SET007 Axioms: SET007+372.ax


%------------------------------------------------------------------------------
% File     : SET007+372 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Product and the Determinant of Matrices with Entries in a Field
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   62 (   2 unit)
%            Number of atoms       : 1094 ( 104 equality)
%            Maximal formula depth :   25 (  17 average)
%            Number of connectives : 1166 ( 134 ~  ;   7  |; 724  &)
%                                         (   7 <=>; 294 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-4 arity)
%            Number of functors    :   64 (   5 constant; 0-6 arity)
%            Number of variables   :  296 (   0 singleton; 296 !;   0 ?)
%            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(d1_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_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] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_matrix_3(A,B,C) = k4_finseqop(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,u1_struct_0(A))),B,k4_finseqop(u1_struct_0(A),C,k1_rlvect_1(A))) ) ) ) )).

fof(d2_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ( C = k2_matrix_3(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(B)
                  & k1_matrix_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(B))
                           => k3_matrix_1(u1_struct_0(A),C,D,E) = k5_rlvect_1(A,k3_matrix_1(u1_struct_0(A),B,D,E)) ) ) ) ) ) ) ) ) )).

fof(d3_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => ( D = k3_matrix_3(A,B,C)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(B)
                      & k1_matrix_1(D) = k1_matrix_1(B)
                      & ! [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(B))
                               => k3_matrix_1(u1_struct_0(A),D,E,F) = k4_rlvect_1(A,k3_matrix_1(u1_struct_0(A),B,E,F),k3_matrix_1(u1_struct_0(A),C,E,F)) ) ) ) ) ) ) ) ) ) )).

fof(t1_matrix_3,axiom,(
    $true )).

fof(t2_matrix_3,axiom,(
    $true )).

fof(t3_matrix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v4_group_1(C)
                & v7_group_1(C)
                & v7_vectsp_1(C)
                & v8_vectsp_1(C)
                & v9_vectsp_1(C)
                & ~ v10_vectsp_1(C)
                & l3_vectsp_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(k1_matrix_3(C,A,B)))
                       => k3_matrix_1(u1_struct_0(C),k1_matrix_3(C,A,B),D,E) = k1_rlvect_1(C) ) ) ) ) ) ) )).

fof(t4_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(B) = k1_matrix_1(C) )
               => k3_matrix_3(A,B,C) = k3_matrix_3(A,C,B) ) ) ) ) )).

fof(t5_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                      & k3_finseq_1(B) = k3_finseq_1(D)
                      & k1_matrix_1(B) = k1_matrix_1(C)
                      & k1_matrix_1(B) = k1_matrix_1(D) )
                   => k3_matrix_3(A,k3_matrix_3(A,B,C),D) = k3_matrix_3(A,B,k3_matrix_3(A,C,D)) ) ) ) ) ) )).

fof(t6_matrix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v4_group_1(C)
                & v7_group_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)
                 => k3_matrix_3(C,D,k1_matrix_3(C,A,B)) = D ) ) ) ) )).

fof(t7_matrix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v4_group_1(C)
                & v7_group_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)
                 => k3_matrix_3(C,D,k2_matrix_3(C,D)) = k1_matrix_3(C,A,B) ) ) ) ) )).

fof(d4_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ( k1_matrix_1(B) = k3_finseq_1(C)
               => ! [D] :
                    ( ( v1_matrix_1(D)
                      & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                   => ( D = k4_matrix_3(A,B,C)
                    <=> ( k3_finseq_1(D) = k3_finseq_1(B)
                        & k1_matrix_1(D) = k1_matrix_1(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(D))
                                 => k3_matrix_1(u1_struct_0(A),D,E,F) = k14_fvsum_1(A,k7_matrix_1(u1_struct_0(A),B,E),k8_matrix_1(u1_struct_0(A),C,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => ( D = k6_matrix_3(A,B,C)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(B)
                      & k1_matrix_1(D) = k1_matrix_1(B)
                      & ! [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(B))
                               => k3_matrix_1(u1_struct_0(A),D,E,F) = k10_group_1(A,C,k3_matrix_1(u1_struct_0(A),B,E,F)) ) ) ) ) ) ) ) ) ) )).

fof(d6_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k7_matrix_3(A,B,C) = k6_matrix_3(A,B,C) ) ) ) )).

