reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th24:
  for R being Ring, M1,M2 being Matrix of n,R
  holds (M1+M2)@ = M1@+M2@
proof
  let R be Ring;
  let M1,M2 be Matrix of n,R;
  for i,j st [i,j] in Indices ((M1+M2)@) holds ((M1+M2)@)*(i,j)=(M1@+M2@)*
  (i,j)
  proof
    let i,j;
    assume [i,j] in Indices ((M1+M2)@);
    then
A1: [i,j] in [:Seg n, Seg n:] by MATRIX_0:24;
    then
A2: [i,j] in Indices (M1@) by MATRIX_0:24;
    i in Seg n & j in Seg n by A1,ZFMISC_1:87;
    then
A3: [j,i] in [:Seg n, Seg n:] by ZFMISC_1:87;
    then
A4: [j,i] in Indices M1 by MATRIX_0:24;
A5: [j,i] in Indices M2 by A3,MATRIX_0:24;
    [j,i] in Indices (M1+M2) by A3,MATRIX_0:24;
    then ((M1+M2)@)*(i,j)=(M1+M2)*(j,i) by MATRIX_0:def 6
      .=M1*(j,i)+M2*(j,i) by A4,MATRIX_3:def 3
      .=(M1@)*(i,j)+M2*(j,i) by A4,MATRIX_0:def 6
      .=(M1@)*(i,j)+(M2@)*(i,j) by A5,MATRIX_0:def 6
      .=(M1@+M2@)*(i,j) by A2,MATRIX_3:def 3;
    hence thesis;
  end;
  hence thesis by MATRIX_0:27;
end;
