
      & ( ~ ( member_nat @ X @ A2 )
       => ( ( image_2499013659892586576at_nat @ ( fun_up8749971096457065944at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_2499013659892586576at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_630_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,F3: produc859450856879609959at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A2 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_6706873702294942047at_nat @ F3 @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A2 )
       => ( ( image_6706873702294942047at_nat @ ( fun_up105607794567514583at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_6706873702294942047at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_631_fun__upd__image,axiom,
    ! [X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,F3: produc859450856879609959at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8206827879206165904at_nat @ X @ A2 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_7637461350386011077at_nat @ F3 @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ) ) )
      & ( ~ ( member8206827879206165904at_nat @ X @ A2 )
       => ( ( image_7637461350386011077at_nat @ ( fun_up8993321861993844797at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_7637461350386011077at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_632_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_5168914502847457605at_nat @ F3 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( image_5168914502847457605at_nat @ ( fun_up6858051064228255165at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_5168914502847457605at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_633_fun__upd__image,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_1672857615394513579at_nat @ F3 @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X @ A2 )
       => ( ( image_1672857615394513579at_nat @ ( fun_up7313344445903829795at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_1672857615394513579at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_634_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F3: produc3843707927480180839at_nat > product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert8211810215607154385at_nat @ Y @ ( image_7325330684212377951at_nat @ F3 @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( image_7325330684212377951at_nat @ ( fun_up3742445477866918359at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_7325330684212377951at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_635_fun__upd__image,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F3: produc3843707927480180839at_nat > produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat] :
      ( ( ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F3 @ X @ Y ) @ A2 )
          = ( insert9069300056098147895at_nat @ Y @ ( image_960689587819713477at_nat @ F3 @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X @ A2 )
       => ( ( image_960689587819713477at_nat @ ( fun_up3138462846764619837at_nat @ F3 @ X @ Y ) @ A2 )
          = ( image_960689587819713477at_nat @ F3 @ A2 ) ) ) ) ).

% fun_upd_image
thf(fact_636_Sup__fin_Ohom__commute,axiom,
    ! [H: nat > nat,N3: set_nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( H @ ( sup_sup_nat @ X4 @ Y3 ) )
          = ( sup_sup_nat @ ( H @ X4 ) @ ( H @ Y3 ) ) )
     => ( ( finite_finite_nat @ N3 )
       => ( ( N3 != bot_bot_set_nat )
         => ( ( H @ ( lattic1093996805478795353in_nat @ N3 ) )
            = ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_637_Sup__fin_Ohom__commute,axiom,
    ! [H: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,N3: set_se5479201105039950375at_nat] :
      ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( H @ ( sup_su5525570899277871387at_nat @ X4 @ Y3 ) )
          = ( sup_su5525570899277871387at_nat @ ( H @ X4 ) @ ( H @ Y3 ) ) )
     => ( ( finite4768004253697144688at_nat @ N3 )
       => ( ( N3 != bot_bo1534825418459847827at_nat )
         => ( ( H @ ( lattic8394993312974875902at_nat @ N3 ) )
            = ( lattic8394993312974875902at_nat @ ( image_4303274092540115909at_nat @ H @ N3 ) ) ) ) ) ) ).

% Sup_fin.hom_commute
thf(fact_638_cov_OExists,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b @ X @ Q @ G )
       => ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
         => ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).

% cov.Exists
thf(fact_639_cov_H_OExists,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b2 @ X @ Q @ G )
       => ( ~ ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ G )
         => ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q ) @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ G ) ) ) ) ) ).

% cov'.Exists
thf(fact_640_finite__Collect__disjI,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X3: list_a] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
        & ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_641_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_642_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X3: product_prod_nat_nat] :
              ( ( P @ X3 )
              | ( Q @ X3 ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_643_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
          @ ^ [X3: list_a] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_644_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
          @ ^ [X3: nat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_645_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
          @ ^ [X3: product_prod_nat_nat] :
              ( ( P @ X3 )
              & ( Q @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_646_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_647_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_648_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( X3 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_649_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( X3 = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_650_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X3: produc3843707927480180839at_nat] : ( X3 = A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_651_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_652_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y6: product_prod_nat_nat,Z4: product_prod_nat_nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_653_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y6: produc3843707927480180839at_nat,Z4: produc3843707927480180839at_nat] : ( Y6 = Z4 )
          @ A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_654_finite__Collect__subsets,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [B7: set_list_a] : ( ord_le8861187494160871172list_a @ B7 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_655_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B7: set_nat] : ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_656_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B7: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B7 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_657_finite__conv__nat__seg__image,axiom,
    ( finite_finite_list_a
    = ( ^ [A6: set_list_a] :
        ? [N4: nat,F4: nat > list_a] :
          ( A6
          = ( image_nat_list_a @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_658_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A6: set_nat] :
        ? [N4: nat,F4: nat > nat] :
          ( A6
          = ( image_nat_nat @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_659_finite__conv__nat__seg__image,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
        ? [N4: nat,F4: nat > product_prod_nat_nat] :
          ( A6
          = ( image_5846123807819985514at_nat @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N4 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_660_nat__seg__image__imp__finite,axiom,
    ! [A2: set_list_a,F3: nat > list_a,N2: nat] :
      ( ( A2
        = ( image_nat_list_a @ F3
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite_finite_list_a @ A2 ) ) ).

% nat_seg_image_imp_finite
thf(fact_661_nat__seg__image__imp__finite,axiom,
    ! [A2: set_nat,F3: nat > nat,N2: nat] :
      ( ( A2
        = ( image_nat_nat @ F3
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite_finite_nat @ A2 ) ) ).

% nat_seg_image_imp_finite
thf(fact_662_nat__seg__image__imp__finite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat,N2: nat] :
      ( ( A2
        = ( image_5846123807819985514at_nat @ F3
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite6177210948735845034at_nat @ A2 ) ) ).

% nat_seg_image_imp_finite
thf(fact_663_image__constant,axiom,
    ! [X: nat,A2: set_nat,C2: product_prod_nat_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( image_5846123807819985514at_nat
          @ ^ [X3: nat] : C2
          @ A2 )
        = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% image_constant
thf(fact_664_image__constant,axiom,
    ! [X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,C2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ X @ A2 )
     => ( ( image_6706873702294942047at_nat
          @ ^ [X3: produc859450856879609959at_nat] : C2
          @ A2 )
        = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% image_constant
thf(fact_665_image__constant,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( image_5168914502847457605at_nat
          @ ^ [X3: product_prod_nat_nat] : C2
          @ A2 )
        = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% image_constant
thf(fact_666_image__constant,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,C2: product_prod_nat_nat] :
      ( ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( image_7325330684212377951at_nat
          @ ^ [X3: produc3843707927480180839at_nat] : C2
          @ A2 )
        = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ).

% image_constant
thf(fact_667_image__constant,axiom,
    ! [X: nat,A2: set_nat,C2: produc3843707927480180839at_nat] :
      ( ( member_nat @ X @ A2 )
     => ( ( image_2499013659892586576at_nat
          @ ^ [X3: nat] : C2
          @ A2 )
        = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ).

% image_constant
thf(fact_668_image__constant,axiom,
    ! [X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,C2: produc3843707927480180839at_nat] :
      ( ( member8206827879206165904at_nat @ X @ A2 )
     => ( ( image_7637461350386011077at_nat
          @ ^ [X3: produc859450856879609959at_nat] : C2
          @ A2 )
        = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ).

% image_constant
thf(fact_669_image__constant,axiom,
    ! [X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,C2: produc3843707927480180839at_nat] :
      ( ( member8440522571783428010at_nat @ X @ A2 )
     => ( ( image_1672857615394513579at_nat
          @ ^ [X3: product_prod_nat_nat] : C2
          @ A2 )
        = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ).

% image_constant
thf(fact_670_image__constant,axiom,
    ! [X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,C2: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ X @ A2 )
     => ( ( image_960689587819713477at_nat
          @ ^ [X3: produc3843707927480180839at_nat] : C2
          @ A2 )
        = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ).

% image_constant
thf(fact_671_image__constant__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( image_5168914502847457605at_nat
            @ ^ [X3: product_prod_nat_nat] : C2
            @ A2 )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( image_5168914502847457605at_nat
            @ ^ [X3: product_prod_nat_nat] : C2
            @ A2 )
          = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% image_constant_conv
thf(fact_672_image__constant__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: produc3843707927480180839at_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( image_1672857615394513579at_nat
            @ ^ [X3: product_prod_nat_nat] : C2
            @ A2 )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( image_1672857615394513579at_nat
            @ ^ [X3: product_prod_nat_nat] : C2
            @ A2 )
          = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% image_constant_conv
thf(fact_673_image__constant__conv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: product_prod_nat_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( image_7325330684212377951at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : C2
            @ A2 )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( image_7325330684212377951at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : C2
            @ A2 )
          = ( insert8211810215607154385at_nat @ C2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% image_constant_conv
thf(fact_674_image__constant__conv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: produc3843707927480180839at_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( image_960689587819713477at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : C2
            @ A2 )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( image_960689587819713477at_nat
            @ ^ [X3: produc3843707927480180839at_nat] : C2
            @ A2 )
          = ( insert9069300056098147895at_nat @ C2 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% image_constant_conv
thf(fact_675_gen__nocp__intros_I1_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen_nocp_intros(1)
thf(fact_676_gen__nocp__intros_I2_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen_nocp_intros(2)
thf(fact_677_gen_H_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen'.intros(8)
thf(fact_678_gen_H_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
          @ G ) ) ) ).

% gen'.intros(9)
thf(fact_679_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F3: nat > nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F3 @ A2 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_680_pigeonhole__infinite,axiom,
    ! [A2: set_list_a,F3: list_a > nat] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_nat @ ( image_list_a_nat @ F3 @ A2 ) )
       => ? [X4: list_a] :
            ( ( member_list_a @ X4 @ A2 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A3: list_a] :
                      ( ( member_list_a @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_681_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F3: nat > list_a] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_list_a @ ( image_nat_list_a @ F3 @ A2 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_682_pigeonhole__infinite,axiom,
    ! [A2: set_list_a,F3: list_a > list_a] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ ( image_list_a_list_a @ F3 @ A2 ) )
       => ? [X4: list_a] :
            ( ( member_list_a @ X4 @ A2 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A3: list_a] :
                      ( ( member_list_a @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_683_pigeonhole__infinite,axiom,
    ! [A2: set_nat,F3: nat > product_prod_nat_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( image_5846123807819985514at_nat @ F3 @ A2 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ( ( member_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_684_pigeonhole__infinite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_nat @ ( image_2486076414777270412at_nat @ F3 @ A2 ) )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A3: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_685_pigeonhole__infinite,axiom,
    ! [A2: set_list_a,F3: list_a > product_prod_nat_nat] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( image_3288686587095672578at_nat @ F3 @ A2 ) )
       => ? [X4: list_a] :
            ( ( member_list_a @ X4 @ A2 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A3: list_a] :
                      ( ( member_list_a @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_686_pigeonhole__infinite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > list_a] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_list_a @ ( image_3765944151573559240list_a @ F3 @ A2 ) )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A3: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_687_pigeonhole__infinite,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( image_5168914502847457605at_nat @ F3 @ A2 ) )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
            & ~ ( finite6177210948735845034at_nat
                @ ( collec3392354462482085612at_nat
                  @ ^ [A3: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_688_pigeonhole__infinite,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,F3: produc859450856879609959at_nat > nat] :
      ( ~ ( finite4392333629123659920at_nat @ A2 )
     => ( ( finite_finite_nat @ ( image_6160980552072806642at_nat @ F3 @ A2 ) )
       => ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A2 )
            & ~ ( finite4392333629123659920at_nat
                @ ( collec7088162979684241874at_nat
                  @ ^ [A3: produc859450856879609959at_nat] :
                      ( ( member8206827879206165904at_nat @ A3 @ A2 )
                      & ( ( F3 @ A3 )
                        = ( F3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_689_pred__subset__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ S ) )
      = ( ord_le3000389064537975527at_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_690_pred__subset__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_691_pred__subset__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ S ) )
      = ( ord_le1268244103169919719at_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_692_pred__subset__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( ord_le3072208448688395470_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ S ) )
      = ( ord_le5997549366648089703at_nat @ R3 @ S ) ) ).

% pred_subset_eq2
thf(fact_693_sup__Un__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( sup_su362511073950362882_nat_o
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ ( sup_su718114333110466843at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_694_sup__Un__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ ( sup_su6327502436637775413at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_695_sup__Un__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( sup_su6164525874564267778_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ ( sup_su5395584912770621083at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_696_sup__Un__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ ( sup_su5525570899277871387at_nat @ R3 @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_697_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_698_bot__empty__eq2,axiom,
    ( bot_bo3364206721330744218_nat_o
    = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ bot_bo4948859079157340979at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_699_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_700_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_701_pred__equals__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 ) )
        = ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_702_pred__equals__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 ) )
        = ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_703_pred__equals__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 ) )
        = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_704_pred__equals__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 ) )
        = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ S ) ) )
      = ( R3 = S ) ) ).

% pred_equals_eq2
thf(fact_705_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B4: set_nat,R3: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_706_pigeonhole__infinite__rel,axiom,
    ! [A2: set_list_a,B4: set_nat,R3: list_a > nat > $o] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A3: list_a] :
                        ( ( member_list_a @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_707_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B4: set_list_a,R3: nat > list_a > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_list_a @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa2: list_a] :
                  ( ( member_list_a @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_708_pigeonhole__infinite__rel,axiom,
    ! [A2: set_list_a,B4: set_list_a,R3: list_a > list_a > $o] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A2 )
             => ? [Xa2: list_a] :
                  ( ( member_list_a @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A3: list_a] :
                        ( ( member_list_a @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_709_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B4: set_Pr1261947904930325089at_nat,R3: nat > product_prod_nat_nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B4 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_710_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_nat,R3: product_prod_nat_nat > nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_711_pigeonhole__infinite__rel,axiom,
    ! [A2: set_list_a,B4: set_Pr1261947904930325089at_nat,R3: list_a > product_prod_nat_nat > $o] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A2 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B4 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A3: list_a] :
                        ( ( member_list_a @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_712_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_list_a,R3: product_prod_nat_nat > list_a > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_list_a @ B4 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A2 )
             => ? [Xa2: list_a] :
                  ( ( member_list_a @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B4 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_713_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A2 )
             => ? [Xa2: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ B4 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_714_pigeonhole__infinite__rel,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_nat,R3: produc859450856879609959at_nat > nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A2 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A2 )
             => ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ B4 )
                  & ( R3 @ X4 @ Xa2 ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B4 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A3 @ A2 )
                        & ( R3 @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_715_not__finite__existsD,axiom,
    ! [P: list_a > $o] :
      ( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ? [X_12: list_a] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_716_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_12: nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_717_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_12: product_prod_nat_nat] : ( P @ X_12 ) ) ).

% not_finite_existsD
thf(fact_718_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X3: nat] : $false ) ) ).

% empty_def
thf(fact_719_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X3: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_720_empty__def,axiom,
    ( bot_bo228742789529271731at_nat
    = ( collec6321179662152712658at_nat
      @ ^ [X3: produc3843707927480180839at_nat] : $false ) ) ).

% empty_def
thf(fact_721_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_722_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_723_Collect__conv__if2,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( A = X3 )
                & ( P @ X3 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_724_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_725_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X3: product_prod_nat_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_726_Collect__conv__if,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X3: produc3843707927480180839at_nat] :
                ( ( X3 = A )
                & ( P @ X3 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_727_subst_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X @ Y )
      = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q2 @ X @ Y ) ) ) ).

% subst.simps(5)
thf(fact_728_eqs__def,axiom,
    ( relational_eqs_a_b
    = ( ^ [X3: nat,G3: set_Re381260168593705685la_a_b] :
          ( collect_nat
          @ ^ [Y4: nat] :
              ( ( X3 != Y4 )
              & ( member4680049679412964150la_a_b @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y4 ) ) @ G3 ) ) ) ) ) ).

% eqs_def
thf(fact_729_gen_Ointros_I9_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen.intros(9)
thf(fact_730_gen_Ointros_I8_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b @ Y @ Q @ G )
     => ( relational_gen_a_b @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen.intros(8)
thf(fact_731_cp_Osimps_I1_J,axiom,
    ! [X: nat,T: relational_term_a] :
      ( ( relational_cp_a_b @ ( relational_Eq_a_b @ X @ T ) )
      = ( relati582353067970734056la_a_b
        @ ^ [A3: a] : ( relational_Eq_a_b @ X @ T )
        @ ^ [Y4: nat] : ( if_Rel1279876242545935705la_a_b @ ( X = Y4 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y4 ) ) )
        @ T ) ) ).

% cp.simps(1)
thf(fact_732_gen_H__cp__intros_I2_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen'_cp_intros(2)
thf(fact_733_gen_H__cp__intros_I1_J,axiom,
    ! [Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,X: nat] :
      ( ( relational_gen_a_b2 @ Y @ Q @ G )
     => ( relational_gen_a_b2 @ X @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) )
        @ ( image_6790371041703824709la_a_b
          @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
          @ G ) ) ) ).

% gen'_cp_intros(1)
thf(fact_734_subst_Osimps_I3_J,axiom,
    ! [Z3: nat,T: relational_term_a,X: nat,Y: nat] :
      ( ( relational_subst_a_b @ ( relational_Eq_a_b @ Z3 @ T ) @ X @ Y )
      = ( relational_Eq_a_b @ ( if_nat @ ( Z3 = X ) @ Y @ Z3 ) @ ( relati7175845559408349773term_a @ T @ X @ Y ) ) ) ).

% subst.simps(3)
thf(fact_735_in__image__insert__iff,axiom,
    ! [B4: set_set_nat,X: nat,A2: set_nat] :
      ( ! [C4: set_nat] :
          ( ( member_set_nat @ C4 @ B4 )
         => ~ ( member_nat @ X @ C4 ) )
     => ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
        = ( ( member_nat @ X @ A2 )
          & ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_736_in__image__insert__iff,axiom,
    ! [B4: set_se6425085446898223655at_nat,X: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ! [C4: set_Pr8693737435421807431at_nat] :
          ( ( member3250002818903848048at_nat @ C4 @ B4 )
         => ~ ( member8206827879206165904at_nat @ X @ C4 ) )
     => ( ( member3250002818903848048at_nat @ A2 @ ( image_1013614488123988933at_nat @ ( insert5050368324300391991at_nat @ X ) @ B4 ) )
        = ( ( member8206827879206165904at_nat @ X @ A2 )
          & ( member3250002818903848048at_nat @ ( minus_8321449233255521966at_nat @ A2 @ ( insert5050368324300391991at_nat @ X @ bot_bo5327735625951526323at_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_737_in__image__insert__iff,axiom,
    ! [B4: set_se7855581050983116737at_nat,X: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ! [C4: set_Pr1261947904930325089at_nat] :
          ( ( member2643936169264416010at_nat @ C4 @ B4 )
         => ~ ( member8440522571783428010at_nat @ X @ C4 ) )
     => ( ( member2643936169264416010at_nat @ A2 @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X ) @ B4 ) )
        = ( ( member8440522571783428010at_nat @ X @ A2 )
          & ( member2643936169264416010at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_738_in__image__insert__iff,axiom,
    ! [B4: set_se5479201105039950375at_nat,X: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ! [C4: set_Pr4329608150637261639at_nat] :
          ( ( member6273874620674182256at_nat @ C4 @ B4 )
         => ~ ( member8757157785044589968at_nat @ X @ C4 ) )
     => ( ( member6273874620674182256at_nat @ A2 @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ X ) @ B4 ) )
        = ( ( member8757157785044589968at_nat @ X @ A2 )
          & ( member6273874620674182256at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_739_cov__nocp__intros,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b @ X @ Q @ G )
       => ( ( relational_gen_a_b @ Y @ Q @ Gy )
         => ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
            @ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
              @ ( image_6790371041703824709la_a_b
                @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
                @ Gy ) ) ) ) ) ) ).

% cov_nocp_intros
thf(fact_740_cov_H_OExists__gen,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b2 @ X @ Q @ G )
       => ( ( relational_gen_a_b @ Y @ Q @ Gy )
         => ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
            @ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
              @ ( image_6790371041703824709la_a_b
                @ ^ [Q3: relational_fmla_a_b] : ( relational_subst_a_b @ Q3 @ Y @ X )
                @ Gy ) ) ) ) ) ) ).

% cov'.Exists_gen
thf(fact_741_cov_OExists__gen,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b @ X @ Q @ G )
       => ( ( relational_gen_a_b @ Y @ Q @ Gy )
         => ( relational_cov_a_b @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
            @ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
              @ ( image_6790371041703824709la_a_b
                @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
                @ Gy ) ) ) ) ) ) ).

% cov.Exists_gen
thf(fact_742_cov_H__cp__intros,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b,G: set_Re381260168593705685la_a_b,Gy: set_Re381260168593705685la_a_b] :
      ( ( X != Y )
     => ( ( relational_cov_a_b2 @ X @ Q @ G )
       => ( ( relational_gen_a_b @ Y @ Q @ Gy )
         => ( relational_cov_a_b2 @ X @ ( relati591517084277583526ts_a_b @ Y @ Q )
            @ ( sup_su5130108678486352897la_a_b @ ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y ) @ ( minus_4077726661957047470la_a_b @ G @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ Y ) ) @ bot_bo4495933725496725865la_a_b ) ) )
              @ ( image_6790371041703824709la_a_b
                @ ^ [Q3: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Q3 @ Y @ X ) )
                @ Gy ) ) ) ) ) ) ).

% cov'_cp_intros
thf(fact_743_Eq__eq__subst__iff,axiom,
    ! [Y: nat,Z3: nat,Q: relational_fmla_a_b,X: nat] :
      ( ( ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z3 ) )
        = ( relational_subst_a_b @ Q @ X @ Y ) )
      = ( ( ( Z3 = X )
         => ( ( X = Y )
            & ( Q
              = ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) ) ) )
        & ( ( Z3 != X )
         => ( ( Q
              = ( relational_Eq_a_b @ X @ ( relational_Var_a @ Z3 ) ) )
            | ( Q
              = ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Z3 ) ) )
            | ( ( Z3 = Y )
              & ( member4680049679412964150la_a_b @ Q @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ X @ ( relational_Var_a @ X ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ Y ) ) @ ( insert7010464514620295119la_a_b @ ( relational_Eq_a_b @ Y @ ( relational_Var_a @ X ) ) @ bot_bo4495933725496725865la_a_b ) ) ) ) ) ) ) ) ) ).