fof(t8_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( k3_finseq_1(B) = k3_finseq_1(C)
               => ( k3_finseq_1(k11_fvsum_1(A,B,C)) = k3_finseq_1(B)
                  & k3_finseq_1(k11_fvsum_1(A,B,C)) = k3_finseq_1(C) ) ) ) ) ) )).

fof(t9_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                      & D = C )
                   => k1_funct_1(k7_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),C),D) = k2_group_1(B) ) ) ) ) ) )).

fof(t10_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                   => ( D = C
                      | k1_funct_1(k7_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),C),D) = k1_rlvect_1(B) ) ) ) ) ) ) )).

fof(t11_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                      & C = D )
                   => k1_funct_1(k8_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),D),C) = k2_group_1(B) ) ) ) ) ) )).

fof(t12_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                   => ( C = D
                      | k1_funct_1(k8_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),D),C) = k1_rlvect_1(B) ) ) ) ) ) ) )).

fof(t13_matrix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & l1_rlvect_1(B) )
         => k9_rlvect_1(B,k4_finseqop(u1_struct_0(B),A,k1_rlvect_1(B))) = k1_rlvect_1(B) ) ) )).

fof(t14_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(C,k3_setwop_2(B))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k3_setwop_2(B))
                       => ( D = C
                          | k1_funct_1(B,D) = k1_rlvect_1(A) ) ) ) )
               => k9_rlvect_1(A,B) = k1_funct_1(B,C) ) ) ) ) )).

fof(t15_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => k3_finseq_1(k11_fvsum_1(A,B,C)) = k3_square_1(k3_finseq_1(B),k3_finseq_1(C)) ) ) ) )).

fof(t16_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(D,k3_setwop_2(B))
                      & k1_funct_1(B,D) = k2_group_1(A)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k3_setwop_2(B))
                           => ( E = D
                              | k1_funct_1(B,E) = k1_rlvect_1(A) ) ) ) )
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k3_setwop_2(k11_fvsum_1(A,B,C)))
                         => ( ( D = E
                             => k1_funct_1(k11_fvsum_1(A,B,C),E) = k1_funct_1(C,D) )
                            & ( D != E
                             => k1_funct_1(k11_fvsum_1(A,B,C),E) = k1_rlvect_1(A) ) ) ) ) ) ) ) ) ) )).

fof(t17_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                   => ( ( C = D
                       => k1_funct_1(k7_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),C),D) = k2_group_1(B) )
                      & ( C != D
                       => k1_funct_1(k7_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),C),D) = k1_rlvect_1(B) ) ) ) ) ) ) ) )).

fof(t18_matrix_3,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)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & v9_vectsp_1(B)
            & ~ v10_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [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(k11_matrix_1(B,A)))
                   => ( ( C = D
                       => k1_funct_1(k8_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),D),C) = k2_group_1(B) )
                      & ( C != D
                       => k1_funct_1(k8_matrix_1(u1_struct_0(B),k11_matrix_1(B,A),D),C) = k1_rlvect_1(B) ) ) ) ) ) ) ) )).

fof(t19_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r2_hidden(D,k3_setwop_2(B))
                      & r2_hidden(D,k3_setwop_2(C))
                      & k1_funct_1(B,D) = k2_group_1(A)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k3_setwop_2(B))
                           => ( E = D
                              | k1_funct_1(B,E) = k1_rlvect_1(A) ) ) ) )
                   => k9_rlvect_1(A,k11_fvsum_1(A,B,C)) = k1_funct_1(C,D) ) ) ) ) ) )).

fof(t20_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => k4_matrix_3(B,k11_matrix_1(B,A),C) = C ) ) ) )).

fof(t21_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => k4_matrix_3(B,C,k11_matrix_1(B,A)) = C ) ) ) )).

fof(t22_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k4_matrix_3(A,k5_matrix_2(u1_struct_0(A),np__1,k4_matrix_2(u1_struct_0(A),B)),k5_matrix_2(u1_struct_0(A),np__1,k4_matrix_2(u1_struct_0(A),C))) = k5_matrix_2(u1_struct_0(A),np__1,k4_matrix_2(u1_struct_0(A),k10_group_1(A,B,C))) ) ) ) )).

