theorem Th3:
  A + B = B + A
proof
A1: Indices A = Indices (A + B) by MATRIX_0:26;
A2: Indices A= Indices B by MATRIX_0:26;
  now
    let i,j;
    assume
A3: [i,j] in Indices (A + B);
    hence (A + B)*(i,j)=A*(i,j) + B*(i,j) by A1,Def5
      .=(B + A)*(i,j) by A2,A1,A3,Def5;
  end;
  hence thesis by MATRIX_0:27;
end;
