
theorem ZZ2:
for F being Field
for p,q being Polynomial of F
for r being monic Polynomial of F holds (r*'p) gcd (r*'q) = r *' (p gcd q)
proof
let F be Field, p,q be Polynomial of F; let r be monic Polynomial of F;
consider a,b being Element of Polynom-Ring F such that
H: a = p & b = q & p gcd q = a gcd b by RING_4:def 12;
per cases;
suppose A: p = 0_.(F) & q = 0_.(F);
  then a gcd b = 0_.(F) by H,RING_4:def 11;
  hence thesis by H,A;
  end;
suppose AS: p <> 0_.(F) or q <> 0_.(F); then
K: (a gcd b) is monic by H,RING_4:def 11;
consider p1 being Polynomial of F such that
E1: p = (p gcd q) *' p1 by RING_4:1,RING_4:52;
   (r *' (p gcd q)) *' p1 = r *' p by E1,POLYNOM3:33; then
A: r *' (p gcd q) divides (r *' p) by RING_4:1;
consider q1 being Polynomial of F such that
E2: q = (p gcd q) *' q1 by RING_4:1,RING_4:52;
   (r *' (p gcd q)) *' q1 = r *' q by E2,POLYNOM3:33; then
r *' (p gcd q) divides (r *' q) by RING_4:1; then
consider u being Polynomial of F such that
C: (r *' p) gcd (r *' q) = (r *' (p gcd q)) *' u
   by A,RING_4:52,RING_4:1;
C1: u is monic
    proof
    consider a,b being Element of Polynom-Ring F such that
    C0: a = r *' p & b = r *' q & (r *' p) gcd (r *' q) = a gcd b
        by RING_4:def 12;
    r *' q <> 0_.F or r *' p <> 0_.F
      proof
      assume r *' q = 0_.(F);
      then q is zero;
      then p is non zero by AS,UPROOTS:def 5;
      hence thesis;
      end; then
    C2: (r *' p) gcd (r *' q) is monic by C0,RING_4:def 11;
    (p gcd q) is monic by AS,H,RING_4:def 11;
    hence thesis by C,C2,ZZ3y;
    end;
  (p gcd q) *' u divides p & (p gcd q) *' u divides q
    proof
    consider v being Polynomial of F such that
    D1: ((r *' (p gcd q)) *' u) *' v = r *' p by C,RING_4:52,RING_4:1;
    ((r *' (p gcd q)) *' u) divides (r *' p) by D1,RING_4:1; then
    (r *' ((p gcd q) *' u)) divides (r *' p) by POLYNOM3:33;
    hence (p gcd q) *' u divides p by ZZ3z;
    consider w being Polynomial of F such that
    D1: ((r *' (p gcd q)) *' u) *' w = r *' q by C,RING_4:52,RING_4:1;
    ((r *' (p gcd q)) *' u) divides (r *' q) by D1,RING_4:1;then
    (r *' ((p gcd q) *' u)) divides (r *' q) by POLYNOM3:33;
    hence (p gcd q) *' u divides q by ZZ3z;
    end; then
E: (p gcd q) *' u divides (p gcd q) by RING_4:52;
(p gcd q) *' (1_.(F)) = p gcd q; then
u = 1_.(F) by C1,ZZ3x,K,H,E,ZZ3z;
hence thesis by C;
end;
end;
