
theorem
for F being Field,
    p,q being Element of the carrier of Polynom-Ring F
holds (p gcd q) divides p & (p gcd q) divides q &
      for r being Element of the carrier of Polynom-Ring F
      st r divides p & r divides q holds r divides (p gcd q)
proof
let F be Field,
    p,q be Element of the carrier of Polynom-Ring F;
set g = p gcd q;
reconsider g1 = p gcd q as Element of Polynom-Ring F;
g1 divides p & g1 divides q by defGCD;
hence g divides p & g divides q;
now let r be Element of the carrier of Polynom-Ring F;
  reconsider r1 = r as Element of Polynom-Ring F;
  assume r divides p & r divides q;
  then r1 divides g1 by defGCD;
  hence r divides g;
  end;
hence thesis;
end;
