SET007 Axioms: SET007+881.ax


%------------------------------------------------------------------------------
% File     : SET007+881 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : A Theory of Matrices of Complex Elements
% 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_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   2 unit)
%            Number of atoms       :  433 ( 111 equality)
%            Maximal formula depth :   26 (   7 average)
%            Number of connectives :  397 (  24 ~  ;  21  |; 210  &)
%                                         (   1 <=>; 141 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   0 propositional; 1-3 arity)
%            Number of functors    :   36 (   8 constant; 0-4 arity)
%            Number of variables   :  118 (   0 singleton; 118 !;   0 ?)
%            Maximal term depth    :    5 (   2 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(t1_matrix_5,axiom,(
    np__1 = k2_group_1(k1_complfld) )).

fof(t2_matrix_5,axiom,(
    k1_rlvect_1(k1_complfld) = np__0 )).

fof(d1_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k1_matrix_5(A) = A ) )).

fof(d2_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
     => k2_matrix_5(A) = A ) )).

fof(t3_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( k1_matrix_5(A) = k1_matrix_5(B)
           => A = B ) ) ) )).

fof(t4_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(u1_struct_0(k1_complfld))) )
         => ( k2_matrix_5(A) = k2_matrix_5(B)
           => A = B ) ) ) )).

fof(t5_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => A = k2_matrix_5(k1_matrix_5(A)) ) )).

fof(t6_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
     => A = k1_matrix_5(k2_matrix_5(A)) ) )).

fof(d3_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => k3_matrix_5(A,B) = k2_matrix_5(k3_matrix_3(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) )).

fof(d4_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k4_matrix_5(A) = k2_matrix_5(k2_matrix_3(k1_complfld,k1_matrix_5(A))) ) )).

fof(d5_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => k5_matrix_5(A,B) = k2_matrix_5(k1_matrix_4(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) )).

fof(d6_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => k6_matrix_5(A,B) = k2_matrix_5(k4_matrix_3(k1_complfld,k1_matrix_5(A),k1_matrix_5(B))) ) ) )).

fof(d7_matrix_5,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( C = k7_matrix_5(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k1_complfld))
                   => ( D = A
                     => C = k2_matrix_5(k6_matrix_3(k1_complfld,k1_matrix_5(B),D)) ) ) ) ) ) ) )).

fof(t7_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( k3_finseq_1(A) = k3_finseq_1(k1_matrix_5(A))
        & k1_matrix_1(A) = k1_matrix_1(k1_matrix_5(A)) ) ) )).

fof(t8_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
     => ( k3_finseq_1(A) = k3_finseq_1(k2_matrix_5(A))
        & k1_matrix_1(A) = k1_matrix_1(k2_matrix_5(A)) ) ) )).

fof(t9_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k4_matrix_5(k4_matrix_5(A)) = A ) ) )).

fof(t10_matrix_5,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))) )
         => k6_matrix_3(A,B,k2_group_1(A)) = B ) ) )).

fof(t11_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k7_matrix_5(np__1,A) = A ) )).

fof(t12_matrix_5,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] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => k6_matrix_3(A,k6_matrix_3(A,D,C),B) = k6_matrix_3(A,D,k10_group_1(A,B,C)) ) ) ) ) )).

fof(t13_matrix_5,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] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(A))) )
                 => k6_matrix_3(A,D,k4_rlvect_1(A,B,C)) = k3_matrix_3(A,k6_matrix_3(A,D,B),k6_matrix_3(A,D,C)) ) ) ) ) )).

fof(t14_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k3_matrix_5(A,A) = k7_matrix_5(np__2,A) ) )).

fof(t15_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k3_matrix_5(k3_matrix_5(A,A),A) = k7_matrix_5(np__3,A) ) )).

fof(d8_matrix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k8_matrix_5(A,B) = k2_matrix_5(k1_matrix_3(k1_complfld,A,B)) ) ) )).

fof(t16_matrix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_matrix_5(k8_matrix_5(A,B)) = k1_matrix_3(k1_complfld,A,B) ) ) )).

