SET007 Axioms: SET007+300.ax


%------------------------------------------------------------------------------
% File     : SET007+300 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Matrices. Abelian Group of Matrices
% 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   77 (   6 unit)
%            Number of atoms       :  506 (  71 equality)
%            Maximal formula depth :   17 (   9 average)
%            Number of connectives :  500 (  71 ~  ;   2  |; 180  &)
%                                         (  17 <=>; 230 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-4 arity)
%            Number of functors    :   49 (  12 constant; 0-4 arity)
%            Number of variables   :  251 (   0 singleton; 240 !;  11 ?)
%            Maximal term depth    :    4 (   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_1,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v1_matrix_1(A) ) )).

fof(rc2_matrix_1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_finseq_1(B,k3_finseq_2(A))
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_finset_1(B)
      & v1_finseq_1(B)
      & v1_matrix_1(B) ) )).

fof(rc3_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_finseq_1(B,k3_finseq_2(A))
          & v1_relat_1(B)
          & ~ v3_relat_1(B)
          & v1_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v1_matrix_1(B) ) ) )).

fof(fc1_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ~ v1_xboole_0(k9_matrix_1(A,B)) ) )).

fof(d1_matrix_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_matrix_1(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ~ ( r2_hidden(C,k2_relat_1(A))
                  & ! [D] :
                      ( ( v1_relat_1(D)
                        & v1_funct_1(D)
                        & v1_finseq_1(D) )
                     => ~ ( D = C
                          & k3_finseq_1(D) = B ) ) ) ) ) ) )).

fof(t1_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => v1_matrix_1(k9_finseq_1(k12_finseq_1(A,B))) ) ) )).

fof(t2_matrix_1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => v1_matrix_1(k2_finseq_2(B,k2_finseq_2(C,A))) ) ) )).

fof(t3_matrix_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => v1_matrix_1(k9_finseq_1(A)) ) )).

fof(t4_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( k3_finseq_1(B) = A
                  & k3_finseq_1(C) = A )
               => v1_matrix_1(k10_finseq_1(B,C)) ) ) ) ) )).

fof(t5_matrix_1,axiom,(
    v1_matrix_1(k1_xboole_0) )).

fof(t6_matrix_1,axiom,(
    v1_matrix_1(k10_finseq_1(k1_xboole_0,k1_xboole_0)) )).

fof(t7_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => v1_matrix_1(k10_finseq_1(k12_finseq_1(A,B),k12_finseq_1(A,C))) ) ) ) )).

fof(t8_matrix_1,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)
                     => v1_matrix_1(k10_finseq_1(k10_finseq_1(B,C),k10_finseq_1(D,E))) ) ) ) ) ) )).

fof(t9_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( v1_matrix_1(B)
              & m2_finseq_1(B,k3_finseq_2(A)) )
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ! [D] :
                    ~ ( r2_hidden(D,k2_relat_1(B))
                      & ! [E] :
                          ( m2_finseq_1(E,A)
                         => ~ ( D = E
                              & k3_finseq_1(E) = C ) ) ) ) ) ) ) )).

fof(d2_matrix_1,axiom,(
    $true )).

fof(d3_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(A)) )
                 => ( m1_matrix_1(D,A,B,C)
                  <=> ( k3_finseq_1(D) = B
                      & ! [E] :
                          ( m2_finseq_1(E,A)
                         => ( r2_hidden(E,k2_relat_1(D))
                           => k3_finseq_1(E) = C ) ) ) ) ) ) ) ) )).

fof(t10_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => m1_matrix_1(k4_finseqop(k4_finseq_2(B,C),A,k4_finseqop(C,B,D)),C,A,B) ) ) ) ) )).

fof(t11_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => m1_matrix_1(k9_finseq_1(B),A,np__1,k3_finseq_1(B)) ) ) )).