% Eq_eq_subst_iff
thf(fact_744_fv__subst,axiom,
    ! [X: nat,Q: relational_fmla_a_b,Y: nat] :
      ( ( ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
          = ( insert_nat @ Y @ ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) )
      & ( ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_fv_a_b @ ( relational_subst_a_b @ Q @ X @ Y ) )
          = ( relational_fv_a_b @ Q ) ) ) ) ).

% fv_subst
thf(fact_745_gen__induct,axiom,
    ! [X1: nat,X2: relational_fmla_a_b,X33: set_Re381260168593705685la_a_b,P: nat > relational_fmla_a_b > set_Re381260168593705685la_a_b > $o] :
      ( ( relational_gen_a_b @ X1 @ X2 @ 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,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                  ( ( relational_gen_a_b @ X4 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G4 )
                 => ( ( P @ X4 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G4 )
                   => ( P @ X4 @ ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) @ G4 ) ) )
             => ( ! [X4: nat,Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b] :
                    ( ( relational_gen_a_b @ X4 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G4 )
                   => ( ( P @ X4 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ G4 )
                     => ( P @ X4 @ ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) @ G4 ) ) )
               => ( ! [X4: nat,Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b] :
                      ( ( relational_gen_a_b @ X4 @ Q12 @ G12 )
                     => ( ( P @ X4 @ Q12 @ G12 )
                       => ! [Q22: relational_fmla_a_b,G22: set_Re381260168593705685la_a_b] :
                            ( ( relational_gen_a_b @ X4 @ Q22 @ G22 )
                           => ( ( P @ X4 @ Q22 @ G22 )
                             => ( P @ X4 @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( sup_su5130108678486352897la_a_b @ G12 @ G22 ) ) ) ) ) )
                 => ( ! [X4: nat,Q12: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                        ( ( ( ( relational_gen_a_b @ X4 @ Q12 @ G4 )
                            & ( P @ X4 @ Q12 @ G4 ) )
                          | ( ( relational_gen_a_b @ X4 @ Q22 @ G4 )
                            & ( P @ X4 @ Q22 @ G4 ) ) )
                       => ( P @ X4 @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ 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 @ X1 @ X2 @ X33 ) ) ) ) ) ) ) ) ) ) ) ) ).

% gen_induct
thf(fact_746_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 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
                 => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
                   => ~ ( relational_gen_a_b2 @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
               => ( ! [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ! [G22: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G12 @ G22 ) )
                         => ( ( relational_gen_a_b2 @ A1 @ Q12 @ G12 )
                           => ~ ( relational_gen_a_b2 @ A1 @ Q22 @ G22 ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                       => ( ( A32 = G4 )
                         => ~ ( ( relational_gen_a_b2 @ A1 @ Q12 @ G4 )
                              | ( relational_gen_a_b2 @ A1 @ Q22 @ 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_747_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 ) ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
              & ( relational_gen_a_b2 @ A12 @ Q3 @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
              & ( relational_gen_a_b2 @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q13 @ Q23 ) )
              & ? [G23: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G13 @ G23 ) )
                  & ( relational_gen_a_b2 @ A12 @ Q13 @ G13 )
                  & ( relational_gen_a_b2 @ A12 @ Q23 @ G23 ) ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q13 @ Q23 ) )
              & ( ( relational_gen_a_b2 @ A12 @ Q13 @ A33 )
                | ( relational_gen_a_b2 @ A12 @ Q23 @ A33 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y4 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y4 @ A12 )
                      @ G3 ) )
                  & ( relational_gen_a_b2 @ Y4 @ Q3 @ G3 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y4 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_subst_a_b @ Qa @ Y4 @ A12 )
                      @ G3 ) )
                  & ( relational_gen_a_b2 @ Y4 @ Q3 @ G3 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y4 @ Q3 ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y4 ) @ G3 ) )
                  & ( A12 != Y4 )
                  & ( relational_gen_a_b2 @ A12 @ Q3 @ G3 ) ) ) ) ) ) ).

% gen'.simps
thf(fact_748_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 ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( A22
                    = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) )
                 => ~ ( relational_gen_a_b @ A1 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( A22
                      = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) )
                   => ~ ( relational_gen_a_b @ A1 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q12 ) @ ( relational_Neg_a_b @ Q22 ) ) @ A32 ) )
               => ( ! [Q12: relational_fmla_a_b,G12: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                      ( ( A22
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ! [G22: set_Re381260168593705685la_a_b] :
                          ( ( A32
                            = ( sup_su5130108678486352897la_a_b @ G12 @ G22 ) )
                         => ( ( relational_gen_a_b @ A1 @ Q12 @ G12 )
                           => ~ ( relational_gen_a_b @ A1 @ Q22 @ G22 ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,G4: set_Re381260168593705685la_a_b,Q22: relational_fmla_a_b] :
                        ( ( A22
                          = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                       => ( ( A32 = G4 )
                         => ~ ( ( relational_gen_a_b @ A1 @ Q12 @ G4 )
                              | ( relational_gen_a_b @ A1 @ Q22 @ 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_749_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 ) ) )
          | ? [Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Neg_a_b @ Q3 ) ) )
              & ( relational_gen_a_b @ A12 @ Q3 @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Disj_a_b @ Q13 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Conj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Neg_a_b @ ( relational_Conj_a_b @ Q13 @ Q23 ) ) )
              & ( relational_gen_a_b @ A12 @ ( relational_Disj_a_b @ ( relational_Neg_a_b @ Q13 ) @ ( relational_Neg_a_b @ Q23 ) ) @ A33 ) )
          | ? [Q13: relational_fmla_a_b,G13: set_Re381260168593705685la_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Disj_a_b @ Q13 @ Q23 ) )
              & ? [G23: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( sup_su5130108678486352897la_a_b @ G13 @ G23 ) )
                  & ( relational_gen_a_b @ A12 @ Q13 @ G13 )
                  & ( relational_gen_a_b @ A12 @ Q23 @ G23 ) ) )
          | ? [Q13: relational_fmla_a_b,Q23: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q13 @ Q23 ) )
              & ( ( relational_gen_a_b @ A12 @ Q13 @ A33 )
                | ( relational_gen_a_b @ A12 @ Q23 @ A33 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ A12 @ ( relational_Var_a @ Y4 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y4 @ A12 ) )
                      @ G3 ) )
                  & ( relational_gen_a_b @ Y4 @ Q3 @ G3 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relational_Conj_a_b @ Q3 @ ( relational_Eq_a_b @ Y4 @ ( relational_Var_a @ A12 ) ) ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b
                      @ ^ [Qa: relational_fmla_a_b] : ( relational_cp_a_b @ ( relational_subst_a_b @ Qa @ Y4 @ A12 ) )
                      @ G3 ) )
                  & ( relational_gen_a_b @ Y4 @ Q3 @ G3 ) ) )
          | ? [Y4: nat,Q3: relational_fmla_a_b] :
              ( ( A23
                = ( relati591517084277583526ts_a_b @ Y4 @ Q3 ) )
              & ? [G3: set_Re381260168593705685la_a_b] :
                  ( ( A33
                    = ( image_6790371041703824709la_a_b @ ( relati3989891337220013914ts_a_b @ Y4 ) @ G3 ) )
                  & ( A12 != Y4 )
                  & ( relational_gen_a_b @ A12 @ Q3 @ G3 ) ) ) ) ) ) ).

% gen.simps
thf(fact_750_finite__UN,axiom,
    ! [A2: set_list_a,B4: list_a > set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B4 @ A2 ) ) )
        = ( ! [X3: list_a] :
              ( ( member_list_a @ X3 @ A2 )
             => ( finite_finite_list_a @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_751_finite__UN,axiom,
    ! [A2: set_list_a,B4: list_a > set_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B4 @ A2 ) ) )
        = ( ! [X3: list_a] :
              ( ( member_list_a @ X3 @ A2 )
             => ( finite_finite_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_752_finite__UN,axiom,
    ! [A2: set_list_a,B4: list_a > set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7840782922268488802at_nat @ B4 @ A2 ) ) )
        = ( ! [X3: list_a] :
              ( ( member_list_a @ X3 @ A2 )
             => ( finite6177210948735845034at_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_753_finite__UN,axiom,
    ! [A2: set_nat,B4: nat > set_list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B4 @ A2 ) ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( finite_finite_list_a @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_754_finite__UN,axiom,
    ! [A2: set_nat,B4: nat > set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A2 ) ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( finite_finite_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_755_finite__UN,axiom,
    ! [A2: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B4 @ A2 ) ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
             => ( finite6177210948735845034at_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_756_finite__UN,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_list_a] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_921225794607697832list_a @ B4 @ A2 ) ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( finite_finite_list_a @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_757_finite__UN,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B4 @ A2 ) ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( finite_finite_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_758_finite__UN,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ A2 ) ) )
        = ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( finite6177210948735845034at_nat @ ( B4 @ X3 ) ) ) ) ) ) ).

% finite_UN
thf(fact_759_finite__Union,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ! [M: set_list_a] :
            ( ( member_set_list_a @ M @ A2 )
           => ( finite_finite_list_a @ M ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A2 ) ) ) ) ).

% finite_Union
thf(fact_760_finite__Union,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [M: set_nat] :
            ( ( member_set_nat @ M @ A2 )
           => ( finite_finite_nat @ M ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).

% finite_Union
thf(fact_761_finite__Union,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A2 )
     => ( ! [M: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ M @ A2 )
           => ( finite6177210948735845034at_nat @ M ) )
       => ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ A2 ) ) ) ) ).

% finite_Union
thf(fact_762_finite__UN__I,axiom,
    ! [A2: set_nat,B4: nat > set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( finite_finite_nat @ ( B4 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_763_finite__UN__I,axiom,
    ! [A2: set_list_a,B4: list_a > set_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ! [A5: list_a] :
            ( ( member_list_a @ A5 @ A2 )
           => ( finite_finite_nat @ ( B4 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_764_finite__UN__I,axiom,
    ! [A2: set_nat,B4: nat > set_list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( finite_finite_list_a @ ( B4 @ A5 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_765_finite__UN__I,axiom,
    ! [A2: set_list_a,B4: list_a > set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ! [A5: list_a] :
            ( ( member_list_a @ A5 @ A2 )
           => ( finite_finite_list_a @ ( B4 @ A5 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_5464838071766335845list_a @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_766_finite__UN__I,axiom,
    ! [A2: set_nat,B4: nat > set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( finite6177210948735845034at_nat @ ( B4 @ A5 ) ) )
       => ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_767_finite__UN__I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( finite_finite_nat @ ( B4 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_15824709712370754et_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_768_finite__UN__I,axiom,
    ! [A2: set_list_a,B4: list_a > set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ! [A5: list_a] :
            ( ( member_list_a @ A5 @ A2 )
           => ( finite6177210948735845034at_nat @ ( B4 @ A5 ) ) )
       => ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_7840782922268488802at_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_769_finite__UN__I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_list_a] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( finite_finite_list_a @ ( B4 @ A5 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_921225794607697832list_a @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_770_finite__UN__I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( finite6177210948735845034at_nat @ ( B4 @ A5 ) ) )
       => ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_771_finite__UN__I,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: produc859450856879609959at_nat > set_nat] :
      ( ( finite4392333629123659920at_nat @ A2 )
     => ( ! [A5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( finite_finite_nat @ ( B4 @ A5 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_2819269535336293544et_nat @ B4 @ A2 ) ) ) ) ) ).

% finite_UN_I
thf(fact_772_fresh2__gt_I3_J,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
     => ( ord_less_nat @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) ) ).

% fresh2_gt(3)
thf(fact_773_finite__UnionD,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A2 ) )
     => ( finite5282473924520328461list_a @ A2 ) ) ).

% finite_UnionD
thf(fact_774_finite__UnionD,axiom,
    ! [A2: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) )
     => ( finite1152437895449049373et_nat @ A2 ) ) ).

% finite_UnionD
thf(fact_775_finite__UnionD,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( finite6177210948735845034at_nat @ ( comple5685304695842803022at_nat @ A2 ) )
     => ( finite9047747110432174090at_nat @ A2 ) ) ).

% finite_UnionD
thf(fact_776_fresh2_I3_J,axiom,
    ! [X: nat,Y: nat,Q: relational_fmla_a_b] :
      ~ ( member_nat @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_fv_a_b @ Q ) ) ).

% fresh2(3)
thf(fact_777_finite__Sup__in,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( A2 != bot_bo1534825418459847827at_nat )
       => ( ! [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
              ( ( member6273874620674182256at_nat @ X4 @ A2 )
             => ( ( member6273874620674182256at_nat @ Y3 @ A2 )
               => ( member6273874620674182256at_nat @ ( sup_su5525570899277871387at_nat @ X4 @ Y3 ) @ A2 ) ) )
         => ( member6273874620674182256at_nat @ ( comple4364388969068068532at_nat @ A2 ) @ A2 ) ) ) ) ).

% finite_Sup_in
thf(fact_778_subst__exists,axiom,
    ! [Z3: nat,Q: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
       => ( ( ( X = Z3 )
           => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
              = ( relati3989891337220013914ts_a_b @ X @ Q ) ) )
          & ( ( X != Z3 )
           => ( ( ( Z3 = Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
                  = ( relati3989891337220013914ts_a_b @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q @ Z3 @ ( relati2677767559083392098h2_a_b @ X @ Y @ Q ) ) @ X @ Y ) ) ) )
              & ( ( Z3 != Y )
               => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
                  = ( relati3989891337220013914ts_a_b @ Z3 @ ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ) ) ) )
      & ( ~ ( member_nat @ Z3 @ ( relational_fv_a_b @ Q ) )
       => ( ( relational_subst_a_b @ ( relati3989891337220013914ts_a_b @ Z3 @ Q ) @ X @ Y )
          = ( relational_subst_a_b @ Q @ X @ Y ) ) ) ) ).

% subst_exists
thf(fact_779_UN__simps_I3_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( sup_su6327502436637775413at_nat @ A2 @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_780_UN__simps_I3_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat,B4: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_781_UN__simps_I3_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: set_Pr1261947904930325089at_nat,B4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su6327502436637775413at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su6327502436637775413at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( sup_su6327502436637775413at_nat @ A2 @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_782_UN__simps_I3_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(3)
thf(fact_783_UN__simps_I2_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat > set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ A2 @ C5 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_784_UN__simps_I2_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat > set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ A2 @ C5 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_785_UN__simps_I2_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su6327502436637775413at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su6327502436637775413at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ A2 @ C5 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_786_UN__simps_I2_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) )
          = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ A2 @ C5 ) ) @ B4 ) ) ) ) ).

% UN_simps(2)
thf(fact_787_UN__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat
        @ ( image_5423882813909314213at_nat
          @ ^ [X3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat )
          @ A2 ) )
      = A2 ) ).

% UN_singleton
thf(fact_788_UN__singleton,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat
        @ ( image_7364710425340685989at_nat
          @ ^ [X3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat )
          @ A2 ) )
      = A2 ) ).

% UN_singleton
thf(fact_789_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( ( comple5685304695842803022at_nat @ A2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A2 )
           => ( X3 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_790_Sup__bot__conv_I1_J,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( ( comple4364388969068068532at_nat @ A2 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A2 )
           => ( X3 = bot_bo228742789529271731at_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_791_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( comple5685304695842803022at_nat @ A2 ) )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A2 )
           => ( X3 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_792_Sup__bot__conv_I2_J,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( comple4364388969068068532at_nat @ A2 ) )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A2 )
           => ( X3 = bot_bo228742789529271731at_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_793_Sup__empty,axiom,
    ( ( comple5685304695842803022at_nat @ bot_bo3083307316010499117at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Sup_empty
thf(fact_794_Sup__empty,axiom,
    ( ( comple4364388969068068532at_nat @ bot_bo1534825418459847827at_nat )
    = bot_bo228742789529271731at_nat ) ).

% Sup_empty
thf(fact_795_UN__constant,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_796_UN__constant,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_797_UN__constant,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_798_UN__constant,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% UN_constant
thf(fact_799_UN__simps_I1_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_800_UN__simps_I1_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat,B4: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_801_UN__simps_I1_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A: product_prod_nat_nat,B4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_802_UN__simps_I1_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) )
          = ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) ) ) ) ) ).

% UN_simps(1)
thf(fact_803_SUP__Sup__eq2,axiom,
    ! [S: set_se6425085446898223655at_nat] :
      ( ( comple8323248879327187099_nat_o
        @ ( image_5230113887261543340_nat_o
          @ ^ [I3: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ I3 )
          @ S ) )
      = ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ ( comple2748534634209756852at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_804_SUP__Sup__eq2,axiom,
    ! [S: set_se7855581050983116737at_nat] :
      ( ( comple3592611370556534995_nat_o
        @ ( image_6923511907955291850_nat_o
          @ ^ [I3: set_Pr1261947904930325089at_nat,X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ I3 )
          @ S ) )
      = ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ ( comple5685304695842803022at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_805_SUP__Sup__eq2,axiom,
    ! [S: set_se5479201105039950375at_nat] :
      ( ( comple493620192902636699_nat_o
        @ ( image_6595146019508340652_nat_o
          @ ^ [I3: set_Pr4329608150637261639at_nat,X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ I3 )
          @ S ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ ( comple4364388969068068532at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_806_SUP__Sup__eq2,axiom,
    ! [S: set_se7592288975875140775at_nat] :
      ( ( comple4502628867940267931_nat_o
        @ ( image_5337237954834410540_nat_o
          @ ^ [I3: set_Pr7459493094073627847at_nat,X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ I3 )
          @ S ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ ( comple7342073961973170484at_nat @ S ) ) ) ) ).

% SUP_Sup_eq2
thf(fact_807_empty__Union__conv,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( comple5685304695842803022at_nat @ A2 ) )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A2 )
           => ( X3 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_808_empty__Union__conv,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( comple4364388969068068532at_nat @ A2 ) )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A2 )
           => ( X3 = bot_bo228742789529271731at_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_809_Union__empty__conv,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( ( comple5685304695842803022at_nat @ A2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A2 )
           => ( X3 = bot_bo2099793752762293965at_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_810_Union__empty__conv,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( ( comple4364388969068068532at_nat @ A2 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A2 )
           => ( X3 = bot_bo228742789529271731at_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_811_Union__empty,axiom,
    ( ( comple5685304695842803022at_nat @ bot_bo3083307316010499117at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Union_empty
thf(fact_812_Union__empty,axiom,
    ( ( comple4364388969068068532at_nat @ bot_bo1534825418459847827at_nat )
    = bot_bo228742789529271731at_nat ) ).

% Union_empty
thf(fact_813_UN__empty,axiom,
    ! [B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ bot_bo2099793752762293965at_nat ) )
      = bot_bo2099793752762293965at_nat ) ).

% UN_empty
thf(fact_814_UN__empty,axiom,
    ! [B4: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ bot_bo2099793752762293965at_nat ) )
      = bot_bo228742789529271731at_nat ) ).

% UN_empty
thf(fact_815_UN__empty,axiom,
    ! [B4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B4 @ bot_bo228742789529271731at_nat ) )
      = bot_bo2099793752762293965at_nat ) ).

% UN_empty
thf(fact_816_UN__empty,axiom,
    ! [B4: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ bot_bo228742789529271731at_nat ) )
      = bot_bo228742789529271731at_nat ) ).

% UN_empty
thf(fact_817_SUP__empty,axiom,
    ! [F3: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ F3 @ bot_bo2099793752762293965at_nat ) )
      = bot_bo2099793752762293965at_nat ) ).

% SUP_empty
thf(fact_818_SUP__empty,axiom,
    ! [F3: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ F3 @ bot_bo2099793752762293965at_nat ) )
      = bot_bo228742789529271731at_nat ) ).

% SUP_empty
thf(fact_819_SUP__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ F3 @ bot_bo228742789529271731at_nat ) )
      = bot_bo2099793752762293965at_nat ) ).

% SUP_empty
thf(fact_820_SUP__empty,axiom,
    ! [F3: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ F3 @ bot_bo228742789529271731at_nat ) )
      = bot_bo228742789529271731at_nat ) ).

% SUP_empty
thf(fact_821_SUP__constant,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_822_SUP__constant,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( A2 = bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo2099793752762293965at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [Y4: product_prod_nat_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_823_SUP__constant,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_824_SUP__constant,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( A2 = bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( A2 != bot_bo228742789529271731at_nat )
       => ( ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [Y4: produc3843707927480180839at_nat] : C2
              @ A2 ) )
          = C2 ) ) ) ).

% SUP_constant
thf(fact_825_UN__extend__simps_I1_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ C5 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ B4 @ C5 ) ) )
          = ( comple5685304695842803022at_nat
            @ ( image_5423882813909314213at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_826_UN__extend__simps_I1_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A: produc3843707927480180839at_nat,B4: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) )
          = ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_827_UN__extend__simps_I1_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A: product_prod_nat_nat,B4: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B4 @ C5 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( insert8211810215607154385at_nat @ A @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ B4 @ C5 ) ) )
          = ( comple5685304695842803022at_nat
            @ ( image_7979982168039674559at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert8211810215607154385at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_828_UN__extend__simps_I1_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( insert9069300056098147895at_nat @ A @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) )
          = ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( insert9069300056098147895at_nat @ A @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(1)
thf(fact_829_UN__extend__simps_I3_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat,B4: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) )
          = A2 ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ B4 @ C5 ) ) )
          = ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_830_UN__extend__simps_I3_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B4: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) )
          = A2 ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ B4 @ C5 ) ) )
          = ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ A2 @ ( B4 @ X3 ) )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(3)
thf(fact_831_UN__extend__simps_I2_J,axiom,
    ! [C5: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat > set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo2099793752762293965at_nat )
       => ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ A2 @ C5 ) ) @ B4 )
          = B4 ) )
      & ( ( C5 != bot_bo2099793752762293965at_nat )
       => ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ A2 @ C5 ) ) @ B4 )
          = ( comple4364388969068068532at_nat
            @ ( image_2893233658358539659at_nat
              @ ^ [X3: product_prod_nat_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_832_UN__extend__simps_I2_J,axiom,
    ! [C5: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( C5 = bot_bo228742789529271731at_nat )
       => ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ A2 @ C5 ) ) @ B4 )
          = B4 ) )
      & ( ( C5 != bot_bo228742789529271731at_nat )
       => ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ A2 @ C5 ) ) @ B4 )
          = ( comple4364388969068068532at_nat
            @ ( image_7364710425340685989at_nat
              @ ^ [X3: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( A2 @ X3 ) @ B4 )
              @ C5 ) ) ) ) ) ).