fof(t17_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k3_matrix_5(A,k4_matrix_5(A)) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) )).

fof(t18_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k5_matrix_5(A,A) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) )).

fof(t19_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(A) = k1_matrix_1(B)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & k3_matrix_5(A,C) = k3_matrix_5(B,C) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | A = B ) ) ) ) ) )).

fof(t20_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
           => k5_matrix_5(A,k4_matrix_5(B)) = k3_matrix_5(A,B) ) ) ) )).

fof(t21_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B)
              & A = k3_matrix_5(A,B) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | B = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ) ) )).

fof(t22_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B)
              & k5_matrix_5(A,B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | A = B ) ) ) ) )).

fof(t23_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B)
              & k3_matrix_5(A,B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | B = k4_matrix_5(A) ) ) ) ) )).

fof(t24_matrix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(A,np__0)
           => k4_matrix_5(k8_matrix_5(A,B)) = k8_matrix_5(A,B) ) ) ) )).

fof(t25_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B)
              & k5_matrix_5(B,A) = B )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | A = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) ) ) )).

fof(t26_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | A = k5_matrix_5(A,k5_matrix_5(B,B)) ) ) ) ) )).

fof(t27_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | k4_matrix_5(k3_matrix_5(A,B)) = k3_matrix_5(k4_matrix_5(A),k4_matrix_5(B)) ) ) ) ) )).

fof(t28_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B) )
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | k5_matrix_5(A,k5_matrix_5(A,B)) = B ) ) ) ) )).

fof(t29_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(A) = k1_matrix_1(B)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & k5_matrix_5(A,C) = k5_matrix_5(B,C) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | A = B ) ) ) ) ) )).

fof(t30_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(A) = k1_matrix_1(B)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & k5_matrix_5(C,A) = k5_matrix_5(C,B) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | A = B ) ) ) ) ) )).

fof(t31_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & k1_matrix_1(A) = k3_finseq_1(B) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | r1_xreal_0(k3_finseq_1(B),np__0)
                  | k6_matrix_5(A,k3_matrix_5(B,C)) = k3_matrix_5(k6_matrix_5(A,B),k6_matrix_5(A,C)) ) ) ) ) ) )).

fof(t32_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & k1_matrix_1(B) = k1_matrix_1(C)
                  & k3_finseq_1(A) = k1_matrix_1(B) )
               => ( r1_xreal_0(k3_finseq_1(B),np__0)
                  | r1_xreal_0(k3_finseq_1(A),np__0)
                  | k6_matrix_5(k3_matrix_5(B,C),A) = k3_matrix_5(k6_matrix_5(B,A),k6_matrix_5(C,A)) ) ) ) ) ) )).

fof(t33_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k1_matrix_1(A) = k1_matrix_1(B) )
           => k3_matrix_5(A,B) = k3_matrix_5(B,A) ) ) ) )).

fof(t34_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(A) = k3_finseq_1(C)
                  & k1_matrix_1(A) = k1_matrix_1(B)
                  & k1_matrix_1(A) = k1_matrix_1(C) )
               => k3_matrix_5(k3_matrix_5(A,B),C) = k3_matrix_5(A,k3_matrix_5(B,C)) ) ) ) ) )).

fof(t35_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k3_matrix_5(A,k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A))) = A ) ) )).

fof(t36_matrix_5,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] :
              ( ( 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(C) = k3_finseq_1(D)
                      & k1_matrix_1(C) = k1_matrix_1(D) )
                   => ( r1_xreal_0(k3_finseq_1(C),np__0)
                      | k6_matrix_3(A,k3_matrix_3(A,C,D),B) = k3_matrix_3(A,k6_matrix_3(A,C,B),k6_matrix_3(A,D,B)) ) ) ) ) ) ) )).

fof(t37_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ! [C] :
              ( v1_xcmplx_0(C)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k1_matrix_1(A) = k1_matrix_1(B) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | k7_matrix_5(C,k3_matrix_5(A,B)) = k3_matrix_5(k7_matrix_5(C,A),k7_matrix_5(C,B)) ) ) ) ) ) )).