fof(t12_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( ( k3_finseq_1(C) = A
                      & k3_finseq_1(D) = A )
                   => m1_matrix_1(k10_finseq_1(C,D),B,np__2,A) ) ) ) ) ) )).

fof(t13_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => m1_matrix_1(k1_xboole_0,B,np__0,A) ) ) )).

fof(t14_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_matrix_1(k9_finseq_1(k1_xboole_0),A,np__1,np__0) ) )).

fof(t15_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => m1_matrix_1(k9_finseq_1(k12_finseq_1(A,B)),A,np__1,np__1) ) ) )).

fof(t16_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_matrix_1(k10_finseq_1(k1_xboole_0,k1_xboole_0),A,np__2,np__0) ) )).

fof(t17_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => m1_matrix_1(k9_finseq_1(k10_finseq_1(B,C)),A,np__1,np__2) ) ) ) )).

fof(t18_matrix_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => m1_matrix_1(k10_finseq_1(k12_finseq_1(A,B),k12_finseq_1(A,C)),A,np__2,np__1) ) ) ) )).

fof(t19_matrix_1,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)
                     => m1_matrix_1(k10_finseq_1(k10_finseq_1(B,C),k10_finseq_1(D,E)),A,np__2,np__2) ) ) ) ) ) )).

fof(d4_matrix_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_matrix_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
             => ( B = k1_matrix_1(A)
              <=> ? [C] :
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v1_finseq_1(C)
                    & r2_hidden(C,k2_relat_1(A))
                    & k3_finseq_1(C) = B ) ) )
            & ( r1_xreal_0(k3_finseq_1(A),np__0)
             => ( B = k1_matrix_1(A)
              <=> B = np__0 ) ) ) ) ) )).

fof(t20_matrix_1,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)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( m1_matrix_1(B,A,k3_finseq_1(B),C)
                <=> C = k1_matrix_1(B) ) ) ) ) ) )).

fof(d5_matrix_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_matrix_1(A) )
     => k2_matrix_1(A) = k2_zfmisc_1(k4_finseq_1(A),k2_finseq_1(k1_matrix_1(A))) ) )).

fof(d6_matrix_1,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(B)
        & m2_finseq_1(B,k3_finseq_2(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => ( r2_hidden(k4_tarski(C,D),k2_matrix_1(B))
               => ! [E] :
                    ( m1_subset_1(E,A)
                   => ( E = k3_matrix_1(A,B,C,D)
                    <=> ? [F] :
                          ( m2_finseq_1(F,A)
                          & F = k1_funct_1(B,C)
                          & E = k1_funct_1(F,D) ) ) ) ) ) ) ) )).

fof(t21_matrix_1,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)) )
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & ! [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(B))
                           => k3_matrix_1(A,B,D,E) = k3_matrix_1(A,C,D,E) ) ) ) )
               => B = C ) ) ) ) )).

fof(t22_matrix_1,axiom,(
    $true )).

fof(t23_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_matrix_1(C,B,np__0,A)
             => ( k3_finseq_1(C) = np__0
                & k1_matrix_1(C) = np__0
                & k2_matrix_1(C) = k1_xboole_0 ) ) ) ) )).

fof(t24_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => ! [D] :
                    ( m1_matrix_1(D,C,A,B)
                   => ( k3_finseq_1(D) = A
                      & k1_matrix_1(D) = B
                      & k2_matrix_1(D) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)) ) ) ) ) ) ) )).

fof(t25_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_matrix_1(C,B,A,A)
             => ( k3_finseq_1(C) = A
                & k1_matrix_1(C) = A
                & k2_matrix_1(C) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(A)) ) ) ) ) )).

fof(t26_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_matrix_1(D,C,A,B)
                 => ( k3_finseq_1(D) = A
                    & k2_matrix_1(D) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(k1_matrix_1(D))) ) ) ) ) ) )).

