SET007 Axioms: SET007+396.ax


%------------------------------------------------------------------------------
% File     : SET007+396 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Products of Many Sorted Algebras
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : pralg_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   73 (  14 unit)
%            Number of atoms       :  465 (  62 equality)
%            Maximal formula depth :   27 (   8 average)
%            Number of connectives :  478 (  86 ~  ;   0  |; 215  &)
%                                         (  17 <=>; 160 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-4 arity)
%            Number of functors    :   53 (   1 constant; 0-9 arity)
%            Number of variables   :  221 (   0 singleton; 216 !;   5 ?)
%            Maximal term depth    :    6 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_pralg_2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_fraenkel(A)
      & v1_pralg_2(A) ) )).

fof(fc1_pralg_2,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ( ~ v1_xboole_0(k4_card_3(B))
        & v1_fraenkel(k4_card_3(B))
        & v1_pralg_2(k4_card_3(B)) ) ) )).

fof(fc2_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( v2_relat_1(B)
        & m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A) )
     => ( v1_relat_1(k11_pboole(A,B,C))
        & v2_relat_1(k11_pboole(A,B,C))
        & v1_funct_1(k11_pboole(A,B,C)) ) ) )).

fof(fc3_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ~ v1_xboole_0(k7_pralg_2(A,B)) ) )).

fof(fc4_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => ~ v1_xboole_0(k8_pralg_2(A,B,C)) ) )).

fof(fc5_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A)
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A) )
     => v4_msualg_1(k9_pralg_2(A,B,C),A) ) )).

fof(fc6_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m2_pralg_2(D,A,B) )
     => ( v1_relat_1(k10_pralg_2(A,B,C,D))
        & v2_relat_1(k10_pralg_2(A,B,C,D))
        & v1_funct_1(k10_pralg_2(A,B,C,D)) ) ) )).

fof(fc7_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => ( v1_relat_1(k11_pralg_2(A,B,C))
        & v2_relat_1(k11_pralg_2(A,B,C))
        & ~ v3_relat_1(k11_pralg_2(A,B,C))
        & v1_funct_1(k11_pralg_2(A,B,C)) ) ) )).

fof(fc8_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => v4_msualg_1(k15_pralg_2(A,B,C),B) ) )).

fof(d1_pralg_2,axiom,(
    ! [A] :
      ( v1_pralg_2(A)
    <=> ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r2_hidden(B,A)
                  & r2_hidden(C,A) )
               => k1_relat_1(B) = k1_relat_1(C) ) ) ) ) )).

fof(t1_pralg_2,axiom,
    ( v1_fraenkel(k1_tarski(k1_xboole_0))
    & v1_pralg_2(k1_tarski(k1_xboole_0)) )).

fof(d2_pralg_2,axiom,(
    ! [A] :
      ( ( v1_fraenkel(A)
        & v1_pralg_2(A) )
     => ! [B] :
          ( ( A != k1_xboole_0
           => ( B = k1_pralg_2(A)
            <=> ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C) )
                 => ( r2_hidden(C,A)
                   => B = k1_relat_1(C) ) ) ) )
          & ( A = k1_xboole_0
           => ( B = k1_pralg_2(A)
            <=> B = k1_xboole_0 ) ) ) ) )).

fof(t2_pralg_2,axiom,(
    ! [A] :
      ( ( v1_fraenkel(A)
        & v1_pralg_2(A) )
     => ( A = k1_tarski(k1_xboole_0)
       => k1_pralg_2(A) = k1_xboole_0 ) ) )).

fof(d3_pralg_2,axiom,(
    $true )).

fof(d4_pralg_2,axiom,(
    $true )).

fof(d5_pralg_2,axiom,(
    $true )).

fof(d6_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k2_pralg_2(A)
          <=> ( ! [C] :
                  ( r2_hidden(C,k1_relat_1(B))
                <=> ? [D] :
                      ( v1_relat_1(D)
                      & v1_funct_1(D)
                      & r2_hidden(D,k1_relat_1(A))
                      & C = k10_funct_6(D) ) )
              & ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C) )
                 => ( r2_hidden(C,k1_relat_1(B))
                   => k1_funct_1(B,C) = k1_funct_1(A,k10_funct_6(C)) ) ) ) ) ) ) )).

fof(t3_pralg_2,axiom,(
    $true )).

fof(t4_pralg_2,axiom,(
    $true )).

fof(t5_pralg_2,axiom,(
    $true )).

