
theorem RelPrimeEx:
  for a, b being non zero Nat st
    not a, b are_coprime holds
      ex k being non zero Nat st k <> 1 & k divides a & k divides b
  proof
    let a, b be non zero Nat;
    assume
Z1: not a, b are_coprime;
    set k = a gcd b;
    k divides a & k divides b by NAT_D:def 5;
    hence thesis by Z1;
  end;