fof(t27_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_matrix_1(D,C,A,B)
                 => ! [E] :
                      ( m1_matrix_1(E,C,A,B)
                     => k2_matrix_1(D) = k2_matrix_1(E) ) ) ) ) ) )).

fof(t28_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_matrix_1(D,C,A,B)
                 => ! [E] :
                      ( m1_matrix_1(E,C,A,B)
                     => ( ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ! [G] :
                                ( m2_subset_1(G,k1_numbers,k5_numbers)
                               => ( r2_hidden(k4_tarski(F,G),k2_matrix_1(D))
                                 => k3_matrix_1(C,D,F,G) = k3_matrix_1(C,E,F,G) ) ) )
                       => D = E ) ) ) ) ) ) )).

fof(t29_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_matrix_1(C,B,A,A)
             => ! [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))
                       => r2_hidden(k4_tarski(E,D),k2_matrix_1(C)) ) ) ) ) ) ) )).

fof(d7_matrix_1,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)) )
             => ( C = k4_matrix_1(A,B)
              <=> ( k3_finseq_1(C) = k1_matrix_1(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))
                          <=> r2_hidden(k4_tarski(E,D),k2_matrix_1(B)) ) ) )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(k4_tarski(E,D),k2_matrix_1(B))
                           => k3_matrix_1(A,C,D,E) = k3_matrix_1(A,B,E,D) ) ) ) ) ) ) ) ) )).

fof(d8_matrix_1,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)
             => ! [D] :
                  ( m2_finseq_1(D,A)
                 => ( D = k5_matrix_1(A,B,C)
                  <=> ( k3_finseq_1(D) = k1_matrix_1(B)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k2_finseq_1(k1_matrix_1(B)))
                           => k1_funct_1(D,E) = k3_matrix_1(A,B,C,E) ) ) ) ) ) ) ) ) )).

fof(d9_matrix_1,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)
             => ! [D] :
                  ( m2_finseq_1(D,A)
                 => ( D = k6_matrix_1(A,B,C)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(B)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k4_finseq_1(B))
                           => k1_funct_1(D,E) = k3_matrix_1(A,B,E,C) ) ) ) ) ) ) ) ) )).

fof(d10_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k9_matrix_1(A,B) = k4_finseq_2(B,k4_finseq_2(B,u1_struct_0(A))) ) ) )).

fof(d11_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k10_matrix_1(A,B) = k4_finseqop(k4_finseq_2(B,u1_struct_0(A)),B,k4_finseqop(u1_struct_0(A),B,k1_rlvect_1(A))) ) ) )).

fof(d12_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(A),B,B)
             => ( C = k11_matrix_1(A,B)
              <=> ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(k4_tarski(D,D),k2_matrix_1(C))
                       => k3_matrix_1(u1_struct_0(A),C,D,D) = k2_group_1(A) ) )
                  & ! [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))
                           => ( D = E
                              | k3_matrix_1(u1_struct_0(A),C,D,E) = k1_rlvect_1(A) ) ) ) ) ) ) ) ) ) )).

fof(d13_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(A),B,B)
             => ! [D] :
                  ( m1_matrix_1(D,u1_struct_0(A),B,B)
                 => ( D = k12_matrix_1(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( r2_hidden(k4_tarski(E,F),k2_matrix_1(C))
                             => k3_matrix_1(u1_struct_0(A),D,E,F) = k5_rlvect_1(A,k3_matrix_1(u1_struct_0(A),C,E,F)) ) ) ) ) ) ) ) ) )).

fof(d14_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(A),B,B)
             => ! [D] :
                  ( m1_matrix_1(D,u1_struct_0(A),B,B)
                 => ! [E] :
                      ( m1_matrix_1(E,u1_struct_0(A),B,B)
                     => ( E = k13_matrix_1(A,B,C,D)
                      <=> ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ! [G] :
                                ( m2_subset_1(G,k1_numbers,k5_numbers)
                               => ( r2_hidden(k4_tarski(F,G),k2_matrix_1(C))
                                 => k3_matrix_1(u1_struct_0(A),E,F,G) = k2_rlvect_1(A,k3_matrix_1(u1_struct_0(A),C,F,G),k3_matrix_1(u1_struct_0(A),D,F,G)) ) ) ) ) ) ) ) ) ) )).