fof(t6_pralg_2,axiom,(
    $true )).

fof(t7_pralg_2,axiom,(
    $true )).

fof(t8_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( k1_relat_1(A) = k1_tarski(k1_xboole_0)
       => k2_pralg_2(A) = A ) ) )).

fof(d7_pralg_2,axiom,(
    $true )).

fof(d8_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => ! [B] :
          ( ( v1_funcop_1(B)
            & m1_pboole(B,k4_card_3(k2_funct_6(A))) )
         => ( B = k3_pralg_2(A)
          <=> ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C) )
               => ( r2_hidden(C,k4_card_3(k2_funct_6(A)))
                 => k1_funct_1(B,C) = k15_pralg_1(A,C) ) ) ) ) ) )).

fof(t9_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_pboole(D,A)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,B,A)
                    & m2_relset_1(E,B,A) )
                 => r6_pboole(B,k7_pboole(B,A,E,k11_pboole(A,C,D)),k11_pboole(B,k7_pboole(B,A,E,C),k7_pboole(B,A,E,D))) ) ) ) ) )).

fof(d9_pralg_2,axiom,(
    $true )).

fof(d10_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( ( v2_relat_1(D)
                & m1_pboole(D,A) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,B,k3_finseq_2(A))
                    & m2_relset_1(E,B,k3_finseq_2(A)) )
                 => ! [F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,B,A)
                        & m2_relset_1(F,B,A) )
                     => ! [G,H] :
                          ( ( v1_funct_1(H)
                            & v1_funct_2(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
                            & m2_relset_1(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G)) )
                         => ! [I] :
                              ( ( v1_funct_1(I)
                                & v1_funct_2(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G))
                                & m2_relset_1(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G)) )
                             => ( r2_hidden(G,B)
                               => ! [J] :
                                    ( ( v1_funct_1(J)
                                      & v1_funct_2(J,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G))
                                      & m2_relset_1(J,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G)) )
                                   => ( J = k4_pralg_2(A,B,C,D,E,F,G,H,I)
                                    <=> ! [K] :
                                          ( ( v1_relat_1(K)
                                            & v1_funct_1(K) )
                                         => ( r2_hidden(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G))
                                           => k1_funct_1(J,K) = k4_tarski(k1_funct_1(H,k15_mcart_1(K)),k1_funct_1(I,k16_mcart_1(K))) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d11_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( v2_relat_1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( ( v2_relat_1(D)
                & m1_pboole(D,A) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,B,k3_finseq_2(A))
                    & m2_relset_1(E,B,k3_finseq_2(A)) )
                 => ! [F] :
                      ( ( v1_funct_1(F)
                        & v1_funct_2(F,B,A)
                        & m2_relset_1(F,B,A) )
                     => ! [G] :
                          ( m3_pboole(G,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),k8_pboole(B,A,F,C))
                         => ! [H] :
                              ( m3_pboole(H,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),k8_pboole(B,A,F,D))
                             => ! [I] :
                                  ( m3_pboole(I,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),k8_pboole(B,A,F,k11_pboole(A,C,D)))
                                 => ( I = k5_pralg_2(A,B,C,D,E,F,G,H)
                                  <=> ! [J] :
                                        ( r2_hidden(J,B)
                                       => ! [K] :
                                            ( ( v1_funct_1(K)
                                              & v1_funct_2(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),J),k1_funct_1(k8_pboole(B,A,F,C),J))
                                              & m2_relset_1(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),J),k1_funct_1(k8_pboole(B,A,F,C),J)) )
                                           => ! [L] :
                                                ( ( v1_funct_1(L)
                                                  & v1_funct_2(L,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),J),k1_funct_1(k8_pboole(B,A,F,D),J))
                                                  & m2_relset_1(L,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),J),k1_funct_1(k8_pboole(B,A,F,D),J)) )
                                               => ( ( K = k1_funct_1(G,J)
                                                    & L = k1_funct_1(H,J) )
                                                 => k1_funct_1(I,J) = k4_pralg_2(A,B,C,D,E,F,J,K,L) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d12_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m1_pboole(C,A)
         => ( m2_pralg_2(C,A,B)
          <=> ! [D] :
                ( r2_hidden(D,A)
               => ( v5_msualg_1(k1_funct_1(C,D),B)
                  & l3_msualg_1(k1_funct_1(C,D),B) ) ) ) ) ) )).

