SET007 Axioms: SET007+921.ax


%------------------------------------------------------------------------------
% File     : SET007+921 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Inner Product and Conjugate of Matrix of Complex Numbers
% 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    : matrixc1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   95 (   0 unit)
%            Number of atoms       :  501 ( 161 equality)
%            Maximal formula depth :   18 (   7 average)
%            Number of connectives :  410 (   4 ~  ;  15  |; 129  &)
%                                         (   7 <=>; 255 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   0 propositional; 1-3 arity)
%            Number of functors    :   66 (   9 constant; 0-6 arity)
%            Number of variables   :  213 (   0 singleton; 212 !;   1 ?)
%            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(d1_matrixc1,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)) )
         => ( B = k1_matrixc1(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & k1_matrix_1(B) = k1_matrix_1(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(A))
                       => k3_matrix_1(k2_numbers,B,C,D) = k15_complex1(k3_matrix_1(k2_numbers,A,C,D)) ) ) ) ) ) ) ) )).

fof(t1_matrixc1,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)) )
             => ( r2_hidden(k4_tarski(A,B),k2_matrix_1(C))
              <=> ( 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(C)) ) ) ) ) ) )).

fof(t2_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k1_matrixc1(k1_matrixc1(A)) = A ) )).

fof(t3_matrixc1,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( k3_finseq_1(k7_matrix_5(A,B)) = k3_finseq_1(B)
            & k1_matrix_1(k7_matrix_5(A,B)) = k1_matrix_1(B) ) ) ) )).

fof(t4_matrixc1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_xcmplx_0(C)
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
                 => ( ( k3_finseq_1(k7_matrix_5(C,D)) = k3_finseq_1(D)
                      & k1_matrix_1(k7_matrix_5(C,D)) = k1_matrix_1(D)
                      & r2_hidden(k4_tarski(A,B),k2_matrix_1(D)) )
                   => k3_matrix_1(k2_numbers,k7_matrix_5(C,D),A,B) = k3_xcmplx_0(C,k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) )).

fof(t5_matrixc1,axiom,(
    ! [A] :
      ( v1_xcmplx_0(A)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => k1_matrixc1(k7_matrix_5(A,B)) = k7_matrix_5(k15_complex1(A),k1_matrixc1(B)) ) ) )).

fof(t6_matrixc1,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(k3_matrix_5(A,B)) = k3_finseq_1(A)
            & k1_matrix_1(k3_matrix_5(A,B)) = k1_matrix_1(A) ) ) ) )).

fof(t7_matrixc1,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)) )
                 => ( ( k3_finseq_1(C) = k3_finseq_1(D)
                      & k1_matrix_1(C) = k1_matrix_1(D)
                      & r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
                   => k3_matrix_1(k2_numbers,k3_matrix_5(C,D),A,B) = k3_binop_2(k3_matrix_1(k2_numbers,C,A,B),k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) )).

fof(t8_matrixc1,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) )
           => k1_matrixc1(k3_matrix_5(A,B)) = k3_matrix_5(k1_matrixc1(A),k1_matrixc1(B)) ) ) ) )).

fof(t9_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( k3_finseq_1(k4_matrix_5(A)) = k3_finseq_1(A)
        & k1_matrix_1(k4_matrix_5(A)) = k1_matrix_1(A) ) ) )).

fof(t10_matrixc1,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)) )
             => ( ( k3_finseq_1(k4_matrix_5(C)) = k3_finseq_1(C)
                  & k1_matrix_1(k4_matrix_5(C)) = k1_matrix_1(C)
                  & r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
               => k3_matrix_1(k2_numbers,k4_matrix_5(C),A,B) = k1_binop_2(k3_matrix_1(k2_numbers,C,A,B)) ) ) ) ) )).

fof(t11_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k7_matrix_5(k7_binop_2(np__1),A) = k4_matrix_5(A) ) )).