% UN_extend_simps(2)
thf(fact_833_UNION__fun__upd,axiom,
    ! [A2: nat > set_Pr1261947904930325089at_nat,I4: nat,B4: set_Pr1261947904930325089at_nat,J: set_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ ( fun_up810672174005406034at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_7178329752028323786at_nat @ A2 @ ( minus_minus_set_nat @ J @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) ) ) @ ( if_set4744415642439091239at_nat @ ( member_nat @ I4 @ J ) @ B4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_834_UNION__fun__upd,axiom,
    ! [A2: produc859450856879609959at_nat > set_Pr1261947904930325089at_nat,I4: produc859450856879609959at_nat,B4: set_Pr1261947904930325089at_nat,J: set_Pr8693737435421807431at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_1384942819424619199at_nat @ ( fun_up6473609724371469111at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_1384942819424619199at_nat @ A2 @ ( minus_8321449233255521966at_nat @ J @ ( insert5050368324300391991at_nat @ I4 @ bot_bo5327735625951526323at_nat ) ) ) ) @ ( if_set4744415642439091239at_nat @ ( member8206827879206165904at_nat @ I4 @ J ) @ B4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_835_UNION__fun__upd,axiom,
    ! [A2: nat > set_Pr4329608150637261639at_nat,I4: nat,B4: set_Pr4329608150637261639at_nat,J: set_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ ( fun_up6340599631146878136at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_1087905878993176368at_nat @ A2 @ ( minus_minus_set_nat @ J @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) ) ) @ ( if_set3208018901896979469at_nat @ ( member_nat @ I4 @ J ) @ B4 @ bot_bo228742789529271731at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_836_UNION__fun__upd,axiom,
    ! [A2: produc859450856879609959at_nat > set_Pr4329608150637261639at_nat,I4: produc859450856879609959at_nat,B4: set_Pr4329608150637261639at_nat,J: set_Pr8693737435421807431at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ ( fun_up2765403931048348445at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_3876558564303283877at_nat @ A2 @ ( minus_8321449233255521966at_nat @ J @ ( insert5050368324300391991at_nat @ I4 @ bot_bo5327735625951526323at_nat ) ) ) ) @ ( if_set3208018901896979469at_nat @ ( member8206827879206165904at_nat @ I4 @ J ) @ B4 @ bot_bo228742789529271731at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_837_UNION__fun__upd,axiom,
    ! [A2: product_prod_nat_nat > set_Pr1261947904930325089at_nat,I4: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ ( fun_up9187585116219995933at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_5423882813909314213at_nat @ A2 @ ( minus_1356011639430497352at_nat @ J @ ( insert8211810215607154385at_nat @ I4 @ bot_bo2099793752762293965at_nat ) ) ) ) @ ( if_set4744415642439091239at_nat @ ( member8440522571783428010at_nat @ I4 @ J ) @ B4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_838_UNION__fun__upd,axiom,
    ! [A2: product_prod_nat_nat > set_Pr4329608150637261639at_nat,I4: product_prod_nat_nat,B4: set_Pr4329608150637261639at_nat,J: set_Pr1261947904930325089at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ ( fun_up4323982136715714051at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ A2 @ ( minus_1356011639430497352at_nat @ J @ ( insert8211810215607154385at_nat @ I4 @ bot_bo2099793752762293965at_nat ) ) ) ) @ ( if_set3208018901896979469at_nat @ ( member8440522571783428010at_nat @ I4 @ J ) @ B4 @ bot_bo228742789529271731at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_839_UNION__fun__upd,axiom,
    ! [A2: produc3843707927480180839at_nat > set_Pr1261947904930325089at_nat,I4: produc3843707927480180839at_nat,B4: set_Pr1261947904930325089at_nat,J: set_Pr4329608150637261639at_nat] :
      ( ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ ( fun_up187358609542073143at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su6327502436637775413at_nat @ ( comple5685304695842803022at_nat @ ( image_7979982168039674559at_nat @ A2 @ ( minus_3314409938677909166at_nat @ J @ ( insert9069300056098147895at_nat @ I4 @ bot_bo228742789529271731at_nat ) ) ) ) @ ( if_set4744415642439091239at_nat @ ( member8757157785044589968at_nat @ I4 @ J ) @ B4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_840_UNION__fun__upd,axiom,
    ! [A2: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat,I4: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,J: set_Pr4329608150637261639at_nat] :
      ( ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ ( fun_up5865045902313378589at_nat @ A2 @ I4 @ B4 ) @ J ) )
      = ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ A2 @ ( minus_3314409938677909166at_nat @ J @ ( insert9069300056098147895at_nat @ I4 @ bot_bo228742789529271731at_nat ) ) ) ) @ ( if_set3208018901896979469at_nat @ ( member8757157785044589968at_nat @ I4 @ J ) @ B4 @ bot_bo228742789529271731at_nat ) ) ) ).

% UNION_fun_upd
thf(fact_841_cSup__eq__non__empty,axiom,
    ! [X5: set_nat,A: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ( ord_less_eq_nat @ X4 @ A ) )
       => ( ! [Y3: nat] :
              ( ! [X6: nat] :
                  ( ( member_nat @ X6 @ X5 )
                 => ( ord_less_eq_nat @ X6 @ Y3 ) )
             => ( ord_less_eq_nat @ A @ Y3 ) )
         => ( ( complete_Sup_Sup_nat @ X5 )
            = A ) ) ) ) ).

% cSup_eq_non_empty
thf(fact_842_cSup__least,axiom,
    ! [X5: set_nat,Z3: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ( ord_less_eq_nat @ X4 @ Z3 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z3 ) ) ) ).

% cSup_least
thf(fact_843_le__cSup__finite,axiom,
    ! [X5: set_nat,X: nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( member_nat @ X @ X5 )
       => ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X5 ) ) ) ) ).

% le_cSup_finite
thf(fact_844_less__cSupD,axiom,
    ! [X5: set_nat,Z3: nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ( ord_less_nat @ Z3 @ ( complete_Sup_Sup_nat @ X5 ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
            & ( ord_less_nat @ Z3 @ X4 ) ) ) ) ).

% less_cSupD
thf(fact_845_less__cSupE,axiom,
    ! [Y: nat,X5: set_nat] :
      ( ( ord_less_nat @ Y @ ( complete_Sup_Sup_nat @ X5 ) )
     => ( ( X5 != bot_bot_set_nat )
       => ~ ! [X4: nat] :
              ( ( member_nat @ X4 @ X5 )
             => ~ ( ord_less_nat @ Y @ X4 ) ) ) ) ).

% less_cSupE
thf(fact_846_finite__imp__Sup__less,axiom,
    ! [X5: set_nat,X: nat,A: nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( member_nat @ X @ X5 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ X5 )
             => ( ord_less_nat @ X4 @ A ) )
         => ( ord_less_nat @ ( complete_Sup_Sup_nat @ X5 ) @ A ) ) ) ) ).

% finite_imp_Sup_less
thf(fact_847_cSUP__least,axiom,
    ! [A2: set_nat,F3: nat > nat,M2: nat] :
      ( ( A2 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F3 @ X4 ) @ M2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F3 @ A2 ) ) @ M2 ) ) ) ).

% cSUP_least
thf(fact_848_cSUP__least,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,F3: produc859450856879609959at_nat > nat,M2: nat] :
      ( ( A2 != bot_bo5327735625951526323at_nat )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F3 @ X4 ) @ M2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_6160980552072806642at_nat @ F3 @ A2 ) ) @ M2 ) ) ) ).

% cSUP_least
thf(fact_849_cSUP__least,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > nat,M2: nat] :
      ( ( A2 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F3 @ X4 ) @ M2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2486076414777270412at_nat @ F3 @ A2 ) ) @ M2 ) ) ) ).

% cSUP_least
thf(fact_850_cSUP__least,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,F3: produc3843707927480180839at_nat > nat,M2: nat] :
      ( ( A2 != bot_bo228742789529271731at_nat )
     => ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F3 @ X4 ) @ M2 ) )
       => ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_3198525501578754290at_nat @ F3 @ A2 ) ) @ M2 ) ) ) ).

% cSUP_least
thf(fact_851_finite__Sup__less__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( X5 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X5 ) @ A )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ X5 )
               => ( ord_less_nat @ X3 @ A ) ) ) ) ) ) ).

% finite_Sup_less_iff
thf(fact_852_cSup__eq__Sup__fin,axiom,
    ! [X5: set_nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( X5 != bot_bot_set_nat )
       => ( ( complete_Sup_Sup_nat @ X5 )
          = ( lattic1093996805478795353in_nat @ X5 ) ) ) ) ).

% cSup_eq_Sup_fin
thf(fact_853_finite__subset__Union,axiom,
    ! [A2: set_list_a,B8: set_set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( ord_le8861187494160871172list_a @ A2 @ ( comple6928918032620976721list_a @ B8 ) )
       => ~ ! [F5: set_set_list_a] :
              ( ( finite5282473924520328461list_a @ F5 )
             => ( ( ord_le8877086941679407844list_a @ F5 @ B8 )
               => ~ ( ord_le8861187494160871172list_a @ A2 @ ( comple6928918032620976721list_a @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_854_finite__subset__Union,axiom,
    ! [A2: set_nat,B8: set_set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
       => ~ ! [F5: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F5 )
             => ( ( ord_le6893508408891458716et_nat @ F5 @ B8 )
               => ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_855_finite__subset__Union,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B8: set_se7855581050983116737at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( comple5685304695842803022at_nat @ B8 ) )
       => ~ ! [F5: set_se7855581050983116737at_nat] :
              ( ( finite9047747110432174090at_nat @ F5 )
             => ( ( ord_le2077887516847798113at_nat @ F5 @ B8 )
               => ~ ( ord_le3146513528884898305at_nat @ A2 @ ( comple5685304695842803022at_nat @ F5 ) ) ) ) ) ) ).

% finite_subset_Union
thf(fact_856_Union__image__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,F3: product_prod_nat_nat > set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_2893233658358539659at_nat @ F3 @ bot_bo2099793752762293965at_nat ) ) )
      = A2 ) ).

% Union_image_empty
thf(fact_857_Union__image__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,F3: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A2 @ ( comple4364388969068068532at_nat @ ( image_7364710425340685989at_nat @ F3 @ bot_bo228742789529271731at_nat ) ) )
      = A2 ) ).

% Union_image_empty
thf(fact_858_subst_Oelims,axiom,
    ! [X: relational_fmla_a_b,Xa: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ! [T2: $o] :
            ( ( X
              = ( relational_Bool_a_b @ T2 ) )
           => ( Y
             != ( relational_Bool_a_b @ T2 ) ) )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
             => ( Y
               != ( relational_Pred_b_a @ P2
                  @ ( map_Re5736185711816362116term_a
                    @ ^ [T7: relational_term_a] : ( relati7175845559408349773term_a @ T7 @ Xa @ Xb )
                    @ Ts ) ) ) )
         => ( ! [Z: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ Z @ T2 ) )
               => ( Y
                 != ( relational_Eq_a_b @ ( if_nat @ ( Z = Xa ) @ Xb @ Z ) @ ( relati7175845559408349773term_a @ T2 @ Xa @ Xb ) ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y
                   != ( relational_Neg_a_b @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( Y
                     != ( relational_Conj_a_b @ ( relational_subst_a_b @ Q12 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa @ Xb ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( Y
                       != ( relational_Disj_a_b @ ( relational_subst_a_b @ Q12 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa @ Xb ) ) ) )
                 => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                       => ~ ( ( ( Xa = Z )
                             => ( Y
                                = ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
                            & ( ( Xa != Z )
                             => ( ( ( Z = Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q4 @ Z @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) ) @ Xa @ Xb ) ) ) )
                                & ( ( Z != Xb )
                                 => ( Y
                                    = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.elims
thf(fact_859_finite__less__ub,axiom,
    ! [F3: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F3 @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F3 @ N4 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_860_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I4: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I4 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_861_subst_Opelims,axiom,
    ! [X: relational_fmla_a_b,Xa: nat,Xb: nat,Y: relational_fmla_a_b] :
      ( ( ( relational_subst_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ X @ ( product_Pair_nat_nat @ Xa @ Xb ) ) )
       => ( ! [T2: $o] :
              ( ( X
                = ( relational_Bool_a_b @ T2 ) )
             => ( ( Y
                  = ( relational_Bool_a_b @ T2 ) )
               => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T2 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
         => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P2 @ Ts ) )
               => ( ( Y
                    = ( relational_Pred_b_a @ P2
                      @ ( map_Re5736185711816362116term_a
                        @ ^ [T7: relational_term_a] : ( relati7175845559408349773term_a @ T7 @ Xa @ Xb )
                        @ Ts ) ) )
                 => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
           => ( ! [Z: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ Z @ T2 ) )
                 => ( ( Y
                      = ( relational_Eq_a_b @ ( if_nat @ ( Z = Xa ) @ Xb @ Z ) @ ( relati7175845559408349773term_a @ T2 @ Xa @ Xb ) ) )
                   => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T2 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y
                        = ( relational_Neg_a_b @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) )
                     => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Neg_a_b @ Q4 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( Y
                          = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q12 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa @ Xb ) ) )
                       => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( Y
                            = ( relational_Disj_a_b @ ( relational_subst_a_b @ Q12 @ Xa @ Xb ) @ ( relational_subst_a_b @ Q22 @ Xa @ Xb ) ) )
                         => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
                   => ~ ! [Z: nat,Q4: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Q4 ) )
                         => ( ( ( ( Xa = Z )
                               => ( Y
                                  = ( relati591517084277583526ts_a_b @ Xa @ Q4 ) ) )
                              & ( ( Xa != Z )
                               => ( ( ( Z = Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) @ ( relational_subst_a_b @ ( relational_subst_a_b @ Q4 @ Z @ ( relati2677767559083392098h2_a_b @ Xa @ Xb @ Q4 ) ) @ Xa @ Xb ) ) ) )
                                  & ( ( Z != Xb )
                                   => ( Y
                                      = ( relati591517084277583526ts_a_b @ Z @ ( relational_subst_a_b @ Q4 @ Xa @ Xb ) ) ) ) ) ) )
                           => ~ ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% subst.pelims
thf(fact_862_subst_Opsimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ X @ Y )
        = ( relational_Conj_a_b @ ( relational_subst_a_b @ Q1 @ X @ Y ) @ ( relational_subst_a_b @ Q2 @ X @ Y ) ) ) ) ).

% subst.psimps(5)
thf(fact_863_subst_Opsimps_I3_J,axiom,
    ! [Z3: nat,T: relational_term_a,X: nat,Y: nat] :
      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z3 @ T ) @ ( product_Pair_nat_nat @ X @ Y ) ) )
     => ( ( relational_subst_a_b @ ( relational_Eq_a_b @ Z3 @ T ) @ X @ Y )
        = ( relational_Eq_a_b @ ( if_nat @ ( Z3 = X ) @ Y @ Z3 ) @ ( relati7175845559408349773term_a @ T @ X @ Y ) ) ) ) ).

% subst.psimps(3)
thf(fact_864_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F6: nat > nat > nat,A5: nat,B: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F6 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_865_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 ) ) )
     => ( ! [T2: $o,X4: nat,Y3: nat] :
            ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Bool_a_b @ T2 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
           => ( P @ ( relational_Bool_a_b @ T2 ) @ X4 @ Y3 ) )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a,X4: nat,Y3: nat] :
              ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Pred_b_a @ P2 @ Ts ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
             => ( P @ ( relational_Pred_b_a @ P2 @ Ts ) @ X4 @ Y3 ) )
         => ( ! [Z: nat,T2: relational_term_a,X4: nat,Y3: nat] :
                ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Eq_a_b @ Z @ T2 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
               => ( P @ ( relational_Eq_a_b @ Z @ T2 ) @ 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 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat,Y3: nat] :
                    ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                   => ( ( P @ Q12 @ X4 @ Y3 )
                     => ( ( P @ Q22 @ X4 @ Y3 )
                       => ( P @ ( relational_Conj_a_b @ Q12 @ Q22 ) @ X4 @ Y3 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b,X4: nat,Y3: nat] :
                      ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                     => ( ( P @ Q12 @ X4 @ Y3 )
                       => ( ( P @ Q22 @ X4 @ Y3 )
                         => ( P @ ( relational_Disj_a_b @ Q12 @ Q22 ) @ X4 @ Y3 ) ) ) )
                 => ( ! [Z: nat,Q4: relational_fmla_a_b,X4: nat,Y3: nat] :
                        ( ( accp_P2470304046166516174at_nat @ relati8369873211719781654el_a_b @ ( produc6913411929637712585at_nat @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
                       => ( ! [Xa2: nat] :
                              ( ( X4 != Z )
                             => ( ( Z = Y3 )
                               => ( ( Xa2
                                    = ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q4 ) )
                                 => ( P @ Q4 @ Z @ Xa2 ) ) ) )
                         => ( ! [Xa2: nat] :
                                ( ( X4 != Z )
                               => ( ( Z = Y3 )
                                 => ( ( Xa2
                                      = ( relati2677767559083392098h2_a_b @ X4 @ Y3 @ Q4 ) )
                                   => ( P @ ( relational_subst_a_b @ Q4 @ Z @ Xa2 ) @ X4 @ Y3 ) ) ) )
                           => ( ( ( X4 != Z )
                               => ( ( Z != Y3 )
                                 => ( P @ Q4 @ X4 @ Y3 ) ) )
                             => ( P @ ( relati591517084277583526ts_a_b @ Z @ Q4 ) @ X4 @ Y3 ) ) ) ) )
                   => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).

% subst.pinduct
thf(fact_866_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N3 )
         => ( ord_less_nat @ X4 @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_867_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_868_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N5 )
         => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_869_less__by__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A2 @ B4 ) ) ).

% less_by_empty
thf(fact_870_less__by__empty,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( A2 = bot_bo228742789529271731at_nat )
     => ( ord_le1268244103169919719at_nat @ A2 @ B4 ) ) ).

% less_by_empty
thf(fact_871_empty__in__Fpow,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ bot_bo2099793752762293965at_nat @ ( finite4599829005654479975at_nat @ A2 ) ) ).

% empty_in_Fpow
thf(fact_872_empty__in__Fpow,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ bot_bo228742789529271731at_nat @ ( finite2966900854383870669at_nat @ A2 ) ) ).

% empty_in_Fpow
thf(fact_873_Fpow__def,axiom,
    ( finite_Fpow_list_a
    = ( ^ [A6: set_list_a] :
          ( collect_set_list_a
          @ ^ [X7: set_list_a] :
              ( ( ord_le8861187494160871172list_a @ X7 @ A6 )
              & ( finite_finite_list_a @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_874_Fpow__def,axiom,
    ( finite_Fpow_nat
    = ( ^ [A6: set_nat] :
          ( collect_set_nat
          @ ^ [X7: set_nat] :
              ( ( ord_less_eq_set_nat @ X7 @ A6 )
              & ( finite_finite_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_875_Fpow__def,axiom,
    ( finite4599829005654479975at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( collec5514110066124741708at_nat
          @ ^ [X7: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X7 @ A6 )
              & ( finite6177210948735845034at_nat @ X7 ) ) ) ) ) ).

% Fpow_def
thf(fact_876_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila4918476307565957903at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat
    @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ Y4 @ X3 )
    @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le7866589430770878221at_nat @ Y4 @ X3 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_877_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ( semila6534579987270727413at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat
    @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ X3 )
    @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le2604355607129572851at_nat @ Y4 @ X3 ) ) ).

% sup_bot.semilattice_neutr_order_axioms
thf(fact_878_cp_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X )
        = Y )
     => ( ! [X4: nat,T2: relational_term_a] :
            ( ( X
              = ( relational_Eq_a_b @ X4 @ T2 ) )
           => ( Y
             != ( relati582353067970734056la_a_b
                @ ^ [A3: a] : ( relational_Eq_a_b @ X4 @ T2 )
                @ ^ [Y4: nat] : ( if_Rel1279876242545935705la_a_b @ ( X4 = Y4 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) )
                @ T2 ) ) )
       => ( ! [Q4: relational_fmla_a_b] :
              ( ( X
                = ( relational_Neg_a_b @ Q4 ) )
             => ( Y
               != ( 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 ) ) ) ) )
         => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Conj_a_b @ Q12 @ Q22 ) )
               => ( Y
                 != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( relational_cp_a_b @ Q22 ) @ ( relational_Bool_a_b @ $false ) ) @ ( 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_cp_a_b @ Q12 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                 => ( Y
                   != ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( 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 @ Q12 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) ) )
             => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                   => ( Y
                     != ( relati3989891337220013914ts_a_b @ X4 @ ( relational_cp_a_b @ Q4 ) ) ) )
               => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                      ( ( X
                        = ( relational_Pred_b_a @ V2 @ Va2 ) )
                     => ( Y
                       != ( relational_Pred_b_a @ V2 @ Va2 ) ) )
                 => ~ ! [V2: $o] :
                        ( ( X
                          = ( relational_Bool_a_b @ V2 ) )
                       => ( Y
                         != ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ).