fof(t23_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(A))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(A))
                                 => ! [I] :
                                      ( m1_subset_1(I,u1_struct_0(A))
                                     => k4_matrix_3(A,k6_matrix_2(u1_struct_0(A),B,D,F,H),k6_matrix_2(u1_struct_0(A),C,E,G,I)) = k6_matrix_2(u1_struct_0(A),k4_rlvect_1(A,k10_group_1(A,B,C),k10_group_1(A,D,G)),k4_rlvect_1(A,k10_group_1(A,B,E),k10_group_1(A,D,I)),k4_rlvect_1(A,k10_group_1(A,F,C),k10_group_1(A,H,G)),k4_rlvect_1(A,k10_group_1(A,F,E),k10_group_1(A,H,I))) ) ) ) ) ) ) ) ) ) )).

fof(t24_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ( k1_matrix_1(B) = k3_finseq_1(C)
               => ( k1_matrix_1(C) = np__0
                  | k4_matrix_1(u1_struct_0(A),k4_matrix_3(A,B,C)) = k4_matrix_3(A,k4_matrix_1(u1_struct_0(A),C),k4_matrix_1(u1_struct_0(A),B)) ) ) ) ) ) )).

fof(t25_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(C,C),C)
                    & m2_relset_1(D,k2_zfmisc_1(C,C),C) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(C,C),C)
                        & m2_relset_1(E,k2_zfmisc_1(C,C),C) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,A,C)
                            & m2_relset_1(F,A,C) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,B,C)
                                & m2_relset_1(G,B,C) )
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(A))
                                 => ! [I] :
                                      ( m1_subset_1(I,k5_finsub_1(B))
                                     => ( ( v1_binop_1(D,C)
                                          & v2_binop_1(D,C)
                                          & v1_binop_1(E,C) )
                                       => ( ( k8_matrix_3(B,A,I,H) = k1_xboole_0
                                            & ~ v1_setwiseo(D,C) )
                                          | k7_setwiseo(k2_zfmisc_1(A,B),C,D,k8_matrix_3(A,B,H,I),k9_matrix_3(A,B,C,E,F,G)) = k7_setwiseo(k2_zfmisc_1(B,A),C,D,k8_matrix_3(B,A,I,H),k9_matrix_3(B,A,C,E,G,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t26_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,C,A)
                                & m2_relset_1(G,C,A) )
                             => ( ( v1_binop_1(D,A)
                                  & v2_binop_1(D,A)
                                  & v1_setwiseo(D,A) )
                               => ! [H] :
                                    ( m1_subset_1(H,B)
                                   => ! [I] :
                                        ( m1_subset_1(I,C)
                                       => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,k2_setwiseo(B,H),k2_setwiseo(C,I)),k9_matrix_3(B,C,A,E,F,G)) = k7_setwiseo(B,A,D,k2_setwiseo(B,H),k7_funcop_1(A,B,E,F,k7_setwiseo(C,A,D,k2_setwiseo(C,I),G))) ) ) ) ) ) ) ) ) ) ) )).

fof(t27_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,C,A)
                                & m2_relset_1(G,C,A) )
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(B))
                                 => ! [I] :
                                      ( m1_subset_1(I,k5_finsub_1(C))
                                     => ( ( v1_binop_1(D,A)
                                          & v2_binop_1(D,A)
                                          & v1_setwiseo(D,A)
                                          & v1_finseqop(D,A)
                                          & r6_binop_1(A,E,D) )
                                       => ! [J] :
                                            ( m1_subset_1(J,B)
                                           => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,k2_setwiseo(B,J),I),k9_matrix_3(B,C,A,E,F,G)) = k7_setwiseo(B,A,D,k2_setwiseo(B,J),k7_funcop_1(A,B,E,F,k7_setwiseo(C,A,D,I,G))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t28_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,C,A)
                                & m2_relset_1(G,C,A) )
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(B))
                                 => ! [I] :
                                      ( m1_subset_1(I,k5_finsub_1(C))
                                     => ( ( v1_setwiseo(D,A)
                                          & v1_binop_1(D,A)
                                          & v2_binop_1(D,A)
                                          & v1_finseqop(D,A)
                                          & r6_binop_1(A,E,D) )
                                       => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,H,I),k9_matrix_3(B,C,A,E,F,G)) = k7_setwiseo(B,A,D,H,k7_funcop_1(A,B,E,F,k7_setwiseo(C,A,D,I,G))) ) ) ) ) ) ) ) ) ) ) )).