fof(t12_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k1_matrixc1(k4_matrix_5(A)) = k4_matrix_5(k1_matrixc1(A)) ) )).

fof(t13_matrixc1,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(k5_matrix_5(A,B)) = k3_finseq_1(A)
            & k1_matrix_1(k5_matrix_5(A,B)) = k1_matrix_1(A) ) ) ) )).

fof(t14_matrixc1,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)) )
                 => ( ( k3_finseq_1(C) = k3_finseq_1(D)
                      & k1_matrix_1(C) = k1_matrix_1(D)
                      & r2_hidden(k4_tarski(A,B),k2_matrix_1(C)) )
                   => k3_matrix_1(k2_numbers,k5_matrix_5(C,D),A,B) = k4_binop_2(k3_matrix_1(k2_numbers,C,A,B),k3_matrix_1(k2_numbers,D,A,B)) ) ) ) ) ) )).

fof(t15_matrixc1,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) )
           => k1_matrixc1(k5_matrix_5(A,B)) = k5_matrix_5(k1_matrixc1(A),k1_matrixc1(B)) ) ) ) )).

fof(d2_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k2_matrixc1(A) = k1_matrixc1(k4_matrix_1(k2_numbers,A)) ) )).

fof(d3_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => ! [B] :
            ( ( v1_matrix_1(B)
              & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
           => ( B = k3_matrixc1(A)
            <=> ( k3_finseq_1(B) = k3_finseq_1(A)
                & k1_matrix_1(B) = np__1
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( r2_hidden(C,k2_finseq_1(k3_finseq_1(A)))
                     => k1_funct_1(B,C) = k13_binarith(k2_numbers,k9_matrix_5(A,C)) ) ) ) ) ) ) ) )).

fof(d4_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k4_matrixc1(A) = k8_matrix_1(k2_numbers,A,np__1) ) )).

fof(d5_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => k5_matrixc1(A,B) = k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k29_binop_2,A,B) ) ) )).

fof(d6_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => k6_matrixc1(A) = k2_finsop_1(k2_numbers,A,k27_binop_2) ) )).

fof(d7_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( C = k7_matrixc1(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k2_finseq_1(k3_finseq_1(A)))
                       => k9_matrix_5(C,D) = k6_matrixc1(k5_matrixc1(k7_matrix_1(k2_numbers,A,D),B)) ) ) ) ) ) ) ) )).

fof(t16_matrixc1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
                 => k6_complsp1(B,k5_matrixc1(C,D)) = k5_matrixc1(k4_complsp2(A,B,C),D) ) ) ) ) )).

fof(d8_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( v1_xcmplx_0(B)
         => k8_matrixc1(A,B) = k7_matrix_5(B,A) ) ) )).

fof(t17_matrixc1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => k1_matrixc1(k8_matrixc1(B,A)) = k7_matrix_5(k15_complex1(A),k1_matrixc1(B)) ) ) )).

fof(t18_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( k3_finseq_1(k5_matrixc1(A,B)) = k3_finseq_1(A)
              & k3_finseq_1(k5_matrixc1(A,B)) = k3_finseq_1(B) ) ) ) ) )).

fof(t19_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(k5_matrixc1(A,B)))
               => k9_matrix_5(k5_matrixc1(A,B),C) = k5_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) )).

fof(t20_matrixc1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k2_numbers,k4_finseq_2(A,k2_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k2_numbers,k4_finseq_2(A,k2_numbers))
                 => k9_matrix_5(k9_matrixc1(A,C,D),B) = k5_binop_2(k9_matrix_5(C,B),k9_matrix_5(D,B)) ) ) ) ) )).

fof(t21_matrixc1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => k15_complex1(k2_binop_1(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,A,k15_complex1(B))) = k2_binop_1(k2_numbers,k2_numbers,k2_numbers,k27_binop_2,k15_complex1(A),B) ) ) )).