% cp.elims
thf(fact_879_cp_Osimps_I3_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_cp_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( 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_cp_a_b @ Q2 ) @ ( relational_Bool_a_b @ $false ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q2 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q2 ) ) @ ( relational_cp_a_b @ Q1 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q1 ) @ ( relational_cp_a_b @ Q2 ) ) ) ) ) ).

% cp.simps(3)
thf(fact_880_fmla_Odisc_I12_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.disc(12)
thf(fact_881_fmla_Odisc_I10_J,axiom,
    ! [X31: nat,X32: relational_term_a] :
      ~ ( relati6551038146797045342ol_a_b @ ( relational_Eq_a_b @ X31 @ X32 ) ) ).

% fmla.disc(10)
thf(fact_882_cpropagated__def,axiom,
    ( relati1591879772219623554ed_a_b
    = ( ^ [Q3: relational_fmla_a_b] :
          ( ( relational_nocp_a_b @ Q3 )
          | ( relati6551038146797045342ol_a_b @ Q3 ) ) ) ) ).

% cpropagated_def
thf(fact_883_cp_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( ( relational_cp_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ X )
       => ( ! [X4: nat,T2: relational_term_a] :
              ( ( X
                = ( relational_Eq_a_b @ X4 @ T2 ) )
             => ( ( Y
                  = ( relati582353067970734056la_a_b
                    @ ^ [A3: a] : ( relational_Eq_a_b @ X4 @ T2 )
                    @ ^ [Y4: nat] : ( if_Rel1279876242545935705la_a_b @ ( X4 = Y4 ) @ ( relational_Bool_a_b @ $true ) @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) )
                    @ T2 ) )
               => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) ) ) )
         => ( ! [Q4: relational_fmla_a_b] :
                ( ( X
                  = ( relational_Neg_a_b @ Q4 ) )
               => ( ( Y
                    = ( 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 ) ) ) )
           => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                 => ( ( Y
                      = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( relational_cp_a_b @ Q22 ) @ ( relational_Bool_a_b @ $false ) ) @ ( 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_cp_a_b @ Q12 ) @ ( relational_Bool_a_b @ $false ) ) @ ( relational_Conj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                   => ( ( Y
                        = ( if_Rel1279876242545935705la_a_b @ ( relati6551038146797045342ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( if_Rel1279876242545935705la_a_b @ ( relati2638701775882563405ol_a_b @ ( relational_cp_a_b @ Q12 ) ) @ ( 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 @ Q12 ) ) @ ( relational_Disj_a_b @ ( relational_cp_a_b @ Q12 ) @ ( relational_cp_a_b @ Q22 ) ) ) ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
               => ( ! [X4: nat,Q4: relational_fmla_a_b] :
                      ( ( X
                        = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                     => ( ( Y
                          = ( relati3989891337220013914ts_a_b @ X4 @ ( relational_cp_a_b @ Q4 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) )
                 => ( ! [V2: b,Va2: list_R6823256787227418703term_a] :
                        ( ( X
                          = ( relational_Pred_b_a @ V2 @ Va2 ) )
                       => ( ( Y
                            = ( relational_Pred_b_a @ V2 @ Va2 ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Pred_b_a @ V2 @ Va2 ) ) ) )
                   => ~ ! [V2: $o] :
                          ( ( X
                            = ( relational_Bool_a_b @ V2 ) )
                         => ( ( Y
                              = ( relational_Bool_a_b @ V2 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati5452380258257131376el_a_b @ ( relational_Bool_a_b @ V2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% cp.pelims
thf(fact_884_subst__term_Opelims,axiom,
    ! [X: relational_term_a,Xa: nat,Xb: nat,Y: relational_term_a] :
      ( ( ( relati7175845559408349773term_a @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ X @ ( product_Pair_nat_nat @ Xa @ Xb ) ) )
       => ( ! [Z: nat] :
              ( ( X
                = ( relational_Var_a @ Z ) )
             => ( ( Y
                  = ( relational_Var_a @ ( if_nat @ ( Xa = Z ) @ Xb @ Z ) ) )
               => ~ ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ ( relational_Var_a @ Z ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) )
         => ~ ! [C: a] :
                ( ( X
                  = ( relational_Const_a @ C ) )
               => ( ( Y
                    = ( relational_Const_a @ C ) )
                 => ~ ( accp_P7512640865500879912at_nat @ relati1089594453538404916_rel_a @ ( produc2180204704594896271at_nat @ ( relational_Const_a @ C ) @ ( product_Pair_nat_nat @ Xa @ Xb ) ) ) ) ) ) ) ) ).

% subst_term.pelims
thf(fact_885_fv__term__set_Opelims,axiom,
    ! [X: relational_term_a,Y: set_nat] :
      ( ( ( relati6004689760767320788_set_a @ X )
        = Y )
     => ( ( accp_R1925461290012245842term_a @ relati7433707222316806765_rel_a @ X )
       => ( ! [N: nat] :
              ( ( X
                = ( relational_Var_a @ N ) )
             => ( ( Y
                  = ( insert_nat @ N @ bot_bot_set_nat ) )
               => ~ ( accp_R1925461290012245842term_a @ relati7433707222316806765_rel_a @ ( relational_Var_a @ N ) ) ) )
         => ~ ! [V2: a] :
                ( ( X
                  = ( relational_Const_a @ V2 ) )
               => ( ( Y = bot_bot_set_nat )
                 => ~ ( accp_R1925461290012245842term_a @ relati7433707222316806765_rel_a @ ( relational_Const_a @ V2 ) ) ) ) ) ) ) ).

% fv_term_set.pelims
thf(fact_886_csts__term_Opelims,axiom,
    ! [X: relati8250976376238479564at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( relati7295153245664512040at_nat @ X )
        = Y )
     => ( ( accp_R5236243021290449749at_nat @ relati8570748485476786143at_nat @ X )
       => ( ! [X4: nat] :
              ( ( X
                = ( relati1023757457726692818at_nat @ X4 ) )
             => ( ( Y = bot_bo2099793752762293965at_nat )
               => ~ ( accp_R5236243021290449749at_nat @ relati8570748485476786143at_nat @ ( relati1023757457726692818at_nat @ X4 ) ) ) )
         => ~ ! [C: product_prod_nat_nat] :
                ( ( X
                  = ( relati6667065761635354038at_nat @ C ) )
               => ( ( Y
                    = ( insert8211810215607154385at_nat @ C @ bot_bo2099793752762293965at_nat ) )
                 => ~ ( accp_R5236243021290449749at_nat @ relati8570748485476786143at_nat @ ( relati6667065761635354038at_nat @ C ) ) ) ) ) ) ) ).

% csts_term.pelims
thf(fact_887_csts__term_Opelims,axiom,
    ! [X: relati2597207553810441522at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( relati8224766339378407694at_nat @ X )
        = Y )
     => ( ( accp_R7163988745762135611at_nat @ relati3755538294459431493at_nat @ X )
       => ( ! [X4: nat] :
              ( ( X
                = ( relati2721891637006993336at_nat @ X4 ) )
             => ( ( Y = bot_bo228742789529271731at_nat )
               => ~ ( accp_R7163988745762135611at_nat @ relati3755538294459431493at_nat @ ( relati2721891637006993336at_nat @ X4 ) ) ) )
         => ~ ! [C: produc3843707927480180839at_nat] :
                ( ( X
                  = ( relati3582544344963639708at_nat @ C ) )
               => ( ( Y
                    = ( insert9069300056098147895at_nat @ C @ bot_bo228742789529271731at_nat ) )
                 => ~ ( accp_R7163988745762135611at_nat @ relati3755538294459431493at_nat @ ( relati3582544344963639708at_nat @ C ) ) ) ) ) ) ) ).

% csts_term.pelims
thf(fact_888_csts__term_Opelims,axiom,
    ! [X: relational_term_a,Y: set_a] :
      ( ( ( relati6638259395341848997term_a @ X )
        = Y )
     => ( ( accp_R1925461290012245842term_a @ relati6793530228708600028_rel_a @ X )
       => ( ! [X4: nat] :
              ( ( X
                = ( relational_Var_a @ X4 ) )
             => ( ( Y = bot_bot_set_a )
               => ~ ( accp_R1925461290012245842term_a @ relati6793530228708600028_rel_a @ ( relational_Var_a @ X4 ) ) ) )
         => ~ ! [C: a] :
                ( ( X
                  = ( relational_Const_a @ C ) )
               => ( ( Y
                    = ( insert_a @ C @ bot_bot_set_a ) )
                 => ~ ( accp_R1925461290012245842term_a @ relati6793530228708600028_rel_a @ ( relational_Const_a @ C ) ) ) ) ) ) ) ).

% csts_term.pelims
thf(fact_889_in__measure,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F3: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur8038558561449204169at_nat @ F3 ) )
      = ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).

% in_measure
thf(fact_890_in__measure,axiom,
    ! [X: nat,Y: nat,F3: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F3 ) )
      = ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).

% in_measure
thf(fact_891_in__measure,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F3: set_Pr1261947904930325089at_nat > nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur1827424007717751593at_nat @ F3 ) )
      = ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).

% in_measure
thf(fact_892_in__measure,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,F3: set_Pr4329608150637261639at_nat > nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( measur4922264856574889999at_nat @ F3 ) )
      = ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).

% in_measure
thf(fact_893_in__finite__psubset,axiom,
    ! [A2: set_list_a,B4: set_list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A2 @ B4 ) @ finite881665802039894343list_a )
      = ( ( ord_less_set_list_a @ A2 @ B4 )
        & ( finite_finite_list_a @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_894_in__finite__psubset,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B4 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A2 @ B4 )
        & ( finite_finite_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_895_in__finite__psubset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B4 ) @ finite469560695537375940at_nat )
      = ( ( ord_le7866589430770878221at_nat @ A2 @ B4 )
        & ( finite6177210948735845034at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_896_in__finite__psubset,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B4 ) @ finite4695646753290404266at_nat )
      = ( ( ord_le2604355607129572851at_nat @ A2 @ B4 )
        & ( finite4343798906461161616at_nat @ B4 ) ) ) ).

% in_finite_psubset
thf(fact_897_fold__union__pair,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( sup_su718114333110466843at_nat
          @ ( comple2748534634209756852at_nat
            @ ( image_5803077947706746251at_nat
              @ ^ [Y4: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y4 ) @ bot_bo5327735625951526323at_nat )
              @ B4 ) )
          @ A2 )
        = ( finite8071398511639125710at_nat
          @ ^ [Y4: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y4 ) )
          @ A2
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_898_fold__union__pair,axiom,
    ! [B4: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat,A2: set_Pr7459493094073627847at_nat] :
      ( ( finite4768004253697144688at_nat @ B4 )
     => ( ( sup_su5395584912770621083at_nat
          @ ( comple7342073961973170484at_nat
            @ ( image_1124969593608450885at_nat
              @ ^ [Y4: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y4 ) @ bot_bo4948859079157340979at_nat )
              @ B4 ) )
          @ A2 )
        = ( finite3577408862935371528at_nat
          @ ^ [Y4: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y4 ) )
          @ A2
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_899_fold__union__pair,axiom,
    ! [B4: set_nat,X: nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( sup_su6327502436637775413at_nat
          @ ( comple5685304695842803022at_nat
            @ ( image_7178329752028323786at_nat
              @ ^ [Y4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ bot_bo2099793752762293965at_nat )
              @ B4 ) )
          @ A2 )
        = ( finite3745491028973389255at_nat
          @ ^ [Y4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) )
          @ A2
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_900_fold__union__pair,axiom,
    ! [B4: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ B4 )
     => ( ( sup_su5525570899277871387at_nat
          @ ( comple4364388969068068532at_nat
            @ ( image_6898982319499645995at_nat
              @ ^ [Y4: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) @ bot_bo228742789529271731at_nat )
              @ B4 ) )
          @ A2 )
        = ( finite3179781904389220718at_nat
          @ ^ [Y4: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y4 ) )
          @ A2
          @ B4 ) ) ) ).

% fold_union_pair
thf(fact_901_Id__on__def,axiom,
    ( id_on_2554058798563519774at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( comple2748534634209756852at_nat
          @ ( image_5803077947706746251at_nat
            @ ^ [X3: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ bot_bo5327735625951526323at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_902_Id__on__def,axiom,
    ( id_on_3275838214495744996at_nat
    = ( ^ [A6: set_se5479201105039950375at_nat] :
          ( comple7342073961973170484at_nat
          @ ( image_1124969593608450885at_nat
            @ ^ [X3: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) @ bot_bo4948859079157340979at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_903_Id__on__def,axiom,
    ( id_on_nat
    = ( ^ [A6: set_nat] :
          ( comple5685304695842803022at_nat
          @ ( image_7178329752028323786at_nat
            @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ bot_bo2099793752762293965at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_904_Id__on__def,axiom,
    ( id_on_6713504163309144702at_nat
    = ( ^ [A6: set_se7855581050983116737at_nat] :
          ( comple4364388969068068532at_nat
          @ ( image_6898982319499645995at_nat
            @ ^ [X3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ bot_bo228742789529271731at_nat )
            @ A6 ) ) ) ) ).

% Id_on_def
thf(fact_905_Id__onI,axiom,
    ! [A: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ A @ A2 )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( id_on_7917584626360399108at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_906_Id__onI,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( id_on_7869290023980668164at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_907_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_908_Id__onI,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_909_Id__onI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member2643936169264416010at_nat @ A @ A2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( id_on_6713504163309144702at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_910_Id__onI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,A2: set_se5479201105039950375at_nat] :
      ( ( member6273874620674182256at_nat @ A @ A2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( id_on_3275838214495744996at_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_911_Id__on__empty,axiom,
    ( ( id_on_nat @ bot_bot_set_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Id_on_empty
thf(fact_912_Id__on__empty,axiom,
    ( ( id_on_6713504163309144702at_nat @ bot_bo3083307316010499117at_nat )
    = bot_bo228742789529271731at_nat ) ).

% Id_on_empty
thf(fact_913_Id__on__empty,axiom,
    ( ( id_on_2554058798563519774at_nat @ bot_bo2099793752762293965at_nat )
    = bot_bo5327735625951526323at_nat ) ).

% Id_on_empty
thf(fact_914_Id__on__empty,axiom,
    ( ( id_on_7869290023980668164at_nat @ bot_bo228742789529271731at_nat )
    = bot_bo7193250159860175667at_nat ) ).

% Id_on_empty
thf(fact_915_Id__on__iff,axiom,
    ! [X: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y ) @ ( id_on_7917584626360399108at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member8206827879206165904at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_916_Id__on__iff,axiom,
    ! [X: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X @ Y ) @ ( id_on_7869290023980668164at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member8757157785044589968at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_917_Id__on__iff,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( id_on_2554058798563519774at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member8440522571783428010at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_918_Id__on__iff,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( id_on_nat @ A2 ) )
      = ( ( X = Y )
        & ( member_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_919_Id__on__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( id_on_6713504163309144702at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member2643936169264416010at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_920_Id__on__iff,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,A2: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( id_on_3275838214495744996at_nat @ A2 ) )
      = ( ( X = Y )
        & ( member6273874620674182256at_nat @ X @ A2 ) ) ) ).

% Id_on_iff
thf(fact_921_Id__on__eqI,axiom,
    ! [A: produc859450856879609959at_nat,B2: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( A = B2 )
     => ( ( member8206827879206165904at_nat @ A @ A2 )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ ( id_on_7917584626360399108at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_922_Id__on__eqI,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( A = B2 )
     => ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ ( id_on_7869290023980668164at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_923_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( A = B2 )
     => ( ( member8440522571783428010at_nat @ A @ A2 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( id_on_2554058798563519774at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_924_Id__on__eqI,axiom,
    ! [A: nat,B2: nat,A2: set_nat] :
      ( ( A = B2 )
     => ( ( member_nat @ A @ A2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( id_on_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_925_Id__on__eqI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( A = B2 )
     => ( ( member2643936169264416010at_nat @ A @ A2 )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( id_on_6713504163309144702at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_926_Id__on__eqI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_se5479201105039950375at_nat] :
      ( ( A = B2 )
     => ( ( member6273874620674182256at_nat @ A @ A2 )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( id_on_3275838214495744996at_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_927_Id__onE,axiom,
    ! [C2: produc349518998152878311at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ C2 @ ( id_on_7917584626360399108at_nat @ A2 ) )
     => ~ ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A2 )
           => ( C2
             != ( produc4662710985925991255at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_928_Id__onE,axiom,
    ! [C2: produc1906917888121415911at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member6009668390874684176at_nat @ C2 @ ( id_on_7869290023980668164at_nat @ A2 ) )
     => ~ ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A2 )
           => ( C2
             != ( produc2335810615602371415at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_929_Id__onE,axiom,
    ! [C2: produc859450856879609959at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( id_on_2554058798563519774at_nat @ A2 ) )
     => ~ ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A2 )
           => ( C2
             != ( produc6161850002892822231at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_930_Id__onE,axiom,
    ! [C2: product_prod_nat_nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A2 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( C2
             != ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_931_Id__onE,axiom,
    ! [C2: produc3843707927480180839at_nat,A2: set_se7855581050983116737at_nat] :
      ( ( member8757157785044589968at_nat @ C2 @ ( id_on_6713504163309144702at_nat @ A2 ) )
     => ~ ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ A2 )
           => ( C2
             != ( produc2922128104949294807at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_932_Id__onE,axiom,
    ! [C2: produc1319942482725812455at_nat,A2: set_se5479201105039950375at_nat] :
      ( ( member1466754251312161552at_nat @ C2 @ ( id_on_3275838214495744996at_nat @ A2 ) )
     => ~ ! [X4: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X4 @ A2 )
           => ( C2
             != ( produc9060074326276436823at_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_933_Id__on__fold,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( id_on_list_a @ A2 )
        = ( finite5942777621280601035list_a
          @ ^ [X3: list_a] : ( insert1856800524785285367list_a @ ( produc6837034575241423639list_a @ X3 @ X3 ) )
          @ bot_bo2955605580254355571list_a
          @ A2 ) ) ) ).

% Id_on_fold
thf(fact_934_Id__on__fold,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( id_on_2554058798563519774at_nat @ A2 )
        = ( finite8071398511639125710at_nat
          @ ^ [X3: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) )
          @ bot_bo5327735625951526323at_nat
          @ A2 ) ) ) ).

% Id_on_fold
thf(fact_935_Id__on__fold,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( id_on_3275838214495744996at_nat @ A2 )
        = ( finite3577408862935371528at_nat
          @ ^ [X3: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ X3 ) )
          @ bot_bo4948859079157340979at_nat
          @ A2 ) ) ) ).

% Id_on_fold
thf(fact_936_Id__on__fold,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( id_on_nat @ A2 )
        = ( finite3745491028973389255at_nat
          @ ^ [X3: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) )
          @ bot_bo2099793752762293965at_nat
          @ A2 ) ) ) ).

% Id_on_fold
thf(fact_937_Id__on__fold,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A2 )
     => ( ( id_on_6713504163309144702at_nat @ A2 )
        = ( finite3179781904389220718at_nat
          @ ^ [X3: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) )
          @ bot_bo228742789529271731at_nat
          @ A2 ) ) ) ).

% Id_on_fold
thf(fact_938_fold__closed__eq,axiom,
    ! [A2: set_nat,B4: set_nat,F3: nat > nat > nat,G5: nat > nat > nat,Z3: nat] :
      ( ! [A5: nat,B: nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite_fold_nat_nat @ F3 @ Z3 @ A2 )
            = ( finite_fold_nat_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_939_fold__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat > product_prod_nat_nat,G5: nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: nat,B: product_prod_nat_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: product_prod_nat_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite7143988214204320871at_nat @ F3 @ Z3 @ A2 )
            = ( finite7143988214204320871at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_940_fold__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_nat,F3: product_prod_nat_nat > nat > nat,G5: product_prod_nat_nat > nat > nat,Z3: nat] :
      ( ! [A5: product_prod_nat_nat,B: nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite3783940821161605769at_nat @ F3 @ Z3 @ A2 )
            = ( finite3783940821161605769at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_941_fold__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite2688938693967789832at_nat @ F3 @ Z3 @ A2 )
            = ( finite2688938693967789832at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_942_fold__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr8693737435421807431at_nat,F3: nat > produc859450856879609959at_nat > produc859450856879609959at_nat,G5: nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat] :
      ( ! [A5: nat,B: produc859450856879609959at_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8206827879206165904at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: produc859450856879609959at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
         => ( ( finite6355506772761797325at_nat @ F3 @ Z3 @ A2 )
            = ( finite6355506772761797325at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_943_fold__closed__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_nat,F3: produc859450856879609959at_nat > nat > nat,G5: produc859450856879609959at_nat > nat > nat,Z3: nat] :
      ( ! [A5: produc859450856879609959at_nat,B: nat] :
          ( ( member8206827879206165904at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc859450856879609959at_nat,B: nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite16511661022317423at_nat @ F3 @ Z3 @ A2 )
            = ( finite16511661022317423at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_944_fold__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr4329608150637261639at_nat,F3: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat,G5: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
      ( ! [A5: nat,B: produc3843707927480180839at_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8757157785044589968at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: produc3843707927480180839at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8757157785044589968at_nat @ B @ B4 )
             => ( member8757157785044589968at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8757157785044589968at_nat @ Z3 @ B4 )
         => ( ( finite2084110811532196557at_nat @ F3 @ Z3 @ A2 )
            = ( finite2084110811532196557at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_945_fold__closed__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,F3: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat,G5: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite9073100149838362274at_nat @ F3 @ Z3 @ A2 )
            = ( finite9073100149838362274at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_946_fold__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr8693737435421807431at_nat,F3: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat,G5: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member8206827879206165904at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
         => ( ( finite3486118685651320814at_nat @ F3 @ Z3 @ A2 )
            = ( finite3486118685651320814at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_947_fold__closed__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_nat,F3: produc3843707927480180839at_nat > nat > nat,G5: produc3843707927480180839at_nat > nat > nat,Z3: nat] :
      ( ! [A5: produc3843707927480180839at_nat,B: nat] :
          ( ( member8757157785044589968at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc3843707927480180839at_nat,B: nat] :
            ( ( member8757157785044589968at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite2783622653218364271at_nat @ F3 @ Z3 @ A2 )
            = ( finite2783622653218364271at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_closed_eq
thf(fact_948_union__fold__insert,axiom,
    ! [A2: set_list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( sup_sup_set_list_a @ A2 @ B4 )
        = ( finite9084501323850038056list_a @ insert_list_a @ B4 @ A2 ) ) ) ).