fof(t38_matrix_5,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)
               => ( r1_xreal_0(k3_finseq_1(B),np__0)
                  | r1_xreal_0(k3_finseq_1(C),np__0)
                  | k4_matrix_3(A,k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(B)),C) = k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(C)) ) ) ) ) ) )).

fof(t39_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( k1_matrix_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | r1_xreal_0(k3_finseq_1(B),np__0)
              | k6_matrix_5(k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)),B) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(B)) ) ) ) ) )).

fof(t40_matrix_5,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))) )
         => ( ~ r1_xreal_0(k3_finseq_1(B),np__0)
           => k6_matrix_3(A,B,k1_rlvect_1(A)) = k1_matrix_3(A,k3_finseq_1(B),k1_matrix_1(B)) ) ) ) )).

fof(t41_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k7_matrix_5(np__0,A) = k8_matrix_5(k3_finseq_1(A),k1_matrix_1(A)) ) ) )).

fof(t42_matrix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
                 => ~ ( ~ r1_xreal_0(k3_finseq_1(C),np__0)
                      & ~ r1_xreal_0(k3_finseq_1(D),np__0)
                      & k1_matrix_1(C) = k3_finseq_1(D)
                      & r1_xreal_0(np__1,A)
                      & r1_xreal_0(A,k3_finseq_1(C))
                      & r1_xreal_0(np__1,B)
                      & r1_xreal_0(B,k1_matrix_1(D))
                      & ! [E] :
                          ( m2_finseq_1(E,k2_numbers)
                         => ~ ( k3_finseq_1(E) = k3_finseq_1(D)
                              & k9_matrix_5(E,np__1) = k5_binop_2(k3_matrix_1(k2_numbers,C,A,np__1),k3_matrix_1(k2_numbers,D,np__1,B))
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( r1_xreal_0(np__1,F)
                                   => ( r1_xreal_0(k3_finseq_1(D),F)
                                      | k9_matrix_5(E,k23_binop_2(F,np__1)) = k3_binop_2(k9_matrix_5(E,F),k5_binop_2(k3_matrix_1(k2_numbers,C,A,k23_binop_2(F,np__1)),k3_matrix_1(k2_numbers,D,k23_binop_2(F,np__1),B))) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k1_matrix_5(A))
        & m2_finseq_1(k1_matrix_5(A),k3_finseq_2(u1_struct_0(k1_complfld))) ) ) )).

fof(dt_k2_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(u1_struct_0(k1_complfld))) )
     => ( v1_matrix_1(k2_matrix_5(A))
        & m2_finseq_1(k2_matrix_5(A),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k3_matrix_5,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers))
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k3_matrix_5(A,B))
        & m2_finseq_1(k3_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k4_matrix_5,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k4_matrix_5(A))
        & m2_finseq_1(k4_matrix_5(A),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k5_matrix_5,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers))
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k5_matrix_5(A,B))
        & m2_finseq_1(k5_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k6_matrix_5,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers))
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k6_matrix_5(A,B))
        & m2_finseq_1(k6_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k7_matrix_5,axiom,(
    ! [A,B] :
      ( ( v1_xcmplx_0(A)
        & v1_matrix_1(B)
        & m1_finseq_1(B,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k7_matrix_5(A,B))
        & m2_finseq_1(k7_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k8_matrix_5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_matrix_1(k8_matrix_5(A,B))
        & m2_finseq_1(k8_matrix_5(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k9_matrix_5,axiom,(
    ! [A,B] :
      ( m1_finseq_1(A,k2_numbers)
     => m1_subset_1(k9_matrix_5(A,B),k2_numbers) ) )).

fof(redefinition_k9_matrix_5,axiom,(
    ! [A,B] :
      ( m1_finseq_1(A,k2_numbers)
     => k9_matrix_5(A,B) = k1_funct_1(A,B) ) )).
%------------------------------------------------------------------------------