fof(t30_matrix_1,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)
                    & l3_vectsp_1(D) )
                 => ( r2_hidden(k4_tarski(A,B),k2_matrix_1(k10_matrix_1(D,C)))
                   => k3_matrix_1(u1_struct_0(D),k10_matrix_1(D,C),A,B) = k1_rlvect_1(D) ) ) ) ) ) )).

fof(t31_matrix_1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] :
          ( ( ~ v3_struct_0(C)
            & l3_vectsp_1(C) )
         => ( m1_subset_1(A,k9_matrix_1(C,B))
          <=> m1_matrix_1(A,u1_struct_0(C),B,B) ) ) ) )).

fof(d15_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(A),B,B)
             => ( m2_matrix_1(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))
                         => ( k3_matrix_1(u1_struct_0(A),C,D,E) = k1_rlvect_1(A)
                            | D = E ) ) ) ) ) ) ) ) )).

fof(t32_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => ! [D] :
                  ( m1_matrix_1(D,u1_struct_0(B),A,A)
                 => k13_matrix_1(B,A,C,D) = k13_matrix_1(B,A,D,C) ) ) ) ) )).

fof(t33_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => ! [D] :
                  ( m1_matrix_1(D,u1_struct_0(B),A,A)
                 => ! [E] :
                      ( m1_matrix_1(E,u1_struct_0(B),A,A)
                     => k13_matrix_1(B,A,k13_matrix_1(B,A,C,D),E) = k13_matrix_1(B,A,C,k13_matrix_1(B,A,D,E)) ) ) ) ) ) )).

fof(t34_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => k13_matrix_1(B,A,C,k10_matrix_1(B,A)) = C ) ) ) )).

fof(t35_matrix_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_matrix_1(C,u1_struct_0(B),A,A)
             => k13_matrix_1(B,A,C,k12_matrix_1(B,A,C)) = k10_matrix_1(B,A) ) ) ) )).

fof(d16_matrix_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v1_rlvect_1(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & l1_rlvect_1(C) )
             => ( C = k14_matrix_1(A,B)
              <=> ( u1_struct_0(C) = k9_matrix_1(A,B)
                  & ! [D] :
                      ( m1_matrix_1(D,u1_struct_0(A),B,B)
                     => ! [E] :
                          ( m1_matrix_1(E,u1_struct_0(A),B,B)
                         => k1_binop_1(u1_rlvect_1(C),D,E) = k13_matrix_1(A,B,D,E) ) )
                  & u2_struct_0(C) = k10_matrix_1(A,B) ) ) ) ) ) )).

fof(s1_matrix_1,axiom,(
    ? [A] :
      ( m1_matrix_1(A,f1_s1_matrix_1,f2_s1_matrix_1,f3_s1_matrix_1)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(k4_tarski(B,C),k2_matrix_1(A))
               => k3_matrix_1(f1_s1_matrix_1,A,B,C) = f4_s1_matrix_1(B,C) ) ) ) ) )).

fof(s2_matrix_1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k2_finseq_1(f2_s2_matrix_1),k2_finseq_1(f3_s2_matrix_1)))
               => ! [C] :
                    ( m1_subset_1(C,f1_s2_matrix_1)
                   => ! [D] :
                        ( m1_subset_1(D,f1_s2_matrix_1)
                       => ( ( p1_s2_matrix_1(A,B,C)
                            & p1_s2_matrix_1(A,B,D) )
                         => C = D ) ) ) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(k2_finseq_1(f2_s2_matrix_1),k2_finseq_1(f3_s2_matrix_1)))
                  & ! [C] :
                      ( m1_subset_1(C,f1_s2_matrix_1)
                     => ~ p1_s2_matrix_1(A,B,C) ) ) ) ) )
   => ? [A] :
        ( m1_matrix_1(A,f1_s2_matrix_1,f2_s2_matrix_1,f3_s2_matrix_1)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(k4_tarski(B,C),k2_matrix_1(A))
                 => p1_s2_matrix_1(B,C,k3_matrix_1(f1_s2_matrix_1,A,B,C)) ) ) ) ) )).

