SET007 Axioms: SET007+423.ax


%------------------------------------------------------------------------------
% File     : SET007+423 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Preliminaries to Circuits, II
% 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    : msafree2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   53 (   0 unit)
%            Number of atoms       :  392 (  31 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  441 ( 102 ~  ;   0  |; 201  &)
%                                         (  12 <=>; 126 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   0 propositional; 1-4 arity)
%            Number of functors    :   41 (   4 constant; 0-5 arity)
%            Number of variables   :  135 (   0 singleton; 125 !;  10 ?)
%            Maximal term depth    :    5 (   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_msafree2,axiom,(
    ? [A] :
      ( l1_msualg_1(A)
      & ~ v3_struct_0(A)
      & ~ v2_msualg_1(A)
      & v1_msafree2(A) ) )).

fof(fc1_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_msafree2(A)
        & l1_msualg_1(A) )
     => ~ v1_xboole_0(k2_msafree2(A)) ) )).

fof(cc1_msafree2,axiom,(
    ! [A] :
      ( l1_msualg_1(A)
     => ( ( ~ v3_struct_0(A)
          & v2_msualg_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_msafree2(A) ) ) ) )).

fof(rc2_msafree2,axiom,(
    ? [A] :
      ( l1_msualg_1(A)
      & ~ v3_struct_0(A)
      & v1_msualg_1(A)
      & ~ v2_msualg_1(A)
      & v2_msafree2(A) ) )).

fof(cc2_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ( ( v5_msualg_1(B,A)
              & v4_msafree2(B,A) )
           => ( v5_msualg_1(B,A)
              & v3_msafree2(B,A) ) ) ) ) )).

fof(rc3_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ? [B] :
          ( l3_msualg_1(B,A)
          & v4_msualg_1(B,A)
          & v5_msualg_1(B,A)
          & v3_msafree2(B,A)
          & v4_msafree2(B,A) ) ) )).

fof(rc4_msafree2,axiom,(
    ? [A] :
      ( l1_msualg_1(A)
      & ~ v3_struct_0(A)
      & v6_group_1(A)
      & ~ v2_msualg_1(A)
      & v2_msafree2(A)
      & v5_msafree2(A) ) )).

fof(cc3_msafree2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ! [D] :
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
         => ( ~ v1_xboole_0(D)
            & v1_relat_1(D)
            & v1_funct_1(D)
            & v1_finset_1(D) ) ) ) )).

fof(rc5_msafree2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ? [D] :
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
          & ~ v1_xboole_0(D)
          & v1_relat_1(D)
          & v1_funct_1(D)
          & v1_finset_1(D) ) ) )).

fof(cc4_msafree2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ! [D] :
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
         => ( ( v1_relat_1(D)
              & v1_funct_1(D) )
           => ( ~ v1_xboole_0(D)
              & v1_relat_1(D)
              & v1_funct_1(D)
              & v1_finset_1(D)
              & v3_trees_2(D) ) ) ) ) )).

fof(rc6_msafree2,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ? [D] :
          ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
          & ~ v1_xboole_0(D)
          & v1_relat_1(D)
          & v1_funct_1(D)
          & v1_finset_1(D)
          & v3_trees_2(D) ) ) )).

fof(d2_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => k2_msafree2(A) = k4_xboole_0(u1_struct_0(A),k5_relset_1(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A))) ) )).

fof(d3_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => k3_msafree2(A) = k5_relset_1(u1_msualg_1(A),u1_struct_0(A),u3_msualg_1(A)) ) )).

fof(t1_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_msualg_1(A)
        & l1_msualg_1(A) )
     => k2_msafree2(A) = u1_struct_0(A) ) )).

fof(t2_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r2_hidden(B,k2_msafree2(A))
           => ! [C] :
                ( m1_subset_1(C,u1_msualg_1(A))
               => k2_msualg_1(A,C) != B ) ) ) ) )).

fof(t3_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => k4_subset_1(u1_struct_0(A),k2_msafree2(A),k3_msafree2(A)) = u1_struct_0(A) ) )).

fof(t4_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => r1_xboole_0(k2_msafree2(A),k3_msafree2(A)) ) )).

fof(t5_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => r1_tarski(k1_msafree2(A),k3_msafree2(A)) ) )).

fof(t6_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => r1_xboole_0(k2_msafree2(A),k1_msafree2(A)) ) )).

fof(d4_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v1_msafree2(A)
      <=> k2_msafree2(A) != k1_xboole_0 ) ) )).

