SET007 Axioms: SET007+334.ax


%------------------------------------------------------------------------------
% File     : SET007+334 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Transpose Matrices and Groups of Permutations
% 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    : matrix_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   77 (   8 unit)
%            Number of atoms       :  578 (  79 equality)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  578 (  77 ~  ;   7  |; 317  &)
%                                         (  15 <=>; 162 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   41 (   1 propositional; 0-4 arity)
%            Number of functors    :   57 (   6 constant; 0-5 arity)
%            Number of variables   :  211 (   0 singleton; 200 !;  11 ?)
%            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_matrix_2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_matrix_2(A) ) )).

fof(fc1_matrix_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k11_matrix_2(A))
        & v1_matrix_2(k11_matrix_2(A)) ) ) )).

fof(fc2_matrix_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k13_matrix_2(A))
        & v1_group_1(k13_matrix_2(A)) ) ) )).

fof(fc3_matrix_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k13_matrix_2(A))
        & v1_group_1(k13_matrix_2(A))
        & v2_group_1(k13_matrix_2(A))
        & v3_group_1(k13_matrix_2(A))
        & v4_group_1(k13_matrix_2(A)) ) ) )).

fof(d1_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : k1_matrix_2(A,B,C) = k2_finseq_2(A,k2_finseq_2(B,C)) ) ) )).

fof(t1_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ! [F] :
                          ( m1_subset_1(F,E)
                         => ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k2_matrix_2(E,C,D,F)))
                           => k3_matrix_1(E,k2_matrix_2(E,C,D,F),A,B) = F ) ) ) ) ) ) ) )).

fof(t2_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k13_matrix_1(B,A,k2_matrix_2(u1_struct_0(B),A,A,C),k2_matrix_2(u1_struct_0(B),A,A,D)) = k2_matrix_2(u1_struct_0(B),A,A,k4_rlvect_1(B,C,D)) ) ) ) ) )).

fof(d2_matrix_2,axiom,(
    ! [A,B,C,D] : k3_matrix_2(A,B,C,D) = k10_finseq_1(k10_finseq_1(A,B),k10_finseq_1(C,D)) )).

fof(t3_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( k3_finseq_1(k3_matrix_2(A,B,C,D)) = np__2
      & k1_matrix_1(k3_matrix_2(A,B,C,D)) = np__2
      & k2_matrix_1(k3_matrix_2(A,B,C,D)) = k2_zfmisc_1(k2_finseq_1(np__2),k2_finseq_1(np__2)) ) )).

fof(t4_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( r2_hidden(k4_tarski(np__1,np__1),k2_matrix_1(k3_matrix_2(A,B,C,D)))
      & r2_hidden(k4_tarski(np__1,np__2),k2_matrix_1(k3_matrix_2(A,B,C,D)))
      & r2_hidden(k4_tarski(np__2,np__1),k2_matrix_1(k3_matrix_2(A,B,C,D)))
      & r2_hidden(k4_tarski(np__2,np__2),k2_matrix_1(k3_matrix_2(A,B,C,D))) ) )).

fof(t5_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( r2_hidden(k4_tarski(np__1,np__1),k2_matrix_1(k5_matrix_2(A,np__1,k4_matrix_2(A,B))))
            & k3_matrix_1(A,k5_matrix_2(A,np__1,k4_matrix_2(A,B)),np__1,np__1) = B ) ) ) )).

fof(t6_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,A)
                     => ( k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__1,np__1) = B
                        & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__1,np__2) = C
                        & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__2,np__1) = D
                        & k3_matrix_1(A,k6_matrix_2(A,B,C,D,E),np__2,np__2) = E ) ) ) ) ) ) )).

fof(d3_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => ( m1_matrix_2(C,A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C))
                         => ( r1_xreal_0(D,E)
                            | k3_matrix_1(u1_struct_0(B),C,D,E) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) )).