fof(t22_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,k2_numbers)
          & m2_relset_1(B,k5_numbers,k2_numbers)
          & ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,k3_finseq_1(A)) )
               => k8_funct_2(k5_numbers,k2_numbers,B,C) = k9_matrix_5(A,C) ) ) ) ) )).

fof(t23_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ( r1_xreal_0(np__1,k3_finseq_1(k1_complsp2(A)))
       => k2_finsop_1(k2_numbers,k1_complsp2(A),k27_binop_2) = k15_complex1(k2_finsop_1(k2_numbers,A,k27_binop_2)) ) ) )).

fof(t24_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ( r1_xreal_0(np__1,k3_finseq_1(A))
       => k6_matrixc1(k1_complsp2(A)) = k15_complex1(k6_matrixc1(A)) ) ) )).

fof(t25_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => k1_complsp2(k5_matrixc1(A,k1_complsp2(B))) = k5_matrixc1(B,k1_complsp2(A)) ) ) ) )).

fof(t26_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ( k3_finseq_1(A) = k3_finseq_1(B)
               => k5_matrixc1(A,k6_complsp1(C,B)) = k6_complsp1(C,k5_matrixc1(A,B)) ) ) ) ) )).

fof(t27_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ( k3_finseq_1(A) = k3_finseq_1(B)
               => k5_matrixc1(k6_complsp1(C,A),B) = k6_complsp1(C,k5_matrixc1(A,B)) ) ) ) ) )).

fof(t28_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => k1_complsp2(k5_matrixc1(A,B)) = k5_matrixc1(k1_complsp2(A),k1_complsp2(B)) ) ) ) )).

fof(t29_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => k6_matrixc1(k6_complsp1(B,A)) = k5_binop_2(B,k6_matrixc1(A)) ) ) )).

fof(d9_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k10_matrixc1(A) = A ) )).

fof(t30_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( ( A = B
              & r1_xreal_0(np__1,k3_finseq_1(A)) )
           => k2_finsop_1(k1_numbers,A,k33_binop_2) = k2_finsop_1(k2_numbers,B,k27_binop_2) ) ) ) )).

fof(t31_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( ( A = B
              & r1_xreal_0(np__1,k3_finseq_1(A)) )
           => k15_rvsum_1(A) = k6_matrixc1(B) ) ) ) )).

fof(t32_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => k6_matrixc1(k4_complsp1(A,B)) = k4_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) ) )).

fof(t33_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(k3_complsp1(A,B)))
               => k9_matrix_5(k3_complsp1(A,B),C) = k3_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) )).

fof(t34_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(k4_complsp1(A,B)))
               => k9_matrix_5(k4_complsp1(A,B),C) = k4_binop_2(k9_matrix_5(A,C),k9_matrix_5(B,C)) ) ) ) ) )).

fof(t35_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => k5_matrixc1(k4_complsp1(A,B),C) = k4_complsp1(k5_matrixc1(A,C),k5_matrixc1(B,C)) ) ) ) ) )).

fof(t36_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => k5_matrixc1(A,k4_complsp1(B,C)) = k4_complsp1(k5_matrixc1(A,B),k5_matrixc1(A,C)) ) ) ) ) )).

fof(t37_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => k5_matrixc1(A,k3_complsp1(B,C)) = k3_complsp1(k5_matrixc1(A,B),k5_matrixc1(A,C)) ) ) ) ) )).

fof(t38_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => k5_matrixc1(k3_complsp1(A,B),C) = k3_complsp1(k5_matrixc1(A,C),k5_matrixc1(B,C)) ) ) ) ) )).

fof(t39_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => k6_matrixc1(k3_complsp1(A,B)) = k3_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) ) )).

fof(t40_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ( ( A = C
                      & B = D
                      & k3_finseq_1(A) = k3_finseq_1(D) )
                   => k1_finseqop(k2_numbers,k2_numbers,k2_numbers,k29_binop_2,A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k35_binop_2,C,D) ) ) ) ) ) )).