fof(d13_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => k7_pralg_2(A,B) = k3_tarski(k2_relat_1(u4_msualg_1(A,B))) ) ) )).

fof(t10_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ( k3_msualg_1(A,C,B) = k4_card_3(k5_relat_1(k1_msualg_1(A,C),u4_msualg_1(A,B)))
                & k1_relat_1(k5_relat_1(k1_msualg_1(A,C),u4_msualg_1(A,B))) = k1_relat_1(k1_msualg_1(A,C))
                & k4_msualg_1(A,C,B) = k1_funct_1(u4_msualg_1(A,B),k2_msualg_1(A,C)) ) ) ) ) )).

fof(t11_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ( k1_msualg_1(A,C) = k1_xboole_0
               => k3_msualg_1(A,C,B) = k1_tarski(k1_xboole_0) ) ) ) ) )).

fof(d15_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & l3_msualg_1(C,A) )
             => k9_pralg_2(A,B,C) = g3_msualg_1(A,k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)),k5_pralg_2(u1_struct_0(A),u1_msualg_1(A),u4_msualg_1(A,B),u4_msualg_1(A,C),u2_msualg_1(A),u3_msualg_1(A),u5_msualg_1(A,B),u5_msualg_1(A,C))) ) ) ) )).

fof(d16_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m2_pralg_2(D,A,B)
             => ! [E] :
                  ( m1_pboole(E,A)
                 => ( ( A != k1_xboole_0
                     => ( E = k10_pralg_2(A,B,C,D)
                      <=> ! [F] :
                            ~ ( r2_hidden(F,A)
                              & ! [G] :
                                  ( l3_msualg_1(G,B)
                                 => ~ ( G = k1_funct_1(D,F)
                                      & k1_funct_1(E,F) = k1_funct_1(u4_msualg_1(B,G),C) ) ) ) ) )
                    & ( A = k1_xboole_0
                     => ( E = k10_pralg_2(A,B,C,D)
                      <=> E = k1_xboole_0 ) ) ) ) ) ) ) )).

fof(d17_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ! [D] :
              ( m1_pboole(D,u1_struct_0(B))
             => ( D = k11_pralg_2(A,B,C)
              <=> ! [E] :
                    ( m1_subset_1(E,u1_struct_0(B))
                   => k1_funct_1(D,E) = k4_card_3(k10_pralg_2(A,B,E,C)) ) ) ) ) ) )).

fof(d18_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ! [D] :
              ( ( v1_funcop_1(D)
                & m1_pboole(D,A) )
             => ( ( A != k1_xboole_0
                 => ( D = k12_pralg_2(A,B,C)
                  <=> ! [E] :
                        ~ ( r2_hidden(E,A)
                          & ! [F] :
                              ( l3_msualg_1(F,B)
                             => ~ ( F = k1_funct_1(C,E)
                                  & k1_funct_1(D,E) = u5_msualg_1(B,F) ) ) ) ) )
                & ( A = k1_xboole_0
                 => ( D = k12_pralg_2(A,B,C)
                  <=> D = k1_xboole_0 ) ) ) ) ) ) )).

fof(t12_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => k1_relat_1(k4_funct_5(k12_pralg_2(A,B,C))) = k2_zfmisc_1(A,u1_msualg_1(B)) ) ) )).

fof(t13_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(B))
                 => r2_hidden(k10_funct_6(k12_pralg_2(A,B,C)),k1_funct_2(u1_msualg_1(B),k1_funct_2(A,k2_relat_1(k4_funct_5(k12_pralg_2(A,B,C)))))) ) ) ) ) )).

fof(d19_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ! [D] :
              ( m1_subset_1(D,u1_msualg_1(B))
             => k13_pralg_2(A,B,C,D) = k1_funct_1(k10_funct_6(k12_pralg_2(A,B,C)),D) ) ) ) )).

fof(t14_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & ~ v2_msualg_1(C)
                & l1_msualg_1(C) )
             => ! [D] :
                  ( m2_pralg_2(D,A,C)
                 => ! [E] :
                      ( m1_subset_1(E,u1_msualg_1(C))
                     => k1_funct_1(k13_pralg_2(A,C,D,E),B) = k5_msualg_1(C,E,k6_pralg_2(A,C,D,B)) ) ) ) ) ) )).

fof(t15_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(B))
                 => ! [E] :
                      ( r2_hidden(E,k2_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
                     => ( v1_relat_1(E)
                        & v1_funct_1(E) ) ) ) ) ) ) )).