fof(d4_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => ( m2_matrix_2(C,A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C))
                         => ( r1_xreal_0(E,D)
                            | k3_matrix_1(u1_struct_0(B),C,D,E) = k1_rlvect_1(B) ) ) ) ) ) ) ) ) )).

fof(t7_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(B)) )
             => ( k3_finseq_1(C) = A
               => m1_matrix_1(C,B,A,k1_matrix_1(C)) ) ) ) ) )).

fof(t8_matrix_2,axiom,(
    $true )).

fof(t9_matrix_2,axiom,(
    $true )).

fof(t10_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v7_vectsp_1(C)
                & v8_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_1(C) )
             => ! [D] :
                  ( m1_matrix_1(D,u1_struct_0(C),A,B)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k2_finseq_1(A))
                       => k1_funct_1(D,E) = k7_matrix_1(u1_struct_0(C),D,E) ) ) ) ) ) ) )).

fof(d5_matrix_2,axiom,(
    $true )).

fof(d6_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
             => ( r2_hidden(A,k2_finseq_1(k1_matrix_1(C)))
               => ! [D] :
                    ( ( v1_matrix_1(D)
                      & m2_finseq_1(D,k3_finseq_2(u1_struct_0(B))) )
                   => ( D = k7_matrix_2(A,B,C)
                    <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( r2_hidden(E,k4_finseq_1(C))
                             => k1_funct_1(D,E) = k2_finseq_3(A,k7_matrix_1(u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ) ) )).

fof(t11_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(A)) )
             => ( ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => B = C ) ) ) ) )).

fof(t12_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ( ~ r1_xreal_0(k1_matrix_1(B),np__0)
           => ( k3_finseq_1(k4_matrix_1(A,B)) = k1_matrix_1(B)
              & k1_matrix_1(k4_matrix_1(A,B)) = k3_finseq_1(B) ) ) ) ) )).

fof(t13_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(A)) )
             => ( ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
                  & k1_matrix_1(k4_matrix_1(A,B)) = k1_matrix_1(k4_matrix_1(A,C)) )
               => ( r1_xreal_0(k1_matrix_1(B),np__0)
                  | r1_xreal_0(k1_matrix_1(C),np__0)
                  | B = C ) ) ) ) ) )).

fof(t14_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(A)) )
             => ~ ( ~ r1_xreal_0(k1_matrix_1(B),np__0)
                  & ~ r1_xreal_0(k1_matrix_1(C),np__0)
                  & ~ ( B = C
                    <=> ( k4_matrix_1(A,B) = k4_matrix_1(A,C)
                        & k1_matrix_1(B) = k1_matrix_1(C) ) ) ) ) ) ) )).

fof(t15_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ~ ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
              & ~ r1_xreal_0(k1_matrix_1(B),np__0)
              & k4_matrix_1(A,k4_matrix_1(A,B)) != B ) ) ) )).

fof(t16_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(B))
               => k7_matrix_1(A,B,C) = k8_matrix_1(A,k4_matrix_1(A,B),C) ) ) ) ) )).

fof(t17_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k2_finseq_1(k1_matrix_1(B)))
               => k7_matrix_1(A,k4_matrix_1(A,B),C) = k8_matrix_1(A,B,C) ) ) ) ) )).

fof(t18_matrix_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(B))
               => k1_funct_1(B,C) = k7_matrix_1(A,B,C) ) ) ) ) )).

