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