% union_fold_insert
thf(fact_949_union__fold__insert,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B4 )
        = ( finite5529483035118572448et_nat @ insert_nat @ B4 @ A2 ) ) ) ).

% union_fold_insert
thf(fact_950_union__fold__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( sup_su6327502436637775413at_nat @ A2 @ B4 )
        = ( finite5618522546757629032at_nat @ insert8211810215607154385at_nat @ B4 @ A2 ) ) ) ).

% union_fold_insert
thf(fact_951_union__fold__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ A2 @ B4 )
        = ( finite2056329533557663592at_nat @ insert9069300056098147895at_nat @ B4 @ A2 ) ) ) ).

% union_fold_insert
thf(fact_952_sup__Sup__fold__sup,axiom,
    ! [A2: set_se5479201105039950375at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( sup_su5525570899277871387at_nat @ ( comple4364388969068068532at_nat @ A2 ) @ B4 )
        = ( finite4349107891793258632at_nat @ sup_su5525570899277871387at_nat @ B4 @ A2 ) ) ) ).

% sup_Sup_fold_sup
thf(fact_953_minus__fold__remove,axiom,
    ! [A2: set_list_a,B4: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( minus_646659088055828811list_a @ B4 @ A2 )
        = ( finite9084501323850038056list_a @ remove_list_a @ B4 @ A2 ) ) ) ).

% minus_fold_remove
thf(fact_954_minus__fold__remove,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( minus_minus_set_nat @ B4 @ A2 )
        = ( finite5529483035118572448et_nat @ remove_nat @ B4 @ A2 ) ) ) ).

% minus_fold_remove
thf(fact_955_minus__fold__remove,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( minus_1356011639430497352at_nat @ B4 @ A2 )
        = ( finite5618522546757629032at_nat @ remove6466555014256735590at_nat @ B4 @ A2 ) ) ) ).

% minus_fold_remove
thf(fact_956_Sup__fold__sup,axiom,
    ! [A2: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A2 )
     => ( ( comple5685304695842803022at_nat @ A2 )
        = ( finite893940847271462664at_nat @ sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ) ) ).

% Sup_fold_sup
thf(fact_957_Sup__fold__sup,axiom,
    ! [A2: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( comple4364388969068068532at_nat @ A2 )
        = ( finite4349107891793258632at_nat @ sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 ) ) ) ).

% Sup_fold_sup
thf(fact_958_Sup__fin_Oeq__fold,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A2 ) )
        = ( finite_fold_nat_nat @ sup_sup_nat @ X @ A2 ) ) ) ).

% Sup_fin.eq_fold
thf(fact_959_Sup__fin_Oeq__fold,axiom,
    ! [A2: set_se5479201105039950375at_nat,X: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( lattic8394993312974875902at_nat @ ( insert2290795007606087959at_nat @ X @ A2 ) )
        = ( finite4349107891793258632at_nat @ sup_su5525570899277871387at_nat @ X @ A2 ) ) ) ).

% Sup_fin.eq_fold
thf(fact_960_image__fold__insert,axiom,
    ! [A2: set_list_a,F3: list_a > product_prod_nat_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( image_3288686587095672578at_nat @ F3 @ A2 )
        = ( finite6430252142798181925at_nat
          @ ^ [K2: list_a] : ( insert8211810215607154385at_nat @ ( F3 @ K2 ) )
          @ bot_bo2099793752762293965at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_961_image__fold__insert,axiom,
    ! [A2: set_nat,F3: nat > product_prod_nat_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( image_5846123807819985514at_nat @ F3 @ A2 )
        = ( finite3745491028973389255at_nat
          @ ^ [K2: nat] : ( insert8211810215607154385at_nat @ ( F3 @ K2 ) )
          @ bot_bo2099793752762293965at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_962_image__fold__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( image_5168914502847457605at_nat @ F3 @ A2 )
        = ( finite5618522546757629032at_nat
          @ ^ [K2: product_prod_nat_nat] : ( insert8211810215607154385at_nat @ ( F3 @ K2 ) )
          @ bot_bo2099793752762293965at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_963_image__fold__insert,axiom,
    ! [A2: set_list_a,F3: list_a > produc3843707927480180839at_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( image_8960669938008688872at_nat @ F3 @ A2 )
        = ( finite446271417765581067at_nat
          @ ^ [K2: list_a] : ( insert9069300056098147895at_nat @ ( F3 @ K2 ) )
          @ bot_bo228742789529271731at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_964_image__fold__insert,axiom,
    ! [A2: set_nat,F3: nat > produc3843707927480180839at_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( image_2499013659892586576at_nat @ F3 @ A2 )
        = ( finite9145620488728376749at_nat
          @ ^ [K2: nat] : ( insert9069300056098147895at_nat @ ( F3 @ K2 ) )
          @ bot_bo228742789529271731at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_965_image__fold__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > produc3843707927480180839at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( image_1672857615394513579at_nat @ F3 @ A2 )
        = ( finite8280934518123415246at_nat
          @ ^ [K2: product_prod_nat_nat] : ( insert9069300056098147895at_nat @ ( F3 @ K2 ) )
          @ bot_bo228742789529271731at_nat
          @ A2 ) ) ) ).

% image_fold_insert
thf(fact_966_Set__filter__fold,axiom,
    ! [A2: set_list_a,P: list_a > $o] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( filter_list_a @ P @ A2 )
        = ( finite9084501323850038056list_a
          @ ^ [X3: list_a,A9: set_list_a] : ( if_set_list_a @ ( P @ X3 ) @ ( insert_list_a @ X3 @ A9 ) @ A9 )
          @ bot_bot_set_list_a
          @ A2 ) ) ) ).

% Set_filter_fold
thf(fact_967_Set__filter__fold,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( filter_nat @ P @ A2 )
        = ( finite5529483035118572448et_nat
          @ ^ [X3: nat,A9: set_nat] : ( if_set_nat @ ( P @ X3 ) @ ( insert_nat @ X3 @ A9 ) @ A9 )
          @ bot_bot_set_nat
          @ A2 ) ) ) ).

% Set_filter_fold
thf(fact_968_Set__filter__fold,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( filter5640266504077782706at_nat @ P @ A2 )
        = ( finite5618522546757629032at_nat
          @ ^ [X3: product_prod_nat_nat,A9: set_Pr1261947904930325089at_nat] : ( if_set4744415642439091239at_nat @ ( P @ X3 ) @ ( insert8211810215607154385at_nat @ X3 @ A9 ) @ A9 )
          @ bot_bo2099793752762293965at_nat
          @ A2 ) ) ) ).

% Set_filter_fold
thf(fact_969_Set__filter__fold,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( filter6899231104951847064at_nat @ P @ A2 )
        = ( finite2056329533557663592at_nat
          @ ^ [X3: produc3843707927480180839at_nat,A9: set_Pr4329608150637261639at_nat] : ( if_set3208018901896979469at_nat @ ( P @ X3 ) @ ( insert9069300056098147895at_nat @ X3 @ A9 ) @ A9 )
          @ bot_bo228742789529271731at_nat
          @ A2 ) ) ) ).

% Set_filter_fold
thf(fact_970_comp__fun__commute__product__fold,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( finite3773130646654257623at_nat
        @ ^ [X3: product_prod_nat_nat,Z5: set_Pr8693737435421807431at_nat] :
            ( finite8071398511639125710at_nat
            @ ^ [Y4: product_prod_nat_nat] : ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) )
            @ Z5
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_971_comp__fun__commute__product__fold,axiom,
    ! [B4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( finite1174777150170982526at_nat
        @ ^ [X3: nat,Z5: set_Pr1261947904930325089at_nat] :
            ( finite3745491028973389255at_nat
            @ ^ [Y4: nat] : ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) )
            @ Z5
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_972_comp__fun__commute__product__fold,axiom,
    ! [B4: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ B4 )
     => ( finite9202498192941607031at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Z5: set_Pr4329608150637261639at_nat] :
            ( finite3179781904389220718at_nat
            @ ^ [Y4: set_Pr1261947904930325089at_nat] : ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) )
            @ Z5
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_973_comp__fun__commute__product__fold,axiom,
    ! [B4: set_se5479201105039950375at_nat] :
      ( ( finite4768004253697144688at_nat @ B4 )
     => ( finite4517476712773419409at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Z5: set_Pr7459493094073627847at_nat] :
            ( finite3577408862935371528at_nat
            @ ^ [Y4: set_Pr4329608150637261639at_nat] : ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) )
            @ Z5
            @ B4 ) ) ) ).

% comp_fun_commute_product_fold
thf(fact_974_comp__fun__commute__filter__fold,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( finite5656229966737060081at_nat
      @ ^ [X3: product_prod_nat_nat,A9: set_Pr1261947904930325089at_nat] : ( if_set4744415642439091239at_nat @ ( P @ X3 ) @ ( insert8211810215607154385at_nat @ X3 @ A9 ) @ A9 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_975_comp__fun__commute__filter__fold,axiom,
    ! [P: produc3843707927480180839at_nat > $o] :
      ( finite6887329086674120945at_nat
      @ ^ [X3: produc3843707927480180839at_nat,A9: set_Pr4329608150637261639at_nat] : ( if_set3208018901896979469at_nat @ ( P @ X3 ) @ ( insert9069300056098147895at_nat @ X3 @ A9 ) @ A9 ) ) ).

% comp_fun_commute_filter_fold
thf(fact_976_finite__filter,axiom,
    ! [S: set_list_a,P: list_a > $o] :
      ( ( finite_finite_list_a @ S )
     => ( finite_finite_list_a @ ( filter_list_a @ P @ S ) ) ) ).

% finite_filter
thf(fact_977_finite__filter,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( finite_finite_nat @ ( filter_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_978_finite__filter,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( finite6177210948735845034at_nat @ ( filter5640266504077782706at_nat @ P @ S ) ) ) ).

% finite_filter
thf(fact_979_comp__fun__commute__Pow__fold,axiom,
    ( finite606358831919644241at_nat
    @ ^ [X3: product_prod_nat_nat,A6: set_se7855581050983116737at_nat] : ( sup_su3642409539654194069at_nat @ A6 @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X3 ) @ A6 ) ) ) ).

% comp_fun_commute_Pow_fold
thf(fact_980_comp__fun__commute__Pow__fold,axiom,
    ( finite1617758837010880465at_nat
    @ ^ [X3: produc3843707927480180839at_nat,A6: set_se5479201105039950375at_nat] : ( sup_su474284620922681339at_nat @ A6 @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ X3 ) @ A6 ) ) ) ).

% comp_fun_commute_Pow_fold
thf(fact_981_Pow__fold,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( pow_list_a @ A2 )
        = ( finite5843373012576554760list_a
          @ ^ [X3: list_a,A6: set_set_list_a] : ( sup_su4537662296134749976list_a @ A6 @ ( image_5749939591322298757list_a @ ( insert_list_a @ X3 ) @ A6 ) )
          @ ( insert_set_list_a @ bot_bot_set_list_a @ bot_bo3186585308812441520list_a )
          @ A2 ) ) ) ).

% Pow_fold
thf(fact_982_Pow__fold,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( pow_nat @ A2 )
        = ( finite4178521680790401110et_nat
          @ ^ [X3: nat,A6: set_set_nat] : ( sup_sup_set_set_nat @ A6 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ A6 ) )
          @ ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
          @ A2 ) ) ) ).

% Pow_fold
thf(fact_983_Pow__fold,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( pow_Pr3466873192417945886at_nat @ A2 )
        = ( finite1811527957910857160at_nat
          @ ^ [X3: product_prod_nat_nat,A6: set_se7855581050983116737at_nat] : ( sup_su3642409539654194069at_nat @ A6 @ ( image_3684629450409544005at_nat @ ( insert8211810215607154385at_nat @ X3 ) @ A6 ) )
          @ ( insert9200635055090092081at_nat @ bot_bo2099793752762293965at_nat @ bot_bo3083307316010499117at_nat )
          @ A2 ) ) ) ).

% Pow_fold
thf(fact_984_Pow__fold,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( pow_Pr1023649790151954692at_nat @ A2 )
        = ( finite2302676418706223688at_nat
          @ ^ [X3: produc3843707927480180839at_nat,A6: set_se5479201105039950375at_nat] : ( sup_su474284620922681339at_nat @ A6 @ ( image_4303274092540115909at_nat @ ( insert9069300056098147895at_nat @ X3 ) @ A6 ) )
          @ ( insert2290795007606087959at_nat @ bot_bo228742789529271731at_nat @ bot_bo1534825418459847827at_nat )
          @ A2 ) ) ) ).

% Pow_fold
thf(fact_985_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_986_inf__bot__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% inf_bot_right
thf(fact_987_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_988_inf__bot__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% inf_bot_left
thf(fact_989_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_990_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ X @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_991_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_992_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ X )
      = bot_bo228742789529271731at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_993_finite__Int,axiom,
    ! [F: set_list_a,G: set_list_a] :
      ( ( ( finite_finite_list_a @ F )
        | ( finite_finite_list_a @ G ) )
     => ( finite_finite_list_a @ ( inf_inf_set_list_a @ F @ G ) ) ) ).

% finite_Int
thf(fact_994_finite__Int,axiom,
    ! [F: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).

% finite_Int
thf(fact_995_finite__Int,axiom,
    ! [F: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F )
        | ( finite6177210948735845034at_nat @ G ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F @ G ) ) ) ).

% finite_Int
thf(fact_996_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B4 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ A2 @ B4 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_997_insert__disjoint_I1_J,axiom,
    ! [A: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ( inf_in4302113700860409141at_nat @ ( insert5050368324300391991at_nat @ A @ A2 ) @ B4 )
        = bot_bo5327735625951526323at_nat )
      = ( ~ ( member8206827879206165904at_nat @ A @ B4 )
        & ( ( inf_in4302113700860409141at_nat @ A2 @ B4 )
          = bot_bo5327735625951526323at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_998_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_999_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1000_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B4 ) )
      = ( ~ ( member_nat @ A @ B4 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1001_insert__disjoint_I2_J,axiom,
    ! [A: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( inf_in4302113700860409141at_nat @ ( insert5050368324300391991at_nat @ A @ A2 ) @ B4 ) )
      = ( ~ ( member8206827879206165904at_nat @ A @ B4 )
        & ( bot_bo5327735625951526323at_nat
          = ( inf_in4302113700860409141at_nat @ A2 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1002_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) @ B4 ) )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A2 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1003_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) @ B4 ) )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A2 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1004_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A: nat,A2: set_nat] :
      ( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A2 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ B4 @ A2 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1005_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( ( inf_in4302113700860409141at_nat @ B4 @ ( insert5050368324300391991at_nat @ A @ A2 ) )
        = bot_bo5327735625951526323at_nat )
      = ( ~ ( member8206827879206165904at_nat @ A @ B4 )
        & ( ( inf_in4302113700860409141at_nat @ B4 @ A2 )
          = bot_bo5327735625951526323at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1006_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B4 @ ( insert8211810215607154385at_nat @ A @ A2 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( ( inf_in2572325071724192079at_nat @ B4 @ A2 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1007_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ B4 @ ( insert9069300056098147895at_nat @ A @ A2 ) )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( ( inf_in7913087082777306421at_nat @ B4 @ A2 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1008_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B2: nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A2 @ ( insert_nat @ B2 @ B4 ) ) )
      = ( ~ ( member_nat @ B2 @ A2 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1009_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B2: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( bot_bo5327735625951526323at_nat
        = ( inf_in4302113700860409141at_nat @ A2 @ ( insert5050368324300391991at_nat @ B2 @ B4 ) ) )
      = ( ~ ( member8206827879206165904at_nat @ B2 @ A2 )
        & ( bot_bo5327735625951526323at_nat
          = ( inf_in4302113700860409141at_nat @ A2 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1010_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A2 @ ( insert8211810215607154385at_nat @ B2 @ B4 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B2 @ A2 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A2 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1011_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ A2 @ ( insert9069300056098147895at_nat @ B2 @ B4 ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B2 @ A2 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A2 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1012_Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B4 @ A2 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_1013_Diff__disjoint,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B4 @ A2 ) )
      = bot_bo228742789529271731at_nat ) ).

% Diff_disjoint
thf(fact_1014_inf__Sup__absorb,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1015_finite__Pow__iff,axiom,
    ! [A2: set_list_a] :
      ( ( finite5282473924520328461list_a @ ( pow_list_a @ A2 ) )
      = ( finite_finite_list_a @ A2 ) ) ).

% finite_Pow_iff
thf(fact_1016_finite__Pow__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ ( pow_nat @ A2 ) )
      = ( finite_finite_nat @ A2 ) ) ).

% finite_Pow_iff
thf(fact_1017_finite__Pow__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ ( pow_Pr3466873192417945886at_nat @ A2 ) )
      = ( finite6177210948735845034at_nat @ A2 ) ) ).

% finite_Pow_iff
thf(fact_1018_Pow__singleton__iff,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat] :
      ( ( ( pow_Pr3466873192417945886at_nat @ X5 )
        = ( insert9200635055090092081at_nat @ Y7 @ bot_bo3083307316010499117at_nat ) )
      = ( ( X5 = bot_bo2099793752762293965at_nat )
        & ( Y7 = bot_bo2099793752762293965at_nat ) ) ) ).

% Pow_singleton_iff
thf(fact_1019_Pow__singleton__iff,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat] :
      ( ( ( pow_Pr1023649790151954692at_nat @ X5 )
        = ( insert2290795007606087959at_nat @ Y7 @ bot_bo1534825418459847827at_nat ) )
      = ( ( X5 = bot_bo228742789529271731at_nat )
        & ( Y7 = bot_bo228742789529271731at_nat ) ) ) ).

% Pow_singleton_iff
thf(fact_1020_Pow__empty,axiom,
    ( ( pow_Pr3466873192417945886at_nat @ bot_bo2099793752762293965at_nat )
    = ( insert9200635055090092081at_nat @ bot_bo2099793752762293965at_nat @ bot_bo3083307316010499117at_nat ) ) ).

% Pow_empty
thf(fact_1021_Pow__empty,axiom,
    ( ( pow_Pr1023649790151954692at_nat @ bot_bo228742789529271731at_nat )
    = ( insert2290795007606087959at_nat @ bot_bo228742789529271731at_nat @ bot_bo1534825418459847827at_nat ) ) ).

% Pow_empty
thf(fact_1022_Fpow__Pow__finite,axiom,
    ( finite_Fpow_list_a
    = ( ^ [A6: set_list_a] : ( inf_in4657809108759609906list_a @ ( pow_list_a @ A6 ) @ ( collect_set_list_a @ finite_finite_list_a ) ) ) ) ).

% Fpow_Pow_finite
thf(fact_1023_Fpow__Pow__finite,axiom,
    ( finite_Fpow_nat
    = ( ^ [A6: set_nat] : ( inf_inf_set_set_nat @ ( pow_nat @ A6 ) @ ( collect_set_nat @ finite_finite_nat ) ) ) ) ).

% Fpow_Pow_finite
thf(fact_1024_Fpow__Pow__finite,axiom,
    ( finite4599829005654479975at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] : ( inf_in767997467307849391at_nat @ ( pow_Pr3466873192417945886at_nat @ A6 ) @ ( collec5514110066124741708at_nat @ finite6177210948735845034at_nat ) ) ) ) ).

% Fpow_Pow_finite
thf(fact_1025_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( comple5685304695842803022at_nat @ B4 ) @ A )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ B4 )
           => ( ( inf_in2572325071724192079at_nat @ X3 @ A )
              = bot_bo2099793752762293965at_nat ) ) ) ) ).

% Sup_inf_eq_bot_iff
thf(fact_1026_Sup__inf__eq__bot__iff,axiom,
    ! [B4: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( comple4364388969068068532at_nat @ B4 ) @ A )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ B4 )
           => ( ( inf_in7913087082777306421at_nat @ X3 @ A )
              = bot_bo228742789529271731at_nat ) ) ) ) ).

% Sup_inf_eq_bot_iff
thf(fact_1027_Union__disjoint,axiom,
    ! [C5: set_se7855581050983116737at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( comple5685304695842803022at_nat @ C5 ) @ A2 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ C5 )
           => ( ( inf_in2572325071724192079at_nat @ X3 @ A2 )
              = bot_bo2099793752762293965at_nat ) ) ) ) ).

% Union_disjoint
thf(fact_1028_Union__disjoint,axiom,
    ! [C5: set_se5479201105039950375at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( comple4364388969068068532at_nat @ C5 ) @ A2 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ C5 )
           => ( ( inf_in7913087082777306421at_nat @ X3 @ A2 )
              = bot_bo228742789529271731at_nat ) ) ) ) ).

% Union_disjoint
thf(fact_1029_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B4 ) @ ( minus_1356011639430497352at_nat @ A2 @ B4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_1030_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ A2 @ B4 ) @ ( minus_3314409938677909166at_nat @ A2 @ B4 ) )
      = bot_bo228742789529271731at_nat ) ).

% Int_Diff_disjoint
thf(fact_1031_Diff__triv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A2 @ B4 )
        = A2 ) ) ).

% Diff_triv
thf(fact_1032_Diff__triv,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat )
     => ( ( minus_3314409938677909166at_nat @ A2 @ B4 )
        = A2 ) ) ).

% Diff_triv
thf(fact_1033_Pow__bottom,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ bot_bo2099793752762293965at_nat @ ( pow_Pr3466873192417945886at_nat @ B4 ) ) ).

% Pow_bottom
thf(fact_1034_Pow__bottom,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] : ( member6273874620674182256at_nat @ bot_bo228742789529271731at_nat @ ( pow_Pr1023649790151954692at_nat @ B4 ) ) ).