fof(t41_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => k10_matrixc1(k13_rvsum_1(A,B)) = k5_matrixc1(k10_matrixc1(A),k10_matrixc1(B)) ) ) ) )).

fof(t42_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(A),np__0)
              | k8_complsp2(A,B) = k6_matrixc1(k5_matrixc1(A,k1_complsp2(B))) ) ) ) ) )).

fof(t43_matrixc1,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) )
           => k2_matrix_1(A) = k2_matrix_1(B) ) ) ) )).

fof(t44_matrixc1,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)) )
                 => ( ( k3_finseq_1(C) = k3_finseq_1(D)
                      & k1_matrix_1(C) = k1_matrix_1(D)
                      & r2_hidden(B,k2_finseq_1(k3_finseq_1(C))) )
                   => k7_matrix_1(k2_numbers,k3_matrix_5(C,D),B) = k3_complsp1(k7_matrix_1(k2_numbers,C,B),k7_matrix_1(k2_numbers,D,B)) ) ) ) ) ) )).

fof(t45_matrixc1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( r2_hidden(A,k2_finseq_1(k3_finseq_1(B)))
           => k7_matrix_1(k2_numbers,B,A) = k1_complsp2(k7_matrix_1(k2_numbers,k1_matrixc1(B),A)) ) ) ) )).

fof(t46_matrixc1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( k3_finseq_1(B) = k1_matrix_1(C)
               => k5_matrixc1(B,k1_complsp2(k7_matrix_1(k2_numbers,k1_matrixc1(C),A))) = k1_complsp2(k5_matrixc1(k7_matrix_1(k2_numbers,k1_matrixc1(C),A),k1_complsp2(B))) ) ) ) ) )).

fof(t47_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( ( v1_matrix_1(B)
            & m2_finseq_1(B,k3_finseq_2(k2_numbers)) )
         => ( ( k3_finseq_1(A) = k1_matrix_1(B)
              & r1_xreal_0(np__1,k3_finseq_1(A)) )
           => k1_complsp2(k7_matrixc1(B,A)) = k7_matrixc1(k1_matrixc1(B),k1_complsp2(A)) ) ) ) )).

fof(t48_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k2_numbers)
             => ( ( k3_finseq_1(A) = k3_finseq_1(B)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => k5_matrixc1(A,k5_matrixc1(B,C)) = k5_matrixc1(k5_matrixc1(A,B),C) ) ) ) ) )).

fof(t49_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => k6_matrixc1(k5_complsp1(A)) = k1_binop_2(k6_matrixc1(A)) ) )).

fof(t50_matrixc1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => k6_matrixc1(k13_binarith(k2_numbers,A)) = A ) )).

fof(t51_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => k6_matrixc1(k8_finseq_1(k2_numbers,A,B)) = k3_binop_2(k6_matrixc1(A),k6_matrixc1(B)) ) ) )).

fof(d10_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( B = k11_matrixc1(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k2_finseq_1(k3_finseq_1(A)))
                   => k9_matrix_5(B,C) = k6_matrixc1(k7_matrix_1(k2_numbers,A,C)) ) ) ) ) ) ) )).

fof(d11_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( B = k12_matrixc1(A)
          <=> ( k3_finseq_1(B) = k1_matrix_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k2_finseq_1(k1_matrix_1(A)))
                   => k9_matrix_5(B,C) = k6_matrixc1(k8_matrix_1(k2_numbers,A,C)) ) ) ) ) ) ) )).

fof(t52_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ( k3_finseq_1(A) = np__1
       => k6_matrixc1(A) = k9_matrix_5(A,np__1) ) ) )).

fof(t53_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( k3_finseq_1(A) = k23_binop_2(C,np__1)
                  & B = k16_finseq_1(k2_numbers,A,C) )
               => k6_matrixc1(A) = k3_binop_2(k6_matrixc1(B),k4_finseq_4(k5_numbers,k2_numbers,A,k3_finseq_1(A))) ) ) ) ) )).

