theorem Th4:
  k divides m & k divides n implies k divides m+n
proof
  assume that
A1: k divides m and
A2: k divides n;
  consider m1 such that
A3: m=k*m1 by A1;
  consider n1 such that
A4: n=k*n1 by A2;
  m+n=k*(m1+n1) by A3,A4;
  hence thesis;
end;