fof(d7_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
             => ( r2_hidden(A,k4_finseq_1(C))
               => ( r1_xreal_0(k1_matrix_1(C),np__0)
                  | ! [D] :
                      ( ( v1_matrix_1(D)
                        & m2_finseq_1(D,k3_finseq_2(u1_struct_0(B))) )
                     => ( ( k3_finseq_1(C) = np__1
                         => ( D = k8_matrix_2(A,B,C)
                          <=> D = k1_xboole_0 ) )
                        & ( k3_finseq_1(C) != np__1
                         => ( D = k8_matrix_2(A,B,C)
                          <=> ( k1_matrix_1(D) = k1_matrix_1(C)
                              & ! [E] :
                                  ( m2_subset_1(E,k1_numbers,k5_numbers)
                                 => ( r2_hidden(E,k2_finseq_1(k1_matrix_1(C)))
                                   => k8_matrix_1(u1_struct_0(B),D,E) = k2_finseq_3(A,k8_matrix_1(u1_struct_0(B),C,E)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d8_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v4_group_1(D)
                    & v7_group_1(D)
                    & v3_rlvect_1(D)
                    & v4_rlvect_1(D)
                    & v5_rlvect_1(D)
                    & v6_rlvect_1(D)
                    & v7_vectsp_1(D)
                    & v8_vectsp_1(D)
                    & v9_vectsp_1(D)
                    & ~ v10_vectsp_1(D)
                    & l3_vectsp_1(D) )
                 => ! [E] :
                      ( m1_matrix_1(E,u1_struct_0(D),C,C)
                     => ( ( C = np__1
                         => k9_matrix_2(A,B,C,D,E) = k1_xboole_0 )
                        & ( C != np__1
                         => k9_matrix_2(A,B,C,D,E) = k7_matrix_2(B,D,k8_matrix_2(A,D,E)) ) ) ) ) ) ) ) )).

fof(d9_matrix_2,axiom,(
    ! [A] :
      ( v1_matrix_2(A)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & ! [C] :
              ( r2_hidden(C,A)
             => ( v1_funct_1(C)
                & v1_funct_2(C,k2_finseq_1(B),k2_finseq_1(B))
                & v3_funct_2(C,k2_finseq_1(B),k2_finseq_1(B))
                & m2_relset_1(C,k2_finseq_1(B),k2_finseq_1(B)) ) ) ) ) )).

fof(d10_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_2(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k10_matrix_2(A)
          <=> ? [C] :
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C)
                & r2_hidden(C,A)
                & B = k3_finseq_1(C) ) ) ) ) )).

fof(t19_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ? [B] :
          ( ~ v1_xboole_0(B)
          & v1_matrix_2(B)
          & k10_matrix_2(B) = A ) ) )).

fof(d11_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( B = k11_matrix_2(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ( v1_funct_1(C)
                & v1_funct_2(C,k2_finseq_1(A),k2_finseq_1(A))
                & v3_funct_2(C,k2_finseq_1(A),k2_finseq_1(A))
                & m2_relset_1(C,k2_finseq_1(A),k2_finseq_1(A)) ) ) ) ) )).

fof(t20_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k10_matrix_2(k11_matrix_2(A)) = A ) )).

fof(t21_matrix_2,axiom,(
    k11_matrix_2(np__1) = k1_tarski(k1_finseq_2(np__1)) )).

fof(d12_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m3_matrix_2(B,k11_matrix_2(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k12_matrix_2(A,B)
              <=> ? [D] :
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D)
                    & D = B
                    & C = k3_finseq_1(D) ) ) ) ) ) )).

fof(t22_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m3_matrix_2(B,k11_matrix_2(A))
         => k12_matrix_2(A,B) = A ) ) )).

fof(d13_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_group_1(B)
            & l1_group_1(B) )
         => ( B = k13_matrix_2(A)
          <=> ( u1_struct_0(B) = k11_matrix_2(A)
              & ! [C] :
                  ( m3_matrix_2(C,k11_matrix_2(A))
                 => ! [D] :
                      ( m3_matrix_2(D,k11_matrix_2(A))
                     => k1_binop_1(u1_group_1(B),C,D) = k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),C,D) ) ) ) ) ) ) )).

fof(t23_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => m1_subset_1(k1_finseq_2(A),u1_struct_0(k13_matrix_2(A))) ) )).

fof(t24_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m3_matrix_2(B,k11_matrix_2(A))
         => ( k5_relat_1(k1_finseq_2(A),B) = B
            & k5_relat_1(B,k1_finseq_2(A)) = B ) ) ) )).