fof(t29_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,C,A)
                                & m2_relset_1(G,C,A) )
                             => ( ( v1_binop_1(D,A)
                                  & v2_binop_1(D,A)
                                  & v1_setwiseo(D,A)
                                  & v1_binop_1(E,A) )
                               => ! [H] :
                                    ( m1_subset_1(H,B)
                                   => ! [I] :
                                        ( m1_subset_1(I,C)
                                       => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,k2_setwiseo(B,H),k2_setwiseo(C,I)),k9_matrix_3(B,C,A,E,F,G)) = k7_setwiseo(C,A,D,k2_setwiseo(C,I),k8_funcop_1(A,C,E,k7_setwiseo(B,A,D,k2_setwiseo(B,H),F),G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t30_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,A),A)
                        & m2_relset_1(E,k2_zfmisc_1(A,A),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,C,A)
                                & m2_relset_1(G,C,A) )
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(B))
                                 => ! [I] :
                                      ( m1_subset_1(I,k5_finsub_1(C))
                                     => ( ( v1_setwiseo(D,A)
                                          & v1_binop_1(D,A)
                                          & v2_binop_1(D,A)
                                          & v1_finseqop(D,A)
                                          & r6_binop_1(A,E,D)
                                          & v1_binop_1(E,A) )
                                       => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,H,I),k9_matrix_3(B,C,A,E,F,G)) = k7_setwiseo(C,A,D,I,k8_funcop_1(A,C,E,k7_setwiseo(B,A,D,H,F),G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t31_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(B,C),A)
                        & m2_relset_1(E,k2_zfmisc_1(B,C),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( m1_subset_1(G,k5_finsub_1(C))
                             => ( ( v1_setwiseo(D,A)
                                  & v1_binop_1(D,A)
                                  & v2_binop_1(D,A)
                                  & v1_finseqop(D,A) )
                               => ! [H] :
                                    ( m1_subset_1(H,B)
                                   => ( ! [I] :
                                          ( m1_subset_1(I,B)
                                         => k8_funct_2(B,A,F,I) = k7_setwiseo(C,A,D,G,k1_cat_2(B,C,A,k1_fraenkel(C,A),k2_cat_2(B,C,A,E),I)) )
                                     => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,k2_setwiseo(B,H),G),E) = k7_setwiseo(B,A,D,k2_setwiseo(B,H),F) ) ) ) ) ) ) ) ) ) ) )).

fof(t32_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(B,C),A)
                        & m2_relset_1(E,k2_zfmisc_1(B,C),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,B,A)
                            & m2_relset_1(F,B,A) )
                         => ! [G] :
                              ( m1_subset_1(G,k5_finsub_1(B))
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(C))
                                 => ( ( ! [I] :
                                          ( m1_subset_1(I,B)
                                         => k8_funct_2(B,A,F,I) = k7_setwiseo(C,A,D,H,k1_cat_2(B,C,A,k1_fraenkel(C,A),k2_cat_2(B,C,A,E),I)) )
                                      & v1_setwiseo(D,A)
                                      & v1_binop_1(D,A)
                                      & v2_binop_1(D,A)
                                      & v1_finseqop(D,A) )
                                   => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,G,H),E) = k7_setwiseo(B,A,D,G,F) ) ) ) ) ) ) ) ) ) )).

fof(t33_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(B,C),A)
                        & m2_relset_1(E,k2_zfmisc_1(B,C),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,C,A)
                            & m2_relset_1(F,C,A) )
                         => ! [G] :
                              ( m1_subset_1(G,k5_finsub_1(B))
                             => ( ( v1_setwiseo(D,A)
                                  & v1_binop_1(D,A)
                                  & v2_binop_1(D,A)
                                  & v1_finseqop(D,A) )
                               => ! [H] :
                                    ( m1_subset_1(H,C)
                                   => ( ! [I] :
                                          ( m1_subset_1(I,C)
                                         => k8_funct_2(C,A,F,I) = k7_setwiseo(B,A,D,G,k1_cat_2(C,B,A,k1_fraenkel(B,A),k3_cat_2(B,C,A,E),I)) )
                                     => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,G,k2_setwiseo(C,H)),E) = k7_setwiseo(C,A,D,k2_setwiseo(C,H),F) ) ) ) ) ) ) ) ) ) ) )).