fof(d5_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_pboole(C,k2_msafree2(A))
             => ( m1_msafree2(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,k2_msafree2(A))
                     => r2_hidden(k1_funct_1(C,D),k1_funct_1(u4_msualg_1(A,B),D)) ) ) ) ) ) ) )).

fof(d6_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v2_msafree2(A)
      <=> ! [B] :
            ( ( ~ v3_struct_0(B)
              & ~ v2_msualg_1(B)
              & l1_msualg_1(B) )
           => ( B = A
             => ! [C] :
                  ( m1_subset_1(C,u1_msualg_1(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_msualg_1(B))
                     => ( k2_msualg_1(B,C) = k2_msualg_1(B,D)
                       => C = D ) ) ) ) ) ) ) )).

fof(d7_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & v2_msafree2(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r2_hidden(B,k4_msafree2(A))
           => ! [C] :
                ( m1_subset_1(C,u1_msualg_1(A))
               => ( C = k5_msafree2(A,B)
                <=> k2_msualg_1(A,C) = B ) ) ) ) ) )).

fof(t7_msafree2,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))
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D) )
                 => ( ( k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C))
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k4_finseq_1(D))
                           => r2_hidden(k1_funct_1(D,E),k1_funct_1(u4_msualg_1(A,B),k4_finseq_4(k5_numbers,u1_struct_0(A),k1_msualg_1(A,C),E))) ) ) )
                   => r2_hidden(D,k3_msualg_1(A,C,B)) ) ) ) ) ) )).

fof(d8_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => k6_msafree2(A,B) = k11_msafree(A,u4_msualg_1(A,B)) ) ) )).

fof(t8_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ( v1_msafree(k13_msafree(A,u4_msualg_1(A,B)),A,k6_msafree2(A,B))
            & m1_msafree(k13_msafree(A,u4_msualg_1(A,B)),A,k6_msafree2(A,B)) ) ) ) )).

fof(d9_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m3_pboole(C,u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B))
             => ( C = k7_msafree2(A,B)
              <=> ( r1_msualg_3(A,k6_msafree2(A,B),B,C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E,F] :
                          ( ( r2_hidden(F,k7_msafree(A,u4_msualg_1(A,B),D))
                            & F = k1_trees_4(k4_tarski(E,D))
                            & r2_hidden(E,k1_funct_1(u4_msualg_1(A,B),D)) )
                         => k1_funct_1(k1_msualg_3(u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B),C,D),F) = E ) ) ) ) ) ) ) )).

fof(t9_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => m1_msafree(u4_msualg_1(A,B),A,B) ) ) )).

fof(d10_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ( ( ~ v2_msualg_1(A)
             => ( v3_msafree2(B,A)
              <=> ! [C] :
                    ( ( ~ v3_struct_0(C)
                      & ~ v2_msualg_1(C)
                      & l1_msualg_1(C) )
                   => ( C = A
                     => ! [D] :
                          ( l3_msualg_1(D,C)
                         => ~ ( D = B
                              & ! [E] :
                                  ( m1_msafree(E,C,D)
                                 => ~ v1_pre_circ(E,u1_struct_0(C)) ) ) ) ) ) ) )
            & ( v2_msualg_1(A)
             => ( v3_msafree2(B,A)
              <=> v1_pre_circ(u4_msualg_1(A,B),u1_struct_0(A)) ) ) ) ) ) )).

fof(d11_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ( v4_msafree2(B,A)
          <=> v1_pre_circ(u4_msualg_1(A,B),u1_struct_0(A)) ) ) ) )).

fof(d12_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v4_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ( B = k8_msafree2(A)
          <=> r6_pboole(u1_struct_0(A),u4_msualg_1(A,B),k2_pre_circ(u1_struct_0(A),k1_tarski(np__0))) ) ) ) )).

fof(d13_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v5_msafree2(A)
      <=> ! [B] :
            ( ( v5_msualg_1(B,A)
              & v3_msafree2(B,A)
              & l3_msualg_1(B,A) )
           => v4_msafree2(B,A) ) ) ) )).

fof(t10_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
                 => ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finset_1(D)
                    & v3_trees_2(D) ) ) ) ) ) )).

fof(t11_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & v1_pre_circ(B,u1_struct_0(A))
            & m1_pboole(B,u1_struct_0(A)) )
         => v3_msafree2(k11_msafree(A,B),A) ) ) )).