% Pow_bottom
thf(fact_1035_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A2 )
           => ! [Y4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y4 @ B4 )
               => ( X3 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1036_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A2 )
           => ! [Y4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y4 @ B4 )
               => ( X3 != Y4 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_1037_Int__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_1038_Int__empty__right,axiom,
    ! [A2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A2 @ bot_bo228742789529271731at_nat )
      = bot_bo228742789529271731at_nat ) ).

% Int_empty_right
thf(fact_1039_Int__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B4 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_1040_Int__empty__left,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ bot_bo228742789529271731at_nat @ B4 )
      = bot_bo228742789529271731at_nat ) ).

% Int_empty_left
thf(fact_1041_disjoint__iff,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B4 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
           => ~ ( member_nat @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1042_disjoint__iff,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ( ( inf_in4302113700860409141at_nat @ A2 @ B4 )
        = bot_bo5327735625951526323at_nat )
      = ( ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A2 )
           => ~ ( member8206827879206165904at_nat @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1043_disjoint__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A2 )
           => ~ ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1044_disjoint__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A2 )
           => ~ ( member8757157785044589968at_nat @ X3 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_1045_Int__emptyI,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ~ ( member_nat @ X4 @ B4 ) )
     => ( ( inf_inf_set_nat @ A2 @ B4 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_1046_Int__emptyI,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
      ( ! [X4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ A2 )
         => ~ ( member8206827879206165904at_nat @ X4 @ B4 ) )
     => ( ( inf_in4302113700860409141at_nat @ A2 @ B4 )
        = bot_bo5327735625951526323at_nat ) ) ).

% Int_emptyI
thf(fact_1047_Int__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A2 )
         => ~ ( member8440522571783428010at_nat @ X4 @ B4 ) )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_1048_Int__emptyI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ! [X4: produc3843707927480180839at_nat] :
          ( ( member8757157785044589968at_nat @ X4 @ A2 )
         => ~ ( member8757157785044589968at_nat @ X4 @ B4 ) )
     => ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
        = bot_bo228742789529271731at_nat ) ) ).

% Int_emptyI
thf(fact_1049_insert__partition,axiom,
    ! [X: set_Pr1261947904930325089at_nat,F: set_se7855581050983116737at_nat] :
      ( ~ ( member2643936169264416010at_nat @ X @ F )
     => ( ! [X4: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X4 @ ( insert9200635055090092081at_nat @ X @ F ) )
           => ! [Xa3: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Xa3 @ ( insert9200635055090092081at_nat @ X @ F ) )
               => ( ( X4 != Xa3 )
                 => ( ( inf_in2572325071724192079at_nat @ X4 @ Xa3 )
                    = bot_bo2099793752762293965at_nat ) ) ) )
       => ( ( inf_in2572325071724192079at_nat @ X @ ( comple5685304695842803022at_nat @ F ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_partition
thf(fact_1050_insert__partition,axiom,
    ! [X: set_Pr4329608150637261639at_nat,F: set_se5479201105039950375at_nat] :
      ( ~ ( member6273874620674182256at_nat @ X @ F )
     => ( ! [X4: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X4 @ ( insert2290795007606087959at_nat @ X @ F ) )
           => ! [Xa3: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Xa3 @ ( insert2290795007606087959at_nat @ X @ F ) )
               => ( ( X4 != Xa3 )
                 => ( ( inf_in7913087082777306421at_nat @ X4 @ Xa3 )
                    = bot_bo228742789529271731at_nat ) ) ) )
       => ( ( inf_in7913087082777306421at_nat @ X @ ( comple4364388969068068532at_nat @ F ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_partition
thf(fact_1051_inter__Set__filter,axiom,
    ! [B4: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ B4 )
     => ( ( inf_in4302113700860409141at_nat @ A2 @ B4 )
        = ( filter2058754485963278488at_nat
          @ ^ [X3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X3 @ A2 )
          @ B4 ) ) ) ).

% inter_Set_filter
thf(fact_1052_inter__Set__filter,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ B4 )
     => ( ( inf_in7913087082777306421at_nat @ A2 @ B4 )
        = ( filter6899231104951847064at_nat
          @ ^ [X3: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X3 @ A2 )
          @ B4 ) ) ) ).

% inter_Set_filter
thf(fact_1053_inter__Set__filter,axiom,
    ! [B4: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B4 )
     => ( ( inf_inf_set_list_a @ A2 @ B4 )
        = ( filter_list_a
          @ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 )
          @ B4 ) ) ) ).

% inter_Set_filter
thf(fact_1054_inter__Set__filter,axiom,
    ! [B4: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( inf_inf_set_nat @ A2 @ B4 )
        = ( filter_nat
          @ ^ [X3: nat] : ( member_nat @ X3 @ A2 )
          @ B4 ) ) ) ).

% inter_Set_filter
thf(fact_1055_inter__Set__filter,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( inf_in2572325071724192079at_nat @ A2 @ B4 )
        = ( filter5640266504077782706at_nat
          @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A2 )
          @ B4 ) ) ) ).

% inter_Set_filter
thf(fact_1056_inf__Int__eq2,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( inf_in2858808372837926172_nat_o
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ ( inf_in4302113700860409141at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1057_inf__Int__eq2,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ ( inf_in2572325071724192079at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1058_inf__Int__eq2,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ( inf_in6124848772414083356_nat_o
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ ( inf_in7913087082777306421at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1059_inf__Int__eq2,axiom,
    ! [R3: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
      ( ( inf_in6969519006101653788_nat_o
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 )
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ S ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ ( inf_in2456762090751560885at_nat @ R3 @ S ) ) ) ) ).

% inf_Int_eq2
thf(fact_1060_Field__insert,axiom,
    ! [A: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R2 ) )
      = ( sup_sup_set_nat @ ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_1061_Field__insert,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( field_3145993188331922222at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ R2 ) )
      = ( sup_su3642409539654194069at_nat @ ( insert9200635055090092081at_nat @ A @ ( insert9200635055090092081at_nat @ B2 @ bot_bo3083307316010499117at_nat ) ) @ ( field_3145993188331922222at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_1062_Field__insert,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( field_1428494348145443988at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ R2 ) )
      = ( sup_su474284620922681339at_nat @ ( insert2290795007606087959at_nat @ A @ ( insert2290795007606087959at_nat @ B2 @ bot_bo1534825418459847827at_nat ) ) @ ( field_1428494348145443988at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_1063_Field__insert,axiom,
    ! [A: product_prod_nat_nat,B2: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( field_4706345496283612622at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ R2 ) )
      = ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ bot_bo2099793752762293965at_nat ) ) @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_1064_Field__insert,axiom,
    ! [A: produc3843707927480180839at_nat,B2: produc3843707927480180839at_nat,R2: set_Pr5564308138774400199at_nat] :
      ( ( field_2862844103579723188at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ R2 ) )
      = ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B2 @ bot_bo228742789529271731at_nat ) ) @ ( field_2862844103579723188at_nat @ R2 ) ) ) ).

% Field_insert
thf(fact_1065_Field__empty,axiom,
    ( ( field_4706345496283612622at_nat @ bot_bo5327735625951526323at_nat )
    = bot_bo2099793752762293965at_nat ) ).

% Field_empty
thf(fact_1066_Field__empty,axiom,
    ( ( field_2862844103579723188at_nat @ bot_bo7193250159860175667at_nat )
    = bot_bo228742789529271731at_nat ) ).

% Field_empty
thf(fact_1067_Field__empty,axiom,
    ( ( field_nat @ bot_bo2099793752762293965at_nat )
    = bot_bot_set_nat ) ).

% Field_empty
thf(fact_1068_Field__empty,axiom,
    ( ( field_3145993188331922222at_nat @ bot_bo228742789529271731at_nat )
    = bot_bo3083307316010499117at_nat ) ).

% Field_empty
thf(fact_1069_FieldI2,axiom,
    ! [I4: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I4 @ J2 ) @ R3 )
     => ( member8206827879206165904at_nat @ J2 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1070_FieldI2,axiom,
    ! [I4: produc3843707927480180839at_nat,J2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I4 @ J2 ) @ R3 )
     => ( member8757157785044589968at_nat @ J2 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1071_FieldI2,axiom,
    ! [I4: product_prod_nat_nat,J2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I4 @ J2 ) @ R3 )
     => ( member8440522571783428010at_nat @ J2 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1072_FieldI2,axiom,
    ! [I4: nat,J2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J2 ) @ R3 )
     => ( member_nat @ J2 @ ( field_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1073_FieldI2,axiom,
    ! [I4: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I4 @ J2 ) @ R3 )
     => ( member2643936169264416010at_nat @ J2 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1074_FieldI2,axiom,
    ! [I4: set_Pr4329608150637261639at_nat,J2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I4 @ J2 ) @ R3 )
     => ( member6273874620674182256at_nat @ J2 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI2
thf(fact_1075_FieldI1,axiom,
    ! [I4: produc859450856879609959at_nat,J2: produc859450856879609959at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ I4 @ J2 ) @ R3 )
     => ( member8206827879206165904at_nat @ I4 @ ( field_3044959742847878580at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1076_FieldI1,axiom,
    ! [I4: produc3843707927480180839at_nat,J2: produc3843707927480180839at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ I4 @ J2 ) @ R3 )
     => ( member8757157785044589968at_nat @ I4 @ ( field_2862844103579723188at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1077_FieldI1,axiom,
    ! [I4: product_prod_nat_nat,J2: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ I4 @ J2 ) @ R3 )
     => ( member8440522571783428010at_nat @ I4 @ ( field_4706345496283612622at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1078_FieldI1,axiom,
    ! [I4: nat,J2: nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I4 @ J2 ) @ R3 )
     => ( member_nat @ I4 @ ( field_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1079_FieldI1,axiom,
    ! [I4: set_Pr1261947904930325089at_nat,J2: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ I4 @ J2 ) @ R3 )
     => ( member2643936169264416010at_nat @ I4 @ ( field_3145993188331922222at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1080_FieldI1,axiom,
    ! [I4: set_Pr4329608150637261639at_nat,J2: set_Pr4329608150637261639at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ I4 @ J2 ) @ R3 )
     => ( member6273874620674182256at_nat @ I4 @ ( field_1428494348145443988at_nat @ R3 ) ) ) ).

% FieldI1
thf(fact_1081_fold__graph__closed__eq,axiom,
    ! [A2: set_nat,B4: set_nat,F3: nat > nat > nat,G5: nat > nat > nat,Z3: nat] :
      ( ! [A5: nat,B: nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite1441398328259824232at_nat @ F3 @ Z3 @ A2 )
            = ( finite1441398328259824232at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1082_fold__graph__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat > product_prod_nat_nat,G5: nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: nat,B: product_prod_nat_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: product_prod_nat_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite4517347104285129833at_nat @ F3 @ Z3 @ A2 )
            = ( finite4517347104285129833at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1083_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_nat,F3: product_prod_nat_nat > nat > nat,G5: product_prod_nat_nat > nat > nat,Z3: nat] :
      ( ! [A5: product_prod_nat_nat,B: nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite1157299711242414731at_nat @ F3 @ Z3 @ A2 )
            = ( finite1157299711242414731at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1084_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,G5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite7670012532332807302at_nat @ F3 @ Z3 @ A2 )
            = ( finite7670012532332807302at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1085_fold__graph__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr8693737435421807431at_nat,F3: nat > produc859450856879609959at_nat > produc859450856879609959at_nat,G5: nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat] :
      ( ! [A5: nat,B: produc859450856879609959at_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8206827879206165904at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: produc859450856879609959at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
         => ( ( finite8253601997657688015at_nat @ F3 @ Z3 @ A2 )
            = ( finite8253601997657688015at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1086_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_nat,F3: produc859450856879609959at_nat > nat > nat,G5: produc859450856879609959at_nat > nat > nat,Z3: nat] :
      ( ! [A5: produc859450856879609959at_nat,B: nat] :
          ( ( member8206827879206165904at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc859450856879609959at_nat,B: nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite1914606885918208113at_nat @ F3 @ Z3 @ A2 )
            = ( finite1914606885918208113at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1087_fold__graph__closed__eq,axiom,
    ! [A2: set_nat,B4: set_Pr4329608150637261639at_nat,F3: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat,G5: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat] :
      ( ! [A5: nat,B: produc3843707927480180839at_nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member8757157785044589968at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: nat,B: produc3843707927480180839at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8757157785044589968at_nat @ B @ B4 )
             => ( member8757157785044589968at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8757157785044589968at_nat @ Z3 @ B4 )
         => ( ( finite7143321807302184911at_nat @ F3 @ Z3 @ A2 )
            = ( finite7143321807302184911at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1088_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr8693737435421807431at_nat,B4: set_Pr1261947904930325089at_nat,F3: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat,G5: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat] :
      ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ A5 @ A2 )
         => ( ( member8440522571783428010at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
         => ( ( finite8328062655258085152at_nat @ F3 @ Z3 @ A2 )
            = ( finite8328062655258085152at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1089_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr8693737435421807431at_nat,F3: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat,G5: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
          ( ( member8440522571783428010at_nat @ A5 @ A2 )
         => ( ( member8206827879206165904at_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
         => ( ( finite2741081191071043692at_nat @ F3 @ Z3 @ A2 )
            = ( finite2741081191071043692at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1090_fold__graph__closed__eq,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_nat,F3: produc3843707927480180839at_nat > nat > nat,G5: produc3843707927480180839at_nat > nat > nat,Z3: nat] :
      ( ! [A5: produc3843707927480180839at_nat,B: nat] :
          ( ( member8757157785044589968at_nat @ A5 @ A2 )
         => ( ( member_nat @ B @ B4 )
           => ( ( F3 @ A5 @ B )
              = ( G5 @ A5 @ B ) ) ) )
     => ( ! [A5: produc3843707927480180839at_nat,B: nat] :
            ( ( member8757157785044589968at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
       => ( ( member_nat @ Z3 @ B4 )
         => ( ( finite7842833648988352625at_nat @ F3 @ Z3 @ A2 )
            = ( finite7842833648988352625at_nat @ G5 @ Z3 @ A2 ) ) ) ) ) ).

% fold_graph_closed_eq
thf(fact_1091_fold__graph__closed__lemma,axiom,
    ! [G5: nat > nat > nat,Z3: nat,A2: set_nat,X: nat,B4: set_nat,F3: nat > nat > nat] :
      ( ( finite1441398328259824232at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: nat,B: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: nat,B: nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ( member_nat @ B @ B4 )
               => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member_nat @ Z3 @ B4 )
           => ( ( finite1441398328259824232at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1092_fold__graph__closed__lemma,axiom,
    ! [G5: nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat,A2: set_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,F3: nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite4517347104285129833at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: nat,B: product_prod_nat_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: nat,B: product_prod_nat_nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ( member8440522571783428010at_nat @ B @ B4 )
               => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
           => ( ( finite4517347104285129833at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8440522571783428010at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1093_fold__graph__closed__lemma,axiom,
    ! [G5: product_prod_nat_nat > nat > nat,Z3: nat,A2: set_Pr1261947904930325089at_nat,X: nat,B4: set_nat,F3: product_prod_nat_nat > nat > nat] :
      ( ( finite1157299711242414731at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: product_prod_nat_nat,B: nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: product_prod_nat_nat,B: nat] :
              ( ( member8440522571783428010at_nat @ A5 @ A2 )
             => ( ( member_nat @ B @ B4 )
               => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member_nat @ Z3 @ B4 )
           => ( ( finite1157299711242414731at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1094_fold__graph__closed__lemma,axiom,
    ! [G5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite7670012532332807302at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A5 @ A2 )
             => ( ( member8440522571783428010at_nat @ B @ B4 )
               => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
           => ( ( finite7670012532332807302at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8440522571783428010at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1095_fold__graph__closed__lemma,axiom,
    ! [G5: nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat,A2: set_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,F3: nat > produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ( finite8253601997657688015at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: nat,B: produc859450856879609959at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: nat,B: produc859450856879609959at_nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ( member8206827879206165904at_nat @ B @ B4 )
               => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
           => ( ( finite8253601997657688015at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1096_fold__graph__closed__lemma,axiom,
    ! [G5: produc859450856879609959at_nat > nat > nat,Z3: nat,A2: set_Pr8693737435421807431at_nat,X: nat,B4: set_nat,F3: produc859450856879609959at_nat > nat > nat] :
      ( ( finite1914606885918208113at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: produc859450856879609959at_nat,B: nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: produc859450856879609959at_nat,B: nat] :
              ( ( member8206827879206165904at_nat @ A5 @ A2 )
             => ( ( member_nat @ B @ B4 )
               => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member_nat @ Z3 @ B4 )
           => ( ( finite1914606885918208113at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1097_fold__graph__closed__lemma,axiom,
    ! [G5: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat,Z3: produc3843707927480180839at_nat,A2: set_nat,X: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,F3: nat > produc3843707927480180839at_nat > produc3843707927480180839at_nat] :
      ( ( finite7143321807302184911at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: nat,B: produc3843707927480180839at_nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( member8757157785044589968at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: nat,B: produc3843707927480180839at_nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ( member8757157785044589968at_nat @ B @ B4 )
               => ( member8757157785044589968at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8757157785044589968at_nat @ Z3 @ B4 )
           => ( ( finite7143321807302184911at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8757157785044589968at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1098_fold__graph__closed__lemma,axiom,
    ! [G5: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat,Z3: product_prod_nat_nat,A2: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,F3: produc859450856879609959at_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( finite8328062655258085152at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A2 )
           => ( ( member8440522571783428010at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: produc859450856879609959at_nat,B: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ A5 @ A2 )
             => ( ( member8440522571783428010at_nat @ B @ B4 )
               => ( member8440522571783428010at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8440522571783428010at_nat @ Z3 @ B4 )
           => ( ( finite8328062655258085152at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8440522571783428010at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1099_fold__graph__closed__lemma,axiom,
    ! [G5: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat,Z3: produc859450856879609959at_nat,A2: set_Pr1261947904930325089at_nat,X: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,F3: product_prod_nat_nat > produc859450856879609959at_nat > produc859450856879609959at_nat] :
      ( ( finite2741081191071043692at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A2 )
           => ( ( member8206827879206165904at_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: product_prod_nat_nat,B: produc859450856879609959at_nat] :
              ( ( member8440522571783428010at_nat @ A5 @ A2 )
             => ( ( member8206827879206165904at_nat @ B @ B4 )
               => ( member8206827879206165904at_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member8206827879206165904at_nat @ Z3 @ B4 )
           => ( ( finite2741081191071043692at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member8206827879206165904at_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1100_fold__graph__closed__lemma,axiom,
    ! [G5: produc3843707927480180839at_nat > nat > nat,Z3: nat,A2: set_Pr4329608150637261639at_nat,X: nat,B4: set_nat,F3: produc3843707927480180839at_nat > nat > nat] :
      ( ( finite7842833648988352625at_nat @ G5 @ Z3 @ A2 @ X )
     => ( ! [A5: produc3843707927480180839at_nat,B: nat] :
            ( ( member8757157785044589968at_nat @ A5 @ A2 )
           => ( ( member_nat @ B @ B4 )
             => ( ( F3 @ A5 @ B )
                = ( G5 @ A5 @ B ) ) ) )
       => ( ! [A5: produc3843707927480180839at_nat,B: nat] :
              ( ( member8757157785044589968at_nat @ A5 @ A2 )
             => ( ( member_nat @ B @ B4 )
               => ( member_nat @ ( G5 @ A5 @ B ) @ B4 ) ) )
         => ( ( member_nat @ Z3 @ B4 )
           => ( ( finite7842833648988352625at_nat @ F3 @ Z3 @ A2 @ X )
              & ( member_nat @ X @ B4 ) ) ) ) ) ) ).

% fold_graph_closed_lemma
thf(fact_1101_finite__Field,axiom,
    ! [R2: set_Pr4048851178543822343list_a] :
      ( ( finite1746232179883068240list_a @ R2 )
     => ( finite_finite_list_a @ ( field_list_a @ R2 ) ) ) ).

% finite_Field
thf(fact_1102_finite__Field,axiom,
    ! [R2: set_Pr8693737435421807431at_nat] :
      ( ( finite4392333629123659920at_nat @ R2 )
     => ( finite6177210948735845034at_nat @ ( field_4706345496283612622at_nat @ R2 ) ) ) ).

% finite_Field
thf(fact_1103_finite__Field,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ R2 )
     => ( finite_finite_nat @ ( field_nat @ R2 ) ) ) ).

% finite_Field
thf(fact_1104_transymcl__insert__both__new,axiom,
    ! [A: produc859450856879609959at_nat,Qeq: set_Pr553994874890374343at_nat,B2: produc859450856879609959at_nat] :
      ( ~ ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ Qeq ) )
     => ( ~ ( member8206827879206165904at_nat @ B2 @ ( field_3044959742847878580at_nat @ Qeq ) )
       => ( ( transy7518170843122434953at_nat @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su593301525918095003at_nat @ ( transy7518170843122434953at_nat @ Qeq ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ B2 @ B2 ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ A @ B2 ) @ ( insert5475735164933346999at_nat @ ( produc4662710985925991255at_nat @ B2 @ A ) @ bot_bo8719193631845471027at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1105_transymcl__insert__both__new,axiom,
    ! [A: produc3843707927480180839at_nat,Qeq: set_Pr5564308138774400199at_nat,B2: produc3843707927480180839at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ Qeq ) )
     => ( ~ ( member8757157785044589968at_nat @ B2 @ ( field_2862844103579723188at_nat @ Qeq ) )
       => ( ( transy4668831892376664969at_nat @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su6664651903789602459at_nat @ ( transy4668831892376664969at_nat @ Qeq ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ B2 @ B2 ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ A @ B2 ) @ ( insert398606708087465655at_nat @ ( produc2335810615602371415at_nat @ B2 @ A ) @ bot_bo7193250159860175667at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1106_transymcl__insert__both__new,axiom,
    ! [A: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat,B2: product_prod_nat_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ Qeq ) )
     => ( ~ ( member8440522571783428010at_nat @ B2 @ ( field_4706345496283612622at_nat @ Qeq ) )
       => ( ( transy7867444541631690531at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su718114333110466843at_nat @ ( transy7867444541631690531at_nat @ Qeq ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ B2 ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ A @ B2 ) @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ B2 @ A ) @ bot_bo5327735625951526323at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1107_transymcl__insert__both__new,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ Qeq ) )
     => ( ~ ( member6273874620674182256at_nat @ B2 @ ( field_1428494348145443988at_nat @ Qeq ) )
       => ( ( transy4884439702966519401at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su5395584912770621083at_nat @ ( transy4884439702966519401at_nat @ Qeq ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ B2 ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ A @ B2 ) @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ B2 @ A ) @ bot_bo4948859079157340979at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1108_transymcl__insert__both__new,axiom,
    ! [A: nat,Qeq: set_Pr1261947904930325089at_nat,B2: nat] :
      ( ~ ( member_nat @ A @ ( field_nat @ Qeq ) )
     => ( ~ ( member_nat @ B2 @ ( field_nat @ Qeq ) )
       => ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su6327502436637775413at_nat @ ( transymcl_nat @ Qeq ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ B2 @ A ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1109_transymcl__insert__both__new,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ Qeq ) )
     => ( ~ ( member2643936169264416010at_nat @ B2 @ ( field_3145993188331922222at_nat @ Qeq ) )
       => ( ( transy8868430128242704515at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ Qeq ) )
          = ( sup_su5525570899277871387at_nat @ ( transy8868430128242704515at_nat @ Qeq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ B2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ A @ B2 ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ B2 @ A ) @ bot_bo228742789529271731at_nat ) ) ) ) ) ) ) ) ).

