
theorem
for F being Field
for p being Polynomial of F
for q being non zero Polynomial of F
for s being monic Polynomial of F
holds s = p gcd q iff
      (s divides p & s divides q &
       for r being Polynomial of F st r divides p & r divides q
       holds r divides s)
proof
let F be Field, p be Polynomial of F; let q be non zero Polynomial of F;
let s be monic Polynomial of F;
now assume AS: s divides p & s divides q &
       for r being Polynomial of F st r divides p & r divides q
       holds r divides s;
  reconsider a = p, b = q as Element of Polynom-Ring F by POLYNOM3:def 10;
  reconsider g = s as Element of the carrier of Polynom-Ring F
           by POLYNOM3:def 10;
  B: b <> 0_.(F);
  now let d be Element of Polynom-Ring F;
    assume C: d divides a & d divides b;
    reconsider h = d as Polynomial of F by POLYNOM3:def 10;
    h divides p & h divides q by C;
    then h divides s by AS;
    hence d divides g;
    end;
  then g is a,b-gcd by AS;
  then g = a gcd b by dpg,B;
  hence s = p gcd q by dd;
  end;
hence thesis by G1;
end;