fof(t12_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k6_msafree2(A,B)),C))
                 => ~ ( r2_hidden(C,k2_msafree2(A))
                      & ! [E] :
                          ( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,B),C))
                         => D != k1_trees_4(k4_tarski(E,C)) ) ) ) ) ) ) )).

fof(t13_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D)
                    & v6_trees_3(D) )
                 => ( r2_hidden(k4_trees_4(k4_tarski(C,u1_struct_0(A)),D),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k2_msualg_1(A,C)))
                   => k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C)) ) ) ) ) ) )).

fof(t14_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D)
                    & v6_trees_3(D) )
                 => ( r2_hidden(k4_trees_4(k4_tarski(C,u1_struct_0(A)),D),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k2_msualg_1(A,C)))
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),k1_msualg_1(A,C)))
                         => r2_hidden(k1_funct_1(D,E),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k1_funct_1(k1_msualg_1(A,C),E))) ) ) ) ) ) ) ) )).

fof(t15_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_msualg_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
                     => ~ ( r2_hidden(C,k4_msafree2(A))
                          & k1_funct_1(E,k1_xboole_0) = k4_tarski(D,u1_struct_0(A))
                          & ! [F] :
                              ( ( v1_relat_1(F)
                                & v1_funct_1(F)
                                & v1_finseq_1(F)
                                & v6_trees_3(F) )
                             => ~ ( k3_finseq_1(F) = k3_finseq_1(k1_msualg_1(A,D))
                                  & ! [G] :
                                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                                     => ( r2_hidden(G,k4_finseq_1(F))
                                       => r2_hidden(k1_funct_1(F,G),k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),k1_funct_1(k1_msualg_1(A,D),G))) ) ) ) ) ) ) ) ) ) ) )).

fof(d14_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( E = k9_msafree2(A,B,C,D)
                      <=> ? [F] :
                            ( v1_relat_1(F)
                            & v1_funct_1(F)
                            & v1_finset_1(F)
                            & v3_trees_2(F)
                            & ? [G] :
                                ( ~ v1_xboole_0(G)
                                & v1_finset_1(G)
                                & v1_trees_1(G)
                                & F = D
                                & G = k1_relat_1(F)
                                & E = k6_trees_1(G) ) ) ) ) ) ) ) ) )).

fof(dt_m1_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ! [C] :
          ( m1_msafree2(C,A,B)
         => m1_pboole(C,k2_msafree2(A)) ) ) )).

fof(existence_m1_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ? [C] : m1_msafree2(C,A,B) ) )).

fof(dt_k1_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => m1_subset_1(k1_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k2_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => m1_subset_1(k2_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k3_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => m1_subset_1(k3_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k4_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ( ~ v1_xboole_0(k4_msafree2(A))
        & m1_subset_1(k4_msafree2(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(redefinition_k4_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => k4_msafree2(A) = k3_msafree2(A) ) )).

fof(dt_k5_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & v2_msafree2(A)
        & l1_msualg_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => m1_subset_1(k5_msafree2(A,B),u1_msualg_1(A)) ) )).

fof(dt_k6_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v4_msualg_1(k6_msafree2(A,B),A)
        & v5_msualg_1(k6_msafree2(A,B),A)
        & v2_msafree(k6_msafree2(A,B),A)
        & l3_msualg_1(k6_msafree2(A,B),A) ) ) )).

fof(dt_k7_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => m3_pboole(k7_msafree2(A,B),u1_struct_0(A),u4_msualg_1(A,k6_msafree2(A,B)),u4_msualg_1(A,B)) ) )).

fof(dt_k8_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v4_msualg_1(k8_msafree2(A),A)
        & l3_msualg_1(k8_msafree2(A),A) ) ) )).

fof(dt_k9_msafree2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A))
        & m1_subset_1(D,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C)) )
     => m2_subset_1(k9_msafree2(A,B,C,D),k1_numbers,k5_numbers) ) )).

fof(d1_msafree2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( ( ~ v2_msualg_1(A)
         => k1_msafree2(A) = a_1_0_msafree2(A) )
        & ( v2_msualg_1(A)
         => k1_msafree2(A) = k1_xboole_0 ) ) ) )).

fof(fraenkel_a_1_0_msafree2,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l1_msualg_1(B) )
     => ( r2_hidden(A,a_1_0_msafree2(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(B))
            & A = C
            & v1_msualg_2(C,B) ) ) ) )).
%------------------------------------------------------------------------------