% transymcl_insert_both_new
thf(fact_1110_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,Y7: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( finite4392333629123659920at_nat @ X5 )
     => ( ( finite4392333629123659920at_nat @ Y7 )
       => ( ( Y7 != bot_bo5327735625951526323at_nat )
         => ( ! [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X5 )
               => ? [Xa2: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Xa2 @ Y7 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ X5 @ Y7 ) @ ( max_ex4511810952740877858at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1111_max__ext_Omax__extI,axiom,
    ! [X5: set_list_a,Y7: set_list_a,R3: set_Pr4048851178543822343list_a] :
      ( ( finite_finite_list_a @ X5 )
     => ( ( finite_finite_list_a @ Y7 )
       => ( ( Y7 != bot_bot_set_list_a )
         => ( ! [X4: list_a] :
                ( ( member_list_a @ X4 @ X5 )
               => ? [Xa2: list_a] :
                    ( ( member_list_a @ Xa2 @ Y7 )
                    & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ X5 @ Y7 ) @ ( max_ext_list_a @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1112_max__ext_Omax__extI,axiom,
    ! [X5: set_nat,Y7: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( finite_finite_nat @ Y7 )
       => ( ( Y7 != bot_bot_set_nat )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ X5 )
               => ? [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ Y7 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y7 ) @ ( max_ext_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1113_max__ext_Omax__extI,axiom,
    ! [X5: set_se7855581050983116737at_nat,Y7: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( finite9047747110432174090at_nat @ X5 )
     => ( ( finite9047747110432174090at_nat @ Y7 )
       => ( ( Y7 != bot_bo3083307316010499117at_nat )
         => ( ! [X4: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ X4 @ X5 )
               => ? [Xa2: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ Xa2 @ Y7 )
                    & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X5 @ Y7 ) @ ( max_ex4535816423174879388at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1114_max__ext_Omax__extI,axiom,
    ! [X5: set_se5479201105039950375at_nat,Y7: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( finite4768004253697144688at_nat @ X5 )
     => ( ( finite4768004253697144688at_nat @ Y7 )
       => ( ( Y7 != bot_bo1534825418459847827at_nat )
         => ( ! [X4: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ X4 @ X5 )
               => ? [Xa2: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ Xa2 @ Y7 )
                    & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X5 @ Y7 ) @ ( max_ex4000160833828605186at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1115_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y7: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( ( finite6177210948735845034at_nat @ Y7 )
       => ( ( Y7 != bot_bo2099793752762293965at_nat )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X5 )
               => ? [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ Y7 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1116_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr4329608150637261639at_nat,Y7: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( finite4343798906461161616at_nat @ X5 )
     => ( ( finite4343798906461161616at_nat @ Y7 )
       => ( ( Y7 != bot_bo228742789529271731at_nat )
         => ( ! [X4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X4 @ X5 )
               => ? [Xa2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ Xa2 @ Y7 )
                    & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X4 @ Xa2 ) @ R3 ) ) )
           => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y7 ) @ ( max_ex8833738776070432290at_nat @ R3 ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_1117_transymcl__into_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ).

% transymcl_into(2)
thf(fact_1118_transymcl__into_I2_J,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transymcl_nat @ R2 ) ) ) ).

% transymcl_into(2)
thf(fact_1119_transymcl__into_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ).

% transymcl_into(2)
thf(fact_1120_transymcl__into_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ).

% transymcl_into(2)
thf(fact_1121_transymcl__into_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ).

% transymcl_into(1)
thf(fact_1122_transymcl__into_I1_J,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R2 ) ) ) ).

% transymcl_into(1)
thf(fact_1123_transymcl__into_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ).

% transymcl_into(1)
thf(fact_1124_transymcl__into_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ).

% transymcl_into(1)
thf(fact_1125_transymcl__self_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ).

% transymcl_self(2)
thf(fact_1126_transymcl__self_I2_J,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y ) @ ( transymcl_nat @ R2 ) ) ) ).

% transymcl_self(2)
thf(fact_1127_transymcl__self_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Y ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ).

% transymcl_self(2)
thf(fact_1128_transymcl__self_I2_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Y ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ).

% transymcl_self(2)
thf(fact_1129_transymcl__self_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ).

% transymcl_self(1)
thf(fact_1130_transymcl__self_I1_J,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ ( transymcl_nat @ R2 ) ) ) ).

% transymcl_self(1)
thf(fact_1131_transymcl__self_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R2 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ X ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ).

% transymcl_self(1)
thf(fact_1132_transymcl__self_I1_J,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R2 )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ X ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ).

% transymcl_self(1)
thf(fact_1133_transymcl__sym,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R2 ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ X ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ).

% transymcl_sym
thf(fact_1134_transymcl__sym,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R2 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( transymcl_nat @ R2 ) ) ) ).

% transymcl_sym
thf(fact_1135_transymcl__sym,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R2 ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ X ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ).

% transymcl_sym
thf(fact_1136_transymcl__sym,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R2 ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ X ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ).

% transymcl_sym
thf(fact_1137_transymcl__trans,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Z3: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ R2 ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ ( transy7867444541631690531at_nat @ R2 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Z3 ) @ ( transy7867444541631690531at_nat @ R2 ) ) ) ) ).

% transymcl_trans
thf(fact_1138_transymcl__trans,axiom,
    ! [X: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z3: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ R2 ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ ( transymcl_nat @ R2 ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Z3 ) @ ( transymcl_nat @ R2 ) ) ) ) ).

% transymcl_trans
thf(fact_1139_transymcl__trans,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R2: set_Pr4329608150637261639at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ R2 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Z3 ) @ ( transy8868430128242704515at_nat @ R2 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Z3 ) @ ( transy8868430128242704515at_nat @ R2 ) ) ) ) ).

% transymcl_trans
thf(fact_1140_transymcl__trans,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,R2: set_Pr7459493094073627847at_nat,Z3: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ R2 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ Y @ Z3 ) @ ( transy4884439702966519401at_nat @ R2 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Z3 ) @ ( transy4884439702966519401at_nat @ R2 ) ) ) ) ).

% transymcl_trans
thf(fact_1141_Field__transymcl__self,axiom,
    ! [A: produc859450856879609959at_nat,Qeq: set_Pr553994874890374343at_nat] :
      ( ( member8206827879206165904at_nat @ A @ ( field_3044959742847878580at_nat @ Qeq ) )
     => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ A @ A ) @ ( transy7518170843122434953at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1142_Field__transymcl__self,axiom,
    ! [A: produc3843707927480180839at_nat,Qeq: set_Pr5564308138774400199at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( field_2862844103579723188at_nat @ Qeq ) )
     => ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ A @ A ) @ ( transy4668831892376664969at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1143_Field__transymcl__self,axiom,
    ! [A: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( field_4706345496283612622at_nat @ Qeq ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( transy7867444541631690531at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1144_Field__transymcl__self,axiom,
    ! [A: nat,Qeq: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A @ ( field_nat @ Qeq ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transymcl_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1145_Field__transymcl__self,axiom,
    ! [A: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat] :
      ( ( member2643936169264416010at_nat @ A @ ( field_3145993188331922222at_nat @ Qeq ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ A ) @ ( transy8868430128242704515at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1146_Field__transymcl__self,axiom,
    ! [A: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat] :
      ( ( member6273874620674182256at_nat @ A @ ( field_1428494348145443988at_nat @ Qeq ) )
     => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A @ A ) @ ( transy4884439702966519401at_nat @ Qeq ) ) ) ).

% Field_transymcl_self
thf(fact_1147_transymcl__insert__same__class,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Qeq: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( transy7867444541631690531at_nat @ Qeq ) )
     => ( ( transy7867444541631690531at_nat @ ( insert5050368324300391991at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ Qeq ) )
        = ( transy7867444541631690531at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_1148_transymcl__insert__same__class,axiom,
    ! [X: nat,Y: nat,Qeq: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transymcl_nat @ Qeq ) )
     => ( ( transymcl_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ Qeq ) )
        = ( transymcl_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_1149_transymcl__insert__same__class,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Qeq: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( transy8868430128242704515at_nat @ Qeq ) )
     => ( ( transy8868430128242704515at_nat @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ Qeq ) )
        = ( transy8868430128242704515at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_1150_transymcl__insert__same__class,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Qeq: set_Pr7459493094073627847at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ ( transy4884439702966519401at_nat @ Qeq ) )
     => ( ( transy4884439702966519401at_nat @ ( insert291458823148189367at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ Qeq ) )
        = ( transy4884439702966519401at_nat @ Qeq ) ) ) ).

% transymcl_insert_same_class
thf(fact_1151_max__ext__additive,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat,C5: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A2 @ B4 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ C5 @ D2 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( sup_su6327502436637775413at_nat @ A2 @ C5 ) @ ( sup_su6327502436637775413at_nat @ B4 @ D2 ) ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ).

% max_ext_additive
thf(fact_1152_max__ext__additive,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat,C5: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A2 @ B4 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
     => ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ C5 @ D2 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
       => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ C5 ) @ ( sup_su5525570899277871387at_nat @ B4 @ D2 ) ) @ ( max_ex8833738776070432290at_nat @ R3 ) ) ) ) ).

% max_ext_additive
thf(fact_1153_max__ext_Ocases,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R3 ) )
     => ~ ( ( finite4392333629123659920at_nat @ A1 )
         => ( ( finite4392333629123659920at_nat @ A22 )
           => ( ( A22 != bot_bo5327735625951526323at_nat )
             => ~ ! [X6: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ X6 @ A1 )
                   => ? [Xa3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ Xa3 @ A22 )
                        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1154_max__ext_Ocases,axiom,
    ! [A1: set_list_a,A22: set_list_a,R3: set_Pr4048851178543822343list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A1 @ A22 ) @ ( max_ext_list_a @ R3 ) )
     => ~ ( ( finite_finite_list_a @ A1 )
         => ( ( finite_finite_list_a @ A22 )
           => ( ( A22 != bot_bot_set_list_a )
             => ~ ! [X6: list_a] :
                    ( ( member_list_a @ X6 @ A1 )
                   => ? [Xa3: list_a] :
                        ( ( member_list_a @ Xa3 @ A22 )
                        & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1155_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1156_max__ext_Ocases,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R3 ) )
     => ~ ( ( finite9047747110432174090at_nat @ A1 )
         => ( ( finite9047747110432174090at_nat @ A22 )
           => ( ( A22 != bot_bo3083307316010499117at_nat )
             => ~ ! [X6: set_Pr1261947904930325089at_nat] :
                    ( ( member2643936169264416010at_nat @ X6 @ A1 )
                   => ? [Xa3: set_Pr1261947904930325089at_nat] :
                        ( ( member2643936169264416010at_nat @ Xa3 @ A22 )
                        & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1157_max__ext_Ocases,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R3 ) )
     => ~ ( ( finite4768004253697144688at_nat @ A1 )
         => ( ( finite4768004253697144688at_nat @ A22 )
           => ( ( A22 != bot_bo1534825418459847827at_nat )
             => ~ ! [X6: set_Pr4329608150637261639at_nat] :
                    ( ( member6273874620674182256at_nat @ X6 @ A1 )
                   => ? [Xa3: set_Pr4329608150637261639at_nat] :
                        ( ( member6273874620674182256at_nat @ Xa3 @ A22 )
                        & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1158_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ A1 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1159_max__ext_Ocases,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
     => ~ ( ( finite4343798906461161616at_nat @ A1 )
         => ( ( finite4343798906461161616at_nat @ A22 )
           => ( ( A22 != bot_bo228742789529271731at_nat )
             => ~ ! [X6: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X6 @ A1 )
                   => ? [Xa3: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ Xa3 @ A22 )
                        & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X6 @ Xa3 ) @ R3 ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_1160_max__ext_Osimps,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R3: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R3 ) )
      = ( ( finite4392333629123659920at_nat @ A1 )
        & ( finite4392333629123659920at_nat @ A22 )
        & ( A22 != bot_bo5327735625951526323at_nat )
        & ! [X3: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X3 @ A1 )
           => ? [Y4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y4 @ A22 )
                & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1161_max__ext_Osimps,axiom,
    ! [A1: set_list_a,A22: set_list_a,R3: set_Pr4048851178543822343list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A1 @ A22 ) @ ( max_ext_list_a @ R3 ) )
      = ( ( finite_finite_list_a @ A1 )
        & ( finite_finite_list_a @ A22 )
        & ( A22 != bot_bot_set_list_a )
        & ! [X3: list_a] :
            ( ( member_list_a @ X3 @ A1 )
           => ? [Y4: list_a] :
                ( ( member_list_a @ Y4 @ A22 )
                & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1162_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R3 ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ A1 )
           => ? [Y4: nat] :
                ( ( member_nat @ Y4 @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1163_max__ext_Osimps,axiom,
    ! [A1: set_se7855581050983116737at_nat,A22: set_se7855581050983116737at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ A1 @ A22 ) @ ( max_ex4535816423174879388at_nat @ R3 ) )
      = ( ( finite9047747110432174090at_nat @ A1 )
        & ( finite9047747110432174090at_nat @ A22 )
        & ( A22 != bot_bo3083307316010499117at_nat )
        & ! [X3: set_Pr1261947904930325089at_nat] :
            ( ( member2643936169264416010at_nat @ X3 @ A1 )
           => ? [Y4: set_Pr1261947904930325089at_nat] :
                ( ( member2643936169264416010at_nat @ Y4 @ A22 )
                & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1164_max__ext_Osimps,axiom,
    ! [A1: set_se5479201105039950375at_nat,A22: set_se5479201105039950375at_nat,R3: set_Pr7459493094073627847at_nat] :
      ( ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ A1 @ A22 ) @ ( max_ex4000160833828605186at_nat @ R3 ) )
      = ( ( finite4768004253697144688at_nat @ A1 )
        & ( finite4768004253697144688at_nat @ A22 )
        & ( A22 != bot_bo1534825418459847827at_nat )
        & ! [X3: set_Pr4329608150637261639at_nat] :
            ( ( member6273874620674182256at_nat @ X3 @ A1 )
           => ? [Y4: set_Pr4329608150637261639at_nat] :
                ( ( member6273874620674182256at_nat @ Y4 @ A22 )
                & ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1165_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R3 ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A1 )
           => ? [Y4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y4 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1166_max__ext_Osimps,axiom,
    ! [A1: set_Pr4329608150637261639at_nat,A22: set_Pr4329608150637261639at_nat,R3: set_Pr5564308138774400199at_nat] :
      ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A1 @ A22 ) @ ( max_ex8833738776070432290at_nat @ R3 ) )
      = ( ( finite4343798906461161616at_nat @ A1 )
        & ( finite4343798906461161616at_nat @ A22 )
        & ( A22 != bot_bo228742789529271731at_nat )
        & ! [X3: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X3 @ A1 )
           => ? [Y4: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ Y4 @ A22 )
                & ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y4 ) @ R3 ) ) ) ) ) ).

% max_ext.simps
thf(fact_1167_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr5564308138774400199at_nat] :
      ( ( max_ex6719288698447201402at_nat
        @ ^ [X3: produc3843707927480180839at_nat,Y4: produc3843707927480180839at_nat] : ( member6009668390874684176at_nat @ ( produc2335810615602371415at_nat @ X3 @ Y4 ) @ R3 ) )
      = ( ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ ( max_ex8833738776070432290at_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1168_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X3: nat,Y4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R3 ) )
      = ( ^ [X3: set_nat,Y4: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X3 @ Y4 ) @ ( max_ext_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1169_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr4329608150637261639at_nat] :
      ( ( max_ex4845989503969680116at_nat
        @ ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ R3 ) )
      = ( ^ [X3: set_se7855581050983116737at_nat,Y4: set_se7855581050983116737at_nat] : ( member4300370950623927696at_nat @ ( produc7462898802507495127at_nat @ X3 @ Y4 ) @ ( max_ex4535816423174879388at_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1170_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr7459493094073627847at_nat] :
      ( ( max_ex8849487260381072218at_nat
        @ ^ [X3: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X3 @ Y4 ) @ R3 ) )
      = ( ^ [X3: set_se5479201105039950375at_nat,Y4: set_se5479201105039950375at_nat] : ( member6436561159828024080at_nat @ ( produc2492784117751810903at_nat @ X3 @ Y4 ) @ ( max_ex4000160833828605186at_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1171_max__extp__max__ext__eq,axiom,
    ! [R3: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X3: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R3 ) )
      = ( ^ [X3: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y4 ) @ ( max_ex8135407076693332796at_nat @ R3 ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_1172_csts_Oelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_a] :
      ( ( ( relational_csts_a_b @ X )
        = Y )
     => ( ( ? [B: $o] :
              ( X
              = ( relational_Bool_a_b @ B ) )
         => ( Y != bot_bot_set_a ) )
       => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ P2 @ Ts ) )
             => ( Y
               != ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) ) )
         => ( ! [X4: nat,T2: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T2 ) )
               => ( Y
                 != ( relati6638259395341848997term_a @ T2 ) ) )
           => ( ! [Q4: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Q4 ) )
                 => ( Y
                   != ( relational_csts_a_b @ Q4 ) ) )
             => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                   => ( Y
                     != ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                     => ( Y
                       != ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) ) )
                 => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                       => ( Y
                         != ( relational_csts_a_b @ Q4 ) ) ) ) ) ) ) ) ) ) ).

% csts.elims
thf(fact_1173_csts_Osimps_I5_J,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b] :
      ( ( relational_csts_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) )
      = ( sup_sup_set_a @ ( relational_csts_a_b @ Q1 ) @ ( relational_csts_a_b @ Q2 ) ) ) ).

% csts.simps(5)
thf(fact_1174_fv__terms__setD,axiom,
    ! [M4: nat,Ts2: list_R6823256787227418703term_a] :
      ( ( member_nat @ M4 @ ( relati4569515538964159125_set_a @ Ts2 ) )
     => ( member5466445112152945266term_a @ ( relational_Var_a @ M4 ) @ ( set_Re3569617851344498910term_a @ Ts2 ) ) ) ).

% fv_terms_setD
thf(fact_1175_fv__terms__setI,axiom,
    ! [M4: nat,Ts2: list_R6823256787227418703term_a] :
      ( ( member5466445112152945266term_a @ ( relational_Var_a @ M4 ) @ ( set_Re3569617851344498910term_a @ Ts2 ) )
     => ( member_nat @ M4 @ ( relati4569515538964159125_set_a @ Ts2 ) ) ) ).

% fv_terms_setI
thf(fact_1176_csts_Osimps_I3_J,axiom,
    ! [X: nat,T: relational_term_a] :
      ( ( relational_csts_a_b @ ( relational_Eq_a_b @ X @ T ) )
      = ( relati6638259395341848997term_a @ T ) ) ).

% csts.simps(3)
thf(fact_1177_qp__fresh__val,axiom,
    ! [Q: relational_fmla_a_b,Sigma2: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
      ( ( relational_qp_a_b @ Q )
     => ( ~ ( member_a @ ( Sigma2 @ X ) @ ( relational_adom_b_a @ I ) )
       => ( ~ ( member_a @ ( Sigma2 @ X ) @ ( relational_csts_a_b @ Q ) )
         => ( ( relational_sat_a_b @ Q @ I @ Sigma2 )
           => ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).

% qp_fresh_val
thf(fact_1178_ap__fresh__val,axiom,
    ! [Q: relational_fmla_a_b,Sigma2: nat > a,X: nat,I: product_prod_b_nat > set_list_a] :
      ( ( relational_ap_a_b @ Q )
     => ( ~ ( member_a @ ( Sigma2 @ X ) @ ( relational_adom_b_a @ I ) )
       => ( ~ ( member_a @ ( Sigma2 @ X ) @ ( relational_csts_a_b @ Q ) )
         => ( ( relational_sat_a_b @ Q @ I @ Sigma2 )
           => ~ ( member_nat @ X @ ( relational_fv_a_b @ Q ) ) ) ) ) ) ).