fof(t34_matrix_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,A),A)
                    & m2_relset_1(D,k2_zfmisc_1(A,A),A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(B,C),A)
                        & m2_relset_1(E,k2_zfmisc_1(B,C),A) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,C,A)
                            & m2_relset_1(F,C,A) )
                         => ! [G] :
                              ( m1_subset_1(G,k5_finsub_1(B))
                             => ! [H] :
                                  ( m1_subset_1(H,k5_finsub_1(C))
                                 => ( ( ! [I] :
                                          ( m1_subset_1(I,C)
                                         => k8_funct_2(C,A,F,I) = k7_setwiseo(B,A,D,G,k1_cat_2(C,B,A,k1_fraenkel(B,A),k3_cat_2(B,C,A,E),I)) )
                                      & v1_setwiseo(D,A)
                                      & v1_binop_1(D,A)
                                      & v2_binop_1(D,A)
                                      & v1_finseqop(D,A) )
                                   => k7_setwiseo(k2_zfmisc_1(B,C),A,D,k8_matrix_3(B,C,G,H),E) = k7_setwiseo(C,A,D,H,F) ) ) ) ) ) ) ) ) ) )).

fof(t35_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => ( ( k1_matrix_1(B) = k3_finseq_1(C)
                      & k1_matrix_1(C) = k3_finseq_1(D) )
                   => k4_matrix_3(A,k4_matrix_3(A,B,C),D) = k4_matrix_3(A,B,k4_matrix_3(A,C,D)) ) ) ) ) ) )).

fof(d7_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => ! [D] :
                  ( m3_matrix_2(D,k11_matrix_2(A))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(B))
                     => ( E = k10_matrix_3(A,B,C,D)
                      <=> ( k3_finseq_1(E) = A
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( ( r2_hidden(F,k3_setwop_2(E))
                                      & G = k1_funct_1(D,F) )
                                   => k1_funct_1(E,F) = k3_matrix_1(u1_struct_0(B),C,F,G) ) ) ) ) ) ) ) ) ) ) )).

fof(d8_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k11_matrix_2(A),u1_struct_0(B))
                    & m2_relset_1(D,k11_matrix_2(A),u1_struct_0(B)) )
                 => ( D = k11_matrix_3(A,B,C)
                  <=> ! [E] :
                        ( m3_matrix_2(E,k11_matrix_2(A))
                       => k8_funct_2(k11_matrix_2(A),u1_struct_0(B),D,E) = k14_matrix_2(B,A,k2_finsop_1(u1_struct_0(B),k10_matrix_3(A,B,C,E),u1_group_1(B)),E) ) ) ) ) ) ) )).

fof(d9_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => k12_matrix_3(A,B,C) = k7_setwiseo(k11_matrix_2(A),u1_struct_0(B),u1_rlvect_1(B),k15_matrix_2(k11_matrix_2(A)),k11_matrix_3(A,B,C)) ) ) ) )).

fof(t36_matrix_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k12_matrix_3(np__1,A,k5_matrix_2(u1_struct_0(A),np__1,k4_matrix_2(u1_struct_0(A),B))) = B ) ) )).

fof(d10_matrix_3,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)
            & v4_group_1(B)
            & v7_group_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)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(B))
                 => ( D = k13_matrix_3(A,B,C)
                  <=> ( k3_finseq_1(D) = A
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k2_setwop_2(A))
                           => k1_funct_1(D,E) = k3_matrix_1(u1_struct_0(B),C,E,E) ) ) ) ) ) ) ) ) )).

fof(dt_k1_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_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,k5_numbers) )
     => m1_matrix_1(k1_matrix_3(A,B,C),u1_struct_0(A),B,C) ) )).

fof(dt_k2_matrix_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(u1_struct_0(A))) )
     => ( v1_matrix_1(k2_matrix_3(A,B))
        & m2_finseq_1(k2_matrix_3(A,B),k3_finseq_2(u1_struct_0(A))) ) ) )).

fof(dt_k3_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(u1_struct_0(A)))
        & v1_matrix_1(C)
        & m1_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
     => ( v1_matrix_1(k3_matrix_3(A,B,C))
        & m2_finseq_1(k3_matrix_3(A,B,C),k3_finseq_2(u1_struct_0(A))) ) ) )).

fof(dt_k4_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(u1_struct_0(A)))
        & v1_matrix_1(C)
        & m1_finseq_1(C,k3_finseq_2(u1_struct_0(A))) )
     => ( v1_matrix_1(k4_matrix_3(A,B,C))
        & m2_finseq_1(k4_matrix_3(A,B,C),k3_finseq_2(u1_struct_0(A))) ) ) )).