fof(t54_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k6_matrixc1(k11_matrixc1(A)) = k6_matrixc1(k12_matrixc1(A)) ) ) )).

fof(d12_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k13_matrixc1(A) = k6_matrixc1(k11_matrixc1(A)) ) )).

fof(t55_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => k12_matrixc1(A) = k11_matrixc1(k4_matrix_1(k2_numbers,A)) ) )).

fof(t56_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k3_finseq_1(A),np__0)
       => k13_matrixc1(A) = k13_matrixc1(k4_matrix_1(k2_numbers,A)) ) ) )).

fof(d13_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(C)
                  & k3_finseq_1(B) = k1_matrix_1(C) )
               => ! [D] :
                    ( ( v1_matrix_1(D)
                      & m2_finseq_1(D,k3_finseq_2(k2_numbers)) )
                   => ( D = k14_matrixc1(A,B,C)
                    <=> ( k3_finseq_1(D) = k3_finseq_1(A)
                        & k1_matrix_1(D) = k3_finseq_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(C))
                                 => k3_matrix_1(k2_numbers,D,E,F) = k5_binop_2(k5_binop_2(k9_matrix_5(A,E),k3_matrix_1(k2_numbers,C,E,F)),k15_complex1(k9_matrix_5(B,F))) ) ) ) ) ) ) ) ) ) ) )).

fof(t57_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(C)
                  & k3_finseq_1(B) = k1_matrix_1(C) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | r1_xreal_0(k3_finseq_1(B),np__0)
                  | k4_matrix_1(k2_numbers,k14_matrixc1(A,B,C)) = k1_matrixc1(k14_matrixc1(B,A,k2_matrixc1(C))) ) ) ) ) ) )).

fof(t58_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(C)
                  & k3_finseq_1(B) = k1_matrix_1(C) )
               => k1_matrixc1(k14_matrixc1(A,B,C)) = k14_matrixc1(k1_complsp2(A),k1_complsp2(B),k1_matrixc1(C)) ) ) ) ) )).

fof(t59_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(B),np__0)
              | k8_complsp2(A,B) = k15_complex1(k8_complsp2(B,A)) ) ) ) ) )).

fof(t60_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ( k3_finseq_1(A) = k3_finseq_1(B)
           => ( r1_xreal_0(k3_finseq_1(B),np__0)
              | k15_complex1(k8_complsp2(A,B)) = k8_complsp2(k1_complsp2(A),k1_complsp2(B)) ) ) ) ) )).

fof(t61_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m2_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( ~ r1_xreal_0(k1_matrix_1(A),np__0)
       => k1_matrixc1(k4_matrix_1(k2_numbers,A)) = k4_matrix_1(k2_numbers,k1_matrixc1(A)) ) ) )).

fof(t62_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k1_matrix_1(C)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | r1_xreal_0(k3_finseq_1(B),np__0)
                  | k8_complsp2(A,k7_matrixc1(k2_matrixc1(C),B)) = k13_matrixc1(k14_matrixc1(A,B,k4_matrix_1(k2_numbers,C))) ) ) ) ) ) )).

fof(t63_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & k3_finseq_1(A) = k1_matrix_1(C) )
               => ( r1_xreal_0(k3_finseq_1(A),np__0)
                  | r1_xreal_0(k3_finseq_1(B),np__0)
                  | r1_xreal_0(k3_finseq_1(C),np__0)
                  | k8_complsp2(k7_matrixc1(C,A),B) = k13_matrixc1(k14_matrixc1(A,B,k4_matrix_1(k2_numbers,C))) ) ) ) ) ) )).

fof(t64_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k1_matrix_1(C)
                  & k3_finseq_1(B) = k3_finseq_1(C) )
               => ( r1_xreal_0(k1_matrix_1(C),np__0)
                  | r1_xreal_0(k3_finseq_1(C),np__0)
                  | k8_complsp2(k7_matrixc1(C,A),B) = k8_complsp2(A,k7_matrixc1(k2_matrixc1(C),B)) ) ) ) ) ) )).