fof(t25_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m3_matrix_2(B,k11_matrix_2(A))
         => ( k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B),B) = k1_finseq_2(A)
            & k5_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B,k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B)) = k1_finseq_2(A) ) ) ) )).

fof(t26_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m3_matrix_2(B,k11_matrix_2(A))
         => m1_subset_1(k6_funct_2(k2_finseq_1(k10_matrix_2(k11_matrix_2(A))),B),u1_struct_0(k13_matrix_2(A))) ) ) )).

fof(t27_matrix_2,axiom,(
    $true )).

fof(t28_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_finseq_2(A) = k2_group_1(k13_matrix_2(A)) ) )).

fof(d14_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
            & v3_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
            & m2_relset_1(B,k2_finseq_1(A),k2_finseq_1(A)) )
         => ( v2_matrix_2(B,A)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r2_hidden(C,k1_relat_1(B))
                    & r2_hidden(D,k1_relat_1(B))
                    & C != D
                    & k1_funct_1(B,C) = D
                    & k1_funct_1(B,D) = C
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k1_relat_1(B))
                         => ( E = C
                            | E = D
                            | k1_funct_1(B,E) = E ) ) ) ) ) ) ) ) )).

fof(d15_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
            & v3_funct_2(B,k2_finseq_1(A),k2_finseq_1(A))
            & m2_relset_1(B,k2_finseq_1(A),k2_finseq_1(A)) )
         => ( v3_matrix_2(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_struct_0(k13_matrix_2(A)))
                & k4_nat_1(k3_finseq_1(C),np__2) = np__0
                & B = k3_group_4(k13_matrix_2(A),C)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( r2_hidden(D,k4_finseq_1(C))
                        & ! [E] :
                            ( m3_matrix_2(E,k11_matrix_2(A))
                           => ~ ( k1_funct_1(C,D) = E
                                & v2_matrix_2(E,k10_matrix_2(k11_matrix_2(A))) ) ) ) ) ) ) ) ) )).

fof(t29_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v3_matrix_2(k6_partfun1(k2_finseq_1(A)),A) ) )).

fof(d16_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m3_matrix_2(D,k11_matrix_2(B))
                 => ( ( v3_matrix_2(D,k10_matrix_2(k11_matrix_2(B)))
                     => k14_matrix_2(A,B,C,D) = C )
                    & ( ~ v3_matrix_2(D,k10_matrix_2(k11_matrix_2(B)))
                     => k14_matrix_2(A,B,C,D) = k5_rlvect_1(A,C) ) ) ) ) ) ) )).

fof(d17_matrix_2,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => k15_matrix_2(A) = A ) )).

fof(t30_matrix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_finset_1(k11_matrix_2(A)) ) )).

fof(dt_m1_matrix_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ! [C] :
          ( m1_matrix_2(C,A,B)
         => m1_matrix_1(C,u1_struct_0(B),A,A) ) ) )).

fof(existence_m1_matrix_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ? [C] : m1_matrix_2(C,A,B) ) )).

fof(dt_m2_matrix_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ! [C] :
          ( m2_matrix_2(C,A,B)
         => m1_matrix_1(C,u1_struct_0(B),A,A) ) ) )).

fof(existence_m2_matrix_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ? [C] : m2_matrix_2(C,A,B) ) )).

fof(dt_m3_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_2(A) )
     => ! [B] :
          ( m3_matrix_2(B,A)
         => ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A)))
            & v3_funct_2(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A)))
            & m2_relset_1(B,k2_finseq_1(k10_matrix_2(A)),k2_finseq_1(k10_matrix_2(A))) ) ) ) )).

fof(existence_m3_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_2(A) )
     => ? [B] : m3_matrix_2(B,A) ) )).

fof(redefinition_m3_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_2(A) )
     => ! [B] :
          ( m3_matrix_2(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_k1_matrix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k1_matrix_2(A,B,C))
        & v1_funct_1(k1_matrix_2(A,B,C))
        & v1_finseq_1(k1_matrix_2(A,B,C))
        & v1_matrix_1(k1_matrix_2(A,B,C)) ) ) )).

