theorem
  m divides n implies k gcd m divides k gcd n
proof
  set M = k gcd m;
A1: M divides k by NAT_D:def 5;
  assume
A2: m divides n;
  M divides m by NAT_D:def 5;
  then M divides n by A2,NAT_D:4;
  hence thesis by A1,NAT_D:def 5;
end;