fof(t65_matrixc1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k2_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(k2_numbers)) )
             => ( ( k3_finseq_1(A) = k3_finseq_1(C)
                  & k3_finseq_1(B) = k1_matrix_1(C) )
               => ( r1_xreal_0(k1_matrix_1(C),np__0)
                  | r1_xreal_0(k3_finseq_1(C),np__0)
                  | r1_xreal_0(k3_finseq_1(A),np__0)
                  | k8_complsp2(A,k7_matrixc1(C,B)) = k8_complsp2(k7_matrixc1(k2_matrixc1(C),A),B) ) ) ) ) ) )).

fof(dt_k1_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k1_matrixc1(A))
        & m2_finseq_1(k1_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k2_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k2_matrixc1(A))
        & m2_finseq_1(k2_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k3_matrixc1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k2_numbers)
     => ( v1_matrix_1(k3_matrixc1(A))
        & m2_finseq_1(k3_matrixc1(A),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k4_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => m2_finseq_1(k4_matrixc1(A),k2_numbers) ) )).

fof(dt_k5_matrixc1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k2_numbers)
        & m1_finseq_1(B,k2_numbers) )
     => m2_finseq_1(k5_matrixc1(A,B),k2_numbers) ) )).

fof(commutativity_k5_matrixc1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k2_numbers)
        & m1_finseq_1(B,k2_numbers) )
     => k5_matrixc1(A,B) = k5_matrixc1(B,A) ) )).

fof(dt_k6_matrixc1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k2_numbers)
     => m1_subset_1(k6_matrixc1(A),k2_numbers) ) )).

fof(dt_k7_matrixc1,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers))
        & m1_finseq_1(B,k2_numbers) )
     => m2_finseq_1(k7_matrixc1(A,B),k2_numbers) ) )).

fof(dt_k8_matrixc1,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers))
        & v1_xcmplx_0(B) )
     => ( v1_matrix_1(k8_matrixc1(A,B))
        & m2_finseq_1(k8_matrixc1(A,B),k3_finseq_2(k2_numbers)) ) ) )).

fof(dt_k9_matrixc1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k2_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
     => m2_finseq_2(k9_matrixc1(A,B,C),k2_numbers,k4_finseq_2(A,k2_numbers)) ) )).

fof(commutativity_k9_matrixc1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k2_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
     => k9_matrixc1(A,B,C) = k9_matrixc1(A,C,B) ) )).

fof(redefinition_k9_matrixc1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k2_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k2_numbers)) )
     => k9_matrixc1(A,B,C) = k5_matrixc1(B,C) ) )).

fof(dt_k10_matrixc1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_numbers)
     => m2_finseq_1(k10_matrixc1(A),k2_numbers) ) )).

fof(dt_k11_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => m2_finseq_1(k11_matrixc1(A),k2_numbers) ) )).

fof(dt_k12_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => m2_finseq_1(k12_matrixc1(A),k2_numbers) ) )).

fof(dt_k13_matrixc1,axiom,(
    ! [A] :
      ( ( v1_matrix_1(A)
        & m1_finseq_1(A,k3_finseq_2(k2_numbers)) )
     => m1_subset_1(k13_matrixc1(A),k2_numbers) ) )).

fof(dt_k14_matrixc1,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(A,k2_numbers)
        & m1_finseq_1(B,k2_numbers)
        & v1_matrix_1(C)
        & m1_finseq_1(C,k3_finseq_2(k2_numbers)) )
     => ( v1_matrix_1(k14_matrixc1(A,B,C))
        & m2_finseq_1(k14_matrixc1(A,B,C),k3_finseq_2(k2_numbers)) ) ) )).
%------------------------------------------------------------------------------