fof(dt_k2_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,A) )
     => m1_matrix_1(k2_matrix_2(A,B,C,D),A,B,C) ) )).

fof(redefinition_k2_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,A) )
     => k2_matrix_2(A,B,C,D) = k1_matrix_2(B,C,D) ) )).

fof(dt_k3_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( v1_relat_1(k3_matrix_2(A,B,C,D))
      & v1_funct_1(k3_matrix_2(A,B,C,D))
      & v1_finseq_1(k3_matrix_2(A,B,C,D))
      & v1_matrix_1(k3_matrix_2(A,B,C,D)) ) )).

fof(dt_k4_matrix_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m2_finseq_2(k4_matrix_2(A,B),A,k4_finseq_2(np__1,A)) ) )).

fof(redefinition_k4_matrix_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => k4_matrix_2(A,B) = k5_finseq_1(B) ) )).

fof(dt_k5_matrix_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(B,A)) )
     => m1_matrix_1(k5_matrix_2(A,B,C),A,np__1,B) ) )).

fof(redefinition_k5_matrix_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(B,A)) )
     => k5_matrix_2(A,B,C) = k5_finseq_1(C) ) )).

fof(dt_k6_matrix_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => m1_matrix_1(k6_matrix_2(A,B,C,D,E),A,np__2,np__2) ) )).

fof(redefinition_k6_matrix_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => k6_matrix_2(A,B,C,D,E) = k3_matrix_2(B,C,D,E) ) )).

fof(dt_k7_matrix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & v1_matrix_1(C)
        & m1_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
     => ( v1_matrix_1(k7_matrix_2(A,B,C))
        & m2_finseq_1(k7_matrix_2(A,B,C),k3_finseq_2(u1_struct_0(B))) ) ) )).

fof(dt_k8_matrix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & v1_matrix_1(C)
        & m1_finseq_1(C,k3_finseq_2(u1_struct_0(B))) )
     => ( v1_matrix_1(k8_matrix_2(A,B,C))
        & m2_finseq_1(k8_matrix_2(A,B,C),k3_finseq_2(u1_struct_0(B))) ) ) )).

fof(dt_k9_matrix_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & ~ v3_struct_0(D)
        & v4_group_1(D)
        & v7_group_1(D)
        & v3_rlvect_1(D)
        & v4_rlvect_1(D)
        & v5_rlvect_1(D)
        & v6_rlvect_1(D)
        & v7_vectsp_1(D)
        & v8_vectsp_1(D)
        & v9_vectsp_1(D)
        & ~ v10_vectsp_1(D)
        & l3_vectsp_1(D)
        & m1_matrix_1(E,u1_struct_0(D),C,C) )
     => ( v1_matrix_1(k9_matrix_2(A,B,C,D,E))
        & m2_finseq_1(k9_matrix_2(A,B,C,D,E),k3_finseq_2(u1_struct_0(D))) ) ) )).

fof(dt_k10_matrix_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_2(A) )
     => m2_subset_1(k10_matrix_2(A),k1_numbers,k5_numbers) ) )).

fof(dt_k11_matrix_2,axiom,(
    $true )).

fof(dt_k12_matrix_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k11_matrix_2(A)) )
     => m2_subset_1(k12_matrix_2(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k13_matrix_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_group_1(k13_matrix_2(A))
        & l1_group_1(k13_matrix_2(A)) ) ) )).

fof(dt_k14_matrix_2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,u1_struct_0(A))
        & m1_subset_1(D,k11_matrix_2(B)) )
     => m1_subset_1(k14_matrix_2(A,B,C,D),u1_struct_0(A)) ) )).

fof(dt_k15_matrix_2,axiom,(
    ! [A] : m1_subset_1(k15_matrix_2(A),k5_finsub_1(A)) )).
%------------------------------------------------------------------------------