% ap_fresh_val
thf(fact_1179_csts_Opelims,axiom,
    ! [X: relational_fmla_a_b,Y: set_a] :
      ( ( ( relational_csts_a_b @ X )
        = Y )
     => ( ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ X )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ( ( Y = bot_bot_set_a )
               => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Bool_a_b @ B ) ) ) )
         => ( ! [P2: b,Ts: list_R6823256787227418703term_a] :
                ( ( X
                  = ( relational_Pred_b_a @ P2 @ Ts ) )
               => ( ( Y
                    = ( comple2307003609928055243_set_a @ ( image_1080223610614215258_set_a @ relati6638259395341848997term_a @ ( set_Re3569617851344498910term_a @ Ts ) ) ) )
                 => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Pred_b_a @ P2 @ Ts ) ) ) )
           => ( ! [X4: nat,T2: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T2 ) )
                 => ( ( Y
                      = ( relati6638259395341848997term_a @ T2 ) )
                   => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Eq_a_b @ X4 @ T2 ) ) ) )
             => ( ! [Q4: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Q4 ) )
                   => ( ( Y
                        = ( relational_csts_a_b @ Q4 ) )
                     => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Neg_a_b @ Q4 ) ) ) )
               => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Q12 @ Q22 ) )
                     => ( ( Y
                          = ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) )
                       => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) )
                 => ( ! [Q12: relational_fmla_a_b,Q22: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Q12 @ Q22 ) )
                       => ( ( Y
                            = ( sup_sup_set_a @ ( relational_csts_a_b @ Q12 ) @ ( relational_csts_a_b @ Q22 ) ) )
                         => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relational_Disj_a_b @ Q12 @ Q22 ) ) ) )
                   => ~ ! [X4: nat,Q4: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ X4 @ Q4 ) )
                         => ( ( Y
                              = ( relational_csts_a_b @ Q4 ) )
                           => ~ ( accp_R989495437599811158la_a_b @ relati7137348651719826542el_a_b @ ( relati591517084277583526ts_a_b @ X4 @ Q4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% csts.pelims
thf(fact_1180_List_Ofinite__set,axiom,
    ! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_1181_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1182_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).

% List.finite_set
thf(fact_1183_Id__on__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( id_on_2554058798563519774at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( set_Pr5518436109238095868at_nat
        @ ( map_Pr9105590726435236202at_nat
          @ ^ [X3: product_prod_nat_nat] : ( produc6161850002892822231at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1184_Id__on__set,axiom,
    ! [Xs: list_nat] :
      ( ( id_on_nat @ ( set_nat2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X3: nat] : ( product_Pair_nat_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1185_Id__on__set,axiom,
    ! [Xs: list_s1210847774152347623at_nat] :
      ( ( id_on_6713504163309144702at_nat @ ( set_se5049602875457034614at_nat @ Xs ) )
      = ( set_Pr3765526544606949372at_nat
        @ ( map_se8673859202641857546at_nat
          @ ^ [X3: set_Pr1261947904930325089at_nat] : ( produc2922128104949294807at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1186_Id__on__set,axiom,
    ! [Xs: list_s4242580248156217677at_nat] :
      ( ( id_on_3275838214495744996at_nat @ ( set_se3524105090824187612at_nat @ Xs ) )
      = ( set_Pr2074889473033032060at_nat
        @ ( map_se8572903940320514724at_nat
          @ ^ [X3: set_Pr4329608150637261639at_nat] : ( produc9060074326276436823at_nat @ X3 @ X3 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1187_finite__list,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ? [Xs2: list_list_a] :
          ( ( set_list_a2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1188_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1189_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ? [Xs2: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs2 )
          = A2 ) ) ).

% finite_list
thf(fact_1190_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_1191_fv__cp__foldr__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,Q: relational_fmla_a_b,A2: 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 )
              = A2 ) ) )
     => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( foldr_789212930732525799la_a_b @ relational_Disj_a_b @ Qs @ Q ) ) )
        = A2 ) ) ).

% fv_cp_foldr_Disj
thf(fact_1192_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 ) )
        = ( ! [X3: relational_fmla_a_b] :
              ( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
             => ( relational_sr_a_b @ X3 ) )
          & ( relational_sr_a_b @ Q ) ) ) ) ).

% sr_foldr_Disj
thf(fact_1193_fv__cp__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,A2: 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 )
                = A2 ) ) )
       => ( ( relational_fv_a_b @ ( relational_cp_a_b @ ( relati1724936043867567613la_a_b @ relational_Disj_a_b @ Qs @ ( relational_Bool_a_b @ $false ) ) ) )
          = A2 ) ) ) ).

% fv_cp_foldr1_Disj
thf(fact_1194_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_1195_set__empty2,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty2
thf(fact_1196_set__empty2,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( set_Pr3765526544606949372at_nat @ Xs ) )
      = ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty2
thf(fact_1197_set__empty,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( ( set_Pr5648618587558075414at_nat @ Xs )
        = bot_bo2099793752762293965at_nat )
      = ( Xs = nil_Pr5478986624290739719at_nat ) ) ).

% set_empty
thf(fact_1198_set__empty,axiom,
    ! [Xs: list_P5464809261938338413at_nat] :
      ( ( ( set_Pr3765526544606949372at_nat @ Xs )
        = bot_bo228742789529271731at_nat )
      = ( Xs = nil_Pr8341282898164651117at_nat ) ) ).

% set_empty
thf(fact_1199_empty__set,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).

% empty_set
thf(fact_1200_empty__set,axiom,
    ( bot_bo228742789529271731at_nat
    = ( set_Pr3765526544606949372at_nat @ nil_Pr8341282898164651117at_nat ) ) ).

% empty_set
thf(fact_1201_Pow__set_I1_J,axiom,
    ( ( pow_Pr3466873192417945886at_nat @ ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) )
    = ( insert9200635055090092081at_nat @ bot_bo2099793752762293965at_nat @ bot_bo3083307316010499117at_nat ) ) ).

% Pow_set(1)
thf(fact_1202_Pow__set_I1_J,axiom,
    ( ( pow_Pr1023649790151954692at_nat @ ( set_Pr3765526544606949372at_nat @ nil_Pr8341282898164651117at_nat ) )
    = ( insert2290795007606087959at_nat @ bot_bo228742789529271731at_nat @ bot_bo1534825418459847827at_nat ) ) ).

% Pow_set(1)
thf(fact_1203_sr__foldr1__Disj,axiom,
    ! [Qs: list_R8263082107343818799la_a_b,X5: 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 )
            = X5 ) )
     => ( ( 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 )
           => ! [X3: relational_fmla_a_b] :
                ( ( member4680049679412964150la_a_b @ X3 @ ( set_Re9104216502384355786la_a_b @ Qs ) )
               => ( relational_sr_a_b @ X3 ) ) ) ) ) ) ).

% sr_foldr1_Disj
thf(fact_1204_set__removeAll,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) )
      = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_removeAll
thf(fact_1205_set__removeAll,axiom,
    ! [X: produc3843707927480180839at_nat,Xs: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( remove4908499291543849587at_nat @ X @ Xs ) )
      = ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ).

% set_removeAll
thf(fact_1206_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( linord2614967742042102400et_nat @ A2 )
        = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1207_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
        = A2 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1208_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( linord2614967742042102400et_nat @ A2 )
          = nil_nat )
        = ( A2 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1209_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A2: set_nat,B4: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A2 )
        = ( linord2614967742042102400et_nat @ B4 ) )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B4 )
         => ( A2 = B4 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1210_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( linord2614967742042102400et_nat @ ( insert_nat @ X @ A2 ) )
        = ( linord8961336180081300637at_nat
          @ ^ [X3: nat] : X3
          @ X
          @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_1211_set__rec,axiom,
    ( set_Pr5648618587558075414at_nat
    = ( rec_li5706324831933932950at_nat @ bot_bo2099793752762293965at_nat
      @ ^ [X3: product_prod_nat_nat,Uu3: list_P6011104703257516679at_nat] : ( insert8211810215607154385at_nat @ X3 ) ) ) ).

% set_rec
thf(fact_1212_set__rec,axiom,
    ( set_Pr3765526544606949372at_nat
    = ( rec_li1831689483791029142at_nat @ bot_bo228742789529271731at_nat
      @ ^ [X3: produc3843707927480180839at_nat,Uu3: list_P5464809261938338413at_nat] : ( insert9069300056098147895at_nat @ X3 ) ) ) ).

% set_rec
thf(fact_1213_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( linord2614967742042102400et_nat @ ( insert_nat @ X @ A2 ) )
          = ( linord8961336180081300637at_nat
            @ ^ [X3: nat] : X3
            @ X
            @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_1214_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( linord2614967742042102400et_nat @ A2 )
          = ( linord8961336180081300637at_nat
            @ ^ [X3: nat] : X3
            @ X
            @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
thf(fact_1215_eval__empty__close,axiom,
    ! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_eval_a_b @ ( relati7004044288120072026TS_a_b @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q ) ) @ Q ) @ I )
        = bot_bot_set_list_a )
      = ( ! [Sigma4: nat > a] :
            ~ ( relational_sat_a_b @ Q @ I @ Sigma4 ) ) ) ).

% eval_empty_close
thf(fact_1216_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_1217_fv__EXISTS,axiom,
    ! [Xs: list_nat,Q: relational_fmla_a_b] :
      ( ( relational_fv_a_b @ ( relati7004044288120072026TS_a_b @ Xs @ Q ) )
      = ( minus_minus_set_nat @ ( relational_fv_a_b @ Q ) @ ( set_nat2 @ Xs ) ) ) ).

% fv_EXISTS
thf(fact_1218_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
        = ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1219_lists__empty,axiom,
    ( ( lists_3066315782749251317at_nat @ bot_bo2099793752762293965at_nat )
    = ( insert696505707821619287at_nat @ nil_Pr5478986624290739719at_nat @ bot_bo5403906011208138963at_nat ) ) ).

% lists_empty
thf(fact_1220_lists__empty,axiom,
    ( ( lists_7451723386363959387at_nat @ bot_bo228742789529271731at_nat )
    = ( insert117935917674147901at_nat @ nil_Pr8341282898164651117at_nat @ bot_bo8338020889238773817at_nat ) ) ).

% lists_empty
thf(fact_1221_set__remove1__eq,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( set_Pr5648618587558075414at_nat @ ( remove5766854097993331357at_nat @ X @ Xs ) )
        = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_1222_set__remove1__eq,axiom,
    ! [Xs: list_P5464809261938338413at_nat,X: produc3843707927480180839at_nat] :
      ( ( distin5355857245021852384at_nat @ Xs )
     => ( ( set_Pr3765526544606949372at_nat @ ( remove8895909992688781315at_nat @ X @ Xs ) )
        = ( minus_3314409938677909166at_nat @ ( set_Pr3765526544606949372at_nat @ Xs ) @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ).

% set_remove1_eq
thf(fact_1223_finite__distinct__list,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ? [Xs2: list_list_a] :
          ( ( ( set_list_a2 @ Xs2 )
            = A2 )
          & ( distinct_list_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_1224_finite__distinct__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs2: list_nat] :
          ( ( ( set_nat2 @ Xs2 )
            = A2 )
          & ( distinct_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_1225_finite__distinct__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ? [Xs2: list_P6011104703257516679at_nat] :
          ( ( ( set_Pr5648618587558075414at_nat @ Xs2 )
            = A2 )
          & ( distin6923225563576452346at_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_1226_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_a,N2: nat] :
      ( ( finite_finite_a @ A2 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ( size_size_list_a @ Xs3 )
                = N2 )
              & ( distinct_a @ Xs3 )
              & ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1227_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_list_a,N2: nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ( size_s349497388124573686list_a @ Xs3 )
                = N2 )
              & ( distinct_list_a @ Xs3 )
              & ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1228_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = N2 )
              & ( distinct_nat @ Xs3 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1229_finite__lists__distinct__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs3: list_P6011104703257516679at_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs3 )
                = N2 )
              & ( distin6923225563576452346at_nat @ Xs3 )
              & ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A2 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1230_finite__lists__length__le,axiom,
    ! [A2: set_a,N2: nat] :
      ( ( finite_finite_a @ A2 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_a @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1231_finite__lists__length__le,axiom,
    ! [A2: set_list_a,N2: nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1232_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1233_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs3: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs3 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_1234_finite__maxlen,axiom,
    ! [M2: set_list_a] :
      ( ( finite_finite_list_a @ M2 )
     => ? [N: nat] :
        ! [X6: list_a] :
          ( ( member_list_a @ X6 @ M2 )
         => ( ord_less_nat @ ( size_size_list_a @ X6 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_1235_finite__lists__length__eq,axiom,
    ! [A2: set_a,N2: nat] :
      ( ( finite_finite_a @ A2 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A2 )
              & ( ( size_size_list_a @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1236_finite__lists__length__eq,axiom,
    ! [A2: set_list_a,N2: nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A2 )
              & ( ( size_s349497388124573686list_a @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1237_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
              & ( ( size_size_list_nat @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1238_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs3: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs3 ) @ A2 )
              & ( ( size_s5460976970255530739at_nat @ Xs3 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1239_sat_Oelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts ) )
           => ( Y
              = ( ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ( Y = ~ B ) )
         => ( ! [X4: nat,T3: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T3 ) )
               => ( Y
                  = ( ( Xb @ X4 )
                   != ( relati1177013128715261720term_a @ Xb @ T3 ) ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( Y
                    = ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( Y
                      = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( Y
                        = ( ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ( Y
                          = ( ~ ? [X3: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(1)
thf(fact_1240_sat_Oelims_I2_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X @ Xa @ Xb )
     => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts ) )
           => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => ~ B )
         => ( ! [X4: nat,T3: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T3 ) )
               => ( ( Xb @ X4 )
                 != ( relati1177013128715261720term_a @ Xb @ T3 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                        & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                          | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ~ ? [X4: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X4 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(2)
thf(fact_1241_eval__term_Osimps_I2_J,axiom,
    ! [Sigma2: nat > a,N2: nat] :
      ( ( relati1177013128715261720term_a @ Sigma2 @ ( relational_Var_a @ N2 ) )
      = ( Sigma2 @ N2 ) ) ).

% eval_term.simps(2)
thf(fact_1242_sat_Osimps_I3_J,axiom,
    ! [X: nat,T5: relational_term_a,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Eq_a_b @ X @ T5 ) @ I @ Sigma2 )
      = ( ( Sigma2 @ X )
        = ( relati1177013128715261720term_a @ Sigma2 @ T5 ) ) ) ).

% sat.simps(3)
thf(fact_1243_eval__term_Oelims,axiom,
    ! [X: nat > a,Xa: relational_term_a,Y: a] :
      ( ( ( relati1177013128715261720term_a @ X @ Xa )
        = Y )
     => ( ! [C: a] :
            ( ( Xa
              = ( relational_Const_a @ C ) )
           => ( Y != C ) )
       => ~ ! [N: nat] :
              ( ( Xa
                = ( relational_Var_a @ N ) )
             => ( Y
               != ( X @ N ) ) ) ) ) ).

% eval_term.elims
thf(fact_1244_sat_Oelims_I3_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X @ Xa @ Xb )
     => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
            ( ( X
              = ( relational_Pred_b_a @ R @ Ts ) )
           => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
       => ( ! [B: $o] :
              ( ( X
                = ( relational_Bool_a_b @ B ) )
             => B )
         => ( ! [X4: nat,T3: relational_term_a] :
                ( ( X
                  = ( relational_Eq_a_b @ X4 @ T3 ) )
               => ( ( Xb @ X4 )
                  = ( relati1177013128715261720term_a @ Xb @ T3 ) ) )
           => ( ! [Phi2: relational_fmla_a_b] :
                  ( ( X
                    = ( relational_Neg_a_b @ Phi2 ) )
                 => ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) )
             => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                   => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                      & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                     => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                        | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                 => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                        ( ( X
                          = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                       => ? [X6: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X6 ) ) ) ) ) ) ) ) ) ) ).

% sat.elims(3)
thf(fact_1245_sat_Opelims_I3_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ~ ( relational_sat_a_b @ X @ Xa @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
               => ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                 => B ) )
           => ( ! [X4: nat,T3: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T3 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T3 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X4 )
                      = ( relati1177013128715261720term_a @ Xb @ T3 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                     => ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                       => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                          & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                         => ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                           => ? [X6: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X6 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(3)
thf(fact_1246_sat_Opelims_I2_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a] :
      ( ( relational_sat_a_b @ X @ Xa @ Xb )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts ) )
             => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
               => ~ ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                 => ~ B ) )
           => ( ! [X4: nat,T3: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T3 ) )
                 => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T3 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                   => ( ( Xb @ X4 )
                     != ( relati1177013128715261720term_a @ Xb @ T3 ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                     => ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                       => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                         => ~ ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
                           => ~ ? [X4: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(2)
thf(fact_1247_sat_Opelims_I1_J,axiom,
    ! [X: relational_fmla_a_b,Xa: product_prod_b_nat > set_list_a,Xb: nat > a,Y: $o] :
      ( ( ( relational_sat_a_b @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ X @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) )
       => ( ! [R: b,Ts: list_R6823256787227418703term_a] :
              ( ( X
                = ( relational_Pred_b_a @ R @ Ts ) )
             => ( ( Y
                  = ( member_list_a @ ( relati4772805863405912879erms_a @ Xb @ Ts ) @ ( Xa @ ( product_Pair_b_nat @ R @ ( size_s88622898042387131term_a @ Ts ) ) ) ) )
               => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Pred_b_a @ R @ Ts ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
         => ( ! [B: $o] :
                ( ( X
                  = ( relational_Bool_a_b @ B ) )
               => ( ( Y = B )
                 => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Bool_a_b @ B ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
           => ( ! [X4: nat,T3: relational_term_a] :
                  ( ( X
                    = ( relational_Eq_a_b @ X4 @ T3 ) )
                 => ( ( Y
                      = ( ( Xb @ X4 )
                        = ( relati1177013128715261720term_a @ Xb @ T3 ) ) )
                   => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Eq_a_b @ X4 @ T3 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
             => ( ! [Phi2: relational_fmla_a_b] :
                    ( ( X
                      = ( relational_Neg_a_b @ Phi2 ) )
                   => ( ( Y
                        = ( ~ ( relational_sat_a_b @ Phi2 @ Xa @ Xb ) ) )
                     => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Neg_a_b @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
               => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                      ( ( X
                        = ( relational_Conj_a_b @ Phi2 @ Psi ) )
                     => ( ( Y
                          = ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                            & ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                       => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Conj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
                 => ( ! [Phi2: relational_fmla_a_b,Psi: relational_fmla_a_b] :
                        ( ( X
                          = ( relational_Disj_a_b @ Phi2 @ Psi ) )
                       => ( ( Y
                            = ( ( relational_sat_a_b @ Phi2 @ Xa @ Xb )
                              | ( relational_sat_a_b @ Psi @ Xa @ Xb ) ) )
                         => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relational_Disj_a_b @ Phi2 @ Psi ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) )
                   => ~ ! [Z: nat,Phi2: relational_fmla_a_b] :
                          ( ( X
                            = ( relati591517084277583526ts_a_b @ Z @ Phi2 ) )
                         => ( ( Y
                              = ( ? [X3: a] : ( relational_sat_a_b @ Phi2 @ Xa @ ( fun_upd_nat_a @ Xb @ Z @ X3 ) ) ) )
                           => ~ ( accp_P6721201822162371452_nat_a @ relati4321809963887231473el_a_b @ ( produc6598558901832717687_nat_a @ ( relati591517084277583526ts_a_b @ Z @ Phi2 ) @ ( produc2895298938842563487_nat_a @ Xa @ Xb ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% sat.pelims(1)
thf(fact_1248_eval__term_Opelims,axiom,
    ! [X: nat > a,Xa: relational_term_a,Y: a] :
      ( ( ( relati1177013128715261720term_a @ X @ Xa )
        = Y )
     => ( ( accp_P3077866103536389332term_a @ relati76914634177155881_rel_a @ ( produc8917778089171359291term_a @ X @ Xa ) )
       => ( ! [C: a] :
              ( ( Xa
                = ( relational_Const_a @ C ) )
             => ( ( Y = C )
               => ~ ( accp_P3077866103536389332term_a @ relati76914634177155881_rel_a @ ( produc8917778089171359291term_a @ X @ ( relational_Const_a @ C ) ) ) ) )
         => ~ ! [N: nat] :
                ( ( Xa
                  = ( relational_Var_a @ N ) )
               => ( ( Y
                    = ( X @ N ) )
                 => ~ ( accp_P3077866103536389332term_a @ relati76914634177155881_rel_a @ ( produc8917778089171359291term_a @ X @ ( relational_Var_a @ N ) ) ) ) ) ) ) ) ).

% eval_term.pelims
thf(fact_1249_Inf__fin_Oinsert__remove,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
            = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_1250_sup__Inf__absorb,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
          = A ) ) ) ).

% sup_Inf_absorb
thf(fact_1251_sup__Inf__absorb,axiom,
    ! [A2: set_se5479201105039950375at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( finite4768004253697144688at_nat @ A2 )
     => ( ( member6273874620674182256at_nat @ A @ A2 )
       => ( ( sup_su5525570899277871387at_nat @ ( lattic6378378157708517732at_nat @ A2 ) @ A )
          = A ) ) ) ).

% sup_Inf_absorb
thf(fact_1252_Inf__fin_Oinsert,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A2 ) )
          = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1253_Inf__fin_OcoboundedI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).

% Inf_fin.coboundedI
thf(fact_1254_Inf__fin_Oin__idem,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
          = ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1255_Inf__fin_OboundedE,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
         => ! [A8: nat] :
              ( ( member_nat @ A8 @ A2 )
             => ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_1256_Inf__fin_OboundedI,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [A5: nat] :
              ( ( member_nat @ A5 @ A2 )
             => ( ord_less_eq_nat @ X @ A5 ) )
         => ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_1257_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1258_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_1259_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_1260_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_1261_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B2 @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_1262_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_1263_wmin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_1264_wmax__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X5 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_1265_wmin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_1266_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_1267_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,X5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X @ X5 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_1268_smin__insertI,axiom,
    ! [X: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_1269_smin__emptyI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_1270_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_1271_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_1272_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_1273_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set

% Helper facts (13)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: set_list_a,Y: set_list_a] :
      ( ( if_set_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: set_list_a,Y: set_list_a] :
      ( ( if_set_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J_T,axiom,
    ! [X: relational_fmla_a_b,Y: relational_fmla_a_b] :
      ( ( if_Rel1279876242545935705la_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( if_set4744415642439091239at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( if_set3208018901896979469at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( if_set3208018901896979469at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ q @ ( relational_Eq_a_b @ x @ ( relational_Var_a @ y ) ) ) @ i ) ) ).

%------------------------------------------------------------------------------