fof(t16_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(B))
                 => ! [E] :
                      ( ( v1_relat_1(E)
                        & v1_funct_1(E) )
                     => ( r2_hidden(E,k2_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
                       => ( k1_relat_1(E) = A
                          & ! [F] :
                              ( m1_subset_1(F,A)
                             => r2_hidden(k1_funct_1(E,F),k4_msualg_1(B,D,k6_pralg_2(A,B,C,F))) ) ) ) ) ) ) ) ) )).

fof(t17_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(B))
                 => ! [E] :
                      ( ( v1_relat_1(E)
                        & v1_funct_1(E) )
                     => ( r2_hidden(E,k1_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
                       => ( k1_relat_1(E) = A
                          & ! [F] :
                              ( m1_subset_1(F,A)
                             => r2_hidden(k1_funct_1(E,F),k3_msualg_1(B,D,k6_pralg_2(A,B,C,F))) )
                          & r1_tarski(k2_relat_1(E),k1_fraenkel(k1_relat_1(k1_msualg_1(B,D)),k8_pralg_2(A,B,C))) ) ) ) ) ) ) ) )).

fof(t18_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & ~ v2_msualg_1(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(B))
                 => ( k1_relat_1(k2_funct_6(k13_pralg_2(A,B,C,D))) = A
                    & ! [E] :
                        ( m1_subset_1(E,A)
                       => k1_funct_1(k2_funct_6(k13_pralg_2(A,B,C,D)),E) = k3_msualg_1(B,D,k6_pralg_2(A,B,C,E)) ) ) ) ) ) ) )).

fof(d20_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ! [D] :
              ( m3_pboole(D,u1_msualg_1(B),k8_pboole(u1_msualg_1(B),k3_finseq_2(u1_struct_0(B)),u2_msualg_1(B),k6_pboole(u1_struct_0(B),k11_pralg_2(A,B,C))),k8_pboole(u1_msualg_1(B),u1_struct_0(B),u3_msualg_1(B),k11_pralg_2(A,B,C)))
             => ( ( A != k1_xboole_0
                 => ( D = k14_pralg_2(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_msualg_1(B))
                       => k1_funct_1(D,E) = k1_cqc_lang(k1_msualg_1(B,E),k1_xboole_0,k10_funct_6(k13_pralg_2(A,B,C,E)),k2_pralg_2(k3_pralg_2(k13_pralg_2(A,B,C,E)))) ) ) )
                & ( A = k1_xboole_0
                 => D = k14_pralg_2(A,B,C) ) ) ) ) ) )).

fof(d21_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => k15_pralg_2(A,B,C) = g3_msualg_1(B,k11_pralg_2(A,B,C),k14_pralg_2(A,B,C)) ) ) )).

fof(t19_pralg_2,axiom,(
    $true )).

fof(t20_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => ( r6_pboole(u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),k11_pralg_2(A,B,C))
            & r6_pboole(u1_msualg_1(B),u5_msualg_1(B,k15_pralg_2(A,B,C)),k14_pralg_2(A,B,C)) ) ) ) )).

fof(dt_m1_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_fraenkel(A)
        & v1_pralg_2(A) )
     => ! [B] :
          ( m1_pralg_2(B,A)
         => m1_pboole(B,k1_pralg_2(A)) ) ) )).

fof(existence_m1_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_fraenkel(A)
        & v1_pralg_2(A) )
     => ? [B] : m1_pralg_2(B,A) ) )).

fof(redefinition_m1_pralg_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_fraenkel(A)
        & v1_pralg_2(A) )
     => ! [B] :
          ( m1_pralg_2(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_m2_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ! [C] :
          ( m2_pralg_2(C,A,B)
         => m1_pboole(C,A) ) ) )).

fof(existence_m2_pralg_2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ? [C] : m2_pralg_2(C,A,B) ) )).

fof(dt_k1_pralg_2,axiom,(
    $true )).

fof(dt_k2_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_pralg_2(A))
        & v1_funct_1(k2_pralg_2(A)) ) ) )).

fof(dt_k3_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => ( v1_funcop_1(k3_pralg_2(A))
        & m1_pboole(k3_pralg_2(A),k4_card_3(k2_funct_6(A))) ) ) )).

fof(redefinition_k3_pralg_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_funcop_1(A) )
     => k3_pralg_2(A) = k7_funct_6(A) ) )).

