theorem
  ((n,n)-->a) + ((n,n)-->b) = (n,n)-->(a+b)
proof
A1: Indices ((n,n)-->a) = Indices ((n,n)-->(a+b)) by MATRIX_0:26;
A2: Indices ((n,n)-->a) = Indices ((n,n)-->b) by MATRIX_0:26;
  now
    let i,j;
    assume
A3: [i,j] in Indices ((n,n)-->(a+b));
    then ((n,n)-->a)*(i,j)=a by A1,MATRIX_0:46;
    then ((n,n)-->a)*(i,j) +((n,n)-->b)*(i,j)=a+b by A2,A1,A3,MATRIX_0:46;
    hence ((n,n)-->a)*(i,j) +((n,n)-->b)*(i,j)=((n,n)-->(a+b))*(i,j)
                by A3,MATRIX_0:46;
  end;
  hence thesis by A1,Def5;
end;
