
theorem G1:
for F being Field,
    p,q being Polynomial of F
holds (p gcd q) divides p & (p gcd q) divides q &
      for r being Polynomial of F
      st r divides p & r divides q holds r divides (p gcd q)
proof
let F be Field, p,q being Polynomial of F;
reconsider a = p, b = q as Element of Polynom-Ring F by POLYNOM3:def 10;
set g = a gcd b;
A0: p gcd q = g by dd;
g divides a by defGCD;
then consider c being Element of Polynom-Ring F such that A1: g * c = a;
reconsider r = c as Polynomial of F by POLYNOM3:def 10;
(p gcd q) *' r = p by A0,A1,POLYNOM3:def 10;
hence p gcd q divides p by T2;
g divides b by defGCD;
then consider c being Element of Polynom-Ring F such that A1: g * c = b;
reconsider r = c as Polynomial of F by POLYNOM3:def 10;
(p gcd q) *' r = q by A0,A1,POLYNOM3:def 10;
hence p gcd q divides q by T2;
now let r be Polynomial of F;
  assume A1: r divides p & r divides q;
  reconsider c = r as Element of Polynom-Ring F by POLYNOM3:def 10;
  consider s being Polynomial of F such that A2: r *' s = p by A1,T2;
  consider t being Polynomial of F such that A3: r *' t = q by A1,T2;
  reconsider x=s, y=t as Element of Polynom-Ring F by POLYNOM3:def 10;
  c * x = a & c * y = b by A2,A3,POLYNOM3:def 10;
  then c divides a & c divides b;
  then c divides a gcd b by defGCD;
  hence r divides (p gcd q) by dd;
  end;
hence thesis;
end;