fof(dt_k5_matrix_3,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & ~ v3_struct_0(D)
        & v3_rlvect_1(D)
        & v4_rlvect_1(D)
        & v5_rlvect_1(D)
        & v6_rlvect_1(D)
        & v4_group_1(D)
        & v7_group_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),A,B)
        & m1_matrix_1(F,u1_struct_0(D),k1_matrix_1(E),C) )
     => m1_matrix_1(k5_matrix_3(A,B,C,D,E,F),u1_struct_0(D),k3_finseq_1(E),k1_matrix_1(F)) ) )).

fof(redefinition_k5_matrix_3,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & ~ v3_struct_0(D)
        & v3_rlvect_1(D)
        & v4_rlvect_1(D)
        & v5_rlvect_1(D)
        & v6_rlvect_1(D)
        & v4_group_1(D)
        & v7_group_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),A,B)
        & m1_matrix_1(F,u1_struct_0(D),k1_matrix_1(E),C) )
     => k5_matrix_3(A,B,C,D,E,F) = k4_matrix_3(D,E,F) ) )).

fof(dt_k6_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_matrix_1(k6_matrix_3(A,B,C))
        & m2_finseq_1(k6_matrix_3(A,B,C),k3_finseq_2(u1_struct_0(A))) ) ) )).

fof(dt_k7_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_matrix_1(k7_matrix_3(A,B,C))
        & m2_finseq_1(k7_matrix_3(A,B,C),k3_finseq_2(u1_struct_0(A))) ) ) )).

fof(dt_k8_matrix_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k5_finsub_1(A))
        & m1_subset_1(D,k5_finsub_1(B)) )
     => m1_subset_1(k8_matrix_3(A,B,C,D),k5_finsub_1(k2_zfmisc_1(A,B))) ) )).

fof(redefinition_k8_matrix_3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k5_finsub_1(A))
        & m1_subset_1(D,k5_finsub_1(B)) )
     => k8_matrix_3(A,B,C,D) = k2_zfmisc_1(C,D) ) )).

fof(dt_k9_matrix_3,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(C,C),C)
        & m1_relset_1(D,k2_zfmisc_1(C,C),C)
        & v1_funct_1(E)
        & v1_funct_2(E,A,C)
        & m1_relset_1(E,A,C)
        & v1_funct_1(F)
        & v1_funct_2(F,B,C)
        & m1_relset_1(F,B,C) )
     => ( v1_funct_1(k9_matrix_3(A,B,C,D,E,F))
        & v1_funct_2(k9_matrix_3(A,B,C,D,E,F),k2_zfmisc_1(A,B),C)
        & m2_relset_1(k9_matrix_3(A,B,C,D,E,F),k2_zfmisc_1(A,B),C) ) ) )).

fof(redefinition_k9_matrix_3,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(C,C),C)
        & m1_relset_1(D,k2_zfmisc_1(C,C),C)
        & v1_funct_1(E)
        & v1_funct_2(E,A,C)
        & m1_relset_1(E,A,C)
        & v1_funct_1(F)
        & v1_funct_2(F,B,C)
        & m1_relset_1(F,B,C) )
     => k9_matrix_3(A,B,C,D,E,F) = k7_finseqop(D,E,F) ) )).

fof(dt_k10_matrix_3,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & m1_matrix_1(C,u1_struct_0(B),A,A)
        & m1_subset_1(D,k11_matrix_2(A)) )
     => m2_finseq_1(k10_matrix_3(A,B,C,D),u1_struct_0(B)) ) )).

fof(dt_k11_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & m1_matrix_1(C,u1_struct_0(B),A,A) )
     => ( v1_funct_1(k11_matrix_3(A,B,C))
        & v1_funct_2(k11_matrix_3(A,B,C),k11_matrix_2(A),u1_struct_0(B))
        & m2_relset_1(k11_matrix_3(A,B,C),k11_matrix_2(A),u1_struct_0(B)) ) ) )).

fof(dt_k12_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & m1_matrix_1(C,u1_struct_0(B),A,A) )
     => m1_subset_1(k12_matrix_3(A,B,C),u1_struct_0(B)) ) )).

fof(dt_k13_matrix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & v9_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & m1_matrix_1(C,u1_struct_0(B),A,A) )
     => m2_finseq_1(k13_matrix_3(A,B,C),u1_struct_0(B)) ) )).
%------------------------------------------------------------------------------