fof(dt_k4_pralg_2,axiom,(
    ! [A,B,C,D,E,F,G,H,I] :
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & v2_relat_1(D)
        & m1_pboole(D,A)
        & v1_funct_1(E)
        & v1_funct_2(E,B,k3_finseq_2(A))
        & m1_relset_1(E,B,k3_finseq_2(A))
        & v1_funct_1(F)
        & v1_funct_2(F,B,A)
        & m1_relset_1(F,B,A)
        & v1_funct_1(H)
        & v1_funct_2(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
        & m1_relset_1(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
        & v1_funct_1(I)
        & v1_funct_2(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G))
        & m1_relset_1(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G)) )
     => ( v1_funct_1(k4_pralg_2(A,B,C,D,E,F,G,H,I))
        & v1_funct_2(k4_pralg_2(A,B,C,D,E,F,G,H,I),k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G))
        & m2_relset_1(k4_pralg_2(A,B,C,D,E,F,G,H,I),k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G)) ) ) )).

fof(dt_k5_pralg_2,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & v2_relat_1(D)
        & m1_pboole(D,A)
        & v1_funct_1(E)
        & v1_funct_2(E,B,k3_finseq_2(A))
        & m1_relset_1(E,B,k3_finseq_2(A))
        & v1_funct_1(F)
        & v1_funct_2(F,B,A)
        & m1_relset_1(F,B,A)
        & m3_pboole(G,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),k8_pboole(B,A,F,C))
        & m3_pboole(H,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),k8_pboole(B,A,F,D)) )
     => m3_pboole(k5_pralg_2(A,B,C,D,E,F,G,H),B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),k8_pboole(B,A,F,k11_pboole(A,C,D))) ) )).

fof(dt_k6_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B)
        & m1_subset_1(D,A) )
     => ( v5_msualg_1(k6_pralg_2(A,B,C,D),B)
        & l3_msualg_1(k6_pralg_2(A,B,C,D),B) ) ) )).

fof(redefinition_k6_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B)
        & m1_subset_1(D,A) )
     => k6_pralg_2(A,B,C,D) = k1_funct_1(C,D) ) )).

fof(dt_k7_pralg_2,axiom,(
    $true )).

fof(dt_k8_pralg_2,axiom,(
    $true )).

fof(dt_k9_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A)
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A) )
     => l3_msualg_1(k9_pralg_2(A,B,C),A) ) )).

fof(dt_k10_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m2_pralg_2(D,A,B) )
     => m1_pboole(k10_pralg_2(A,B,C,D),A) ) )).

fof(dt_k11_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => m1_pboole(k11_pralg_2(A,B,C),u1_struct_0(B)) ) )).

fof(dt_k12_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => ( v1_funcop_1(k12_pralg_2(A,B,C))
        & m1_pboole(k12_pralg_2(A,B,C),A) ) ) )).

fof(dt_k13_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B)
        & m1_subset_1(D,u1_msualg_1(B)) )
     => ( v1_funcop_1(k13_pralg_2(A,B,C,D))
        & m1_pboole(k13_pralg_2(A,B,C,D),A) ) ) )).

fof(dt_k14_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => m3_pboole(k14_pralg_2(A,B,C),u1_msualg_1(B),k8_pboole(u1_msualg_1(B),k3_finseq_2(u1_struct_0(B)),u2_msualg_1(B),k6_pboole(u1_struct_0(B),k11_pralg_2(A,B,C))),k8_pboole(u1_msualg_1(B),u1_struct_0(B),u3_msualg_1(B),k11_pralg_2(A,B,C))) ) )).

fof(dt_k15_pralg_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & ~ v2_msualg_1(B)
        & l1_msualg_1(B)
        & m2_pralg_2(C,A,B) )
     => l3_msualg_1(k15_pralg_2(A,B,C),B) ) )).

fof(d14_pralg_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_msualg_1(B) )
         => ! [C] :
              ( m2_pralg_2(C,A,B)
             => k8_pralg_2(A,B,C) = k3_tarski(a_3_0_pralg_2(A,B,C)) ) ) ) )).

fof(fraenkel_a_3_0_pralg_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & ~ v3_struct_0(C)
        & l1_msualg_1(C)
        & m2_pralg_2(D,B,C) )
     => ( r2_hidden(A,a_3_0_pralg_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = k7_pralg_2(C,k6_pralg_2(B,C,D,E)) ) ) ) )).
%------------------------------------------------------------------------------