fof(dt_m1_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => ! [D] :
          ( m1_matrix_1(D,A,B,C)
         => ( v1_matrix_1(D)
            & m2_finseq_1(D,k3_finseq_2(A)) ) ) ) )).

fof(existence_m1_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => ? [D] : m1_matrix_1(D,A,B,C) ) )).

fof(dt_m2_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ! [C] :
          ( m2_matrix_1(C,A,B)
         => m1_matrix_1(C,u1_struct_0(A),B,B) ) ) )).

fof(existence_m2_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ? [C] : m2_matrix_1(C,A,B) ) )).

fof(dt_k1_matrix_1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_matrix_1(A) )
     => m2_subset_1(k1_matrix_1(A),k1_numbers,k5_numbers) ) )).

fof(dt_k2_matrix_1,axiom,(
    $true )).

fof(dt_k3_matrix_1,axiom,(
    ! [A,B,C,D] :
      ( ( v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers) )
     => m1_subset_1(k3_matrix_1(A,B,C,D),A) ) )).

fof(dt_k4_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A)) )
     => ( v1_matrix_1(k4_matrix_1(A,B))
        & m2_finseq_1(k4_matrix_1(A,B),k3_finseq_2(A)) ) ) )).

fof(dt_k5_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k5_matrix_1(A,B,C),A) ) )).

fof(dt_k6_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k6_matrix_1(A,B,C),A) ) )).

fof(dt_k7_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_2(k7_matrix_1(A,B,C),A,k4_finseq_2(k1_matrix_1(B),A)) ) )).

fof(redefinition_k7_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => k7_matrix_1(A,B,C) = k5_matrix_1(A,B,C) ) )).

fof(dt_k8_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_2(k8_matrix_1(A,B,C),A,k4_finseq_2(k3_finseq_1(B),A)) ) )).

fof(redefinition_k8_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(A))
        & m1_subset_1(C,k5_numbers) )
     => k8_matrix_1(A,B,C) = k6_matrix_1(A,B,C) ) )).

fof(dt_k9_matrix_1,axiom,(
    $true )).

fof(dt_k10_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => m1_matrix_1(k10_matrix_1(A,B),u1_struct_0(A),B,B) ) )).

fof(dt_k11_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => m1_matrix_1(k11_matrix_1(A,B),u1_struct_0(A),B,B) ) )).

fof(dt_k12_matrix_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_matrix_1(C,u1_struct_0(A),B,B) )
     => m1_matrix_1(k12_matrix_1(A,B,C),u1_struct_0(A),B,B) ) )).

fof(dt_k13_matrix_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_matrix_1(C,u1_struct_0(A),B,B)
        & m1_matrix_1(D,u1_struct_0(A),B,B) )
     => m1_matrix_1(k13_matrix_1(A,B,C,D),u1_struct_0(A),B,B) ) )).

fof(dt_k14_matrix_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v3_struct_0(k14_matrix_1(A,B))
        & v1_rlvect_1(k14_matrix_1(A,B))
        & v3_rlvect_1(k14_matrix_1(A,B))
        & v4_rlvect_1(k14_matrix_1(A,B))
        & v5_rlvect_1(k14_matrix_1(A,B))
        & v6_rlvect_1(k14_matrix_1(A,B))
        & l1_rlvect_1(k14_matrix_1(A,B)) ) ) )).
%------------------------------------------------------------------------------