
theorem lemgcdn:
for F being Field
for p being Element of the carrier of Polynom-Ring F
for q being non zero Element of the carrier of Polynom-Ring F
st q divides p holds p gcd q = NormPolynomial q
proof
let F be Field, p be Element of the carrier of Polynom-Ring F;
let q be non zero Element of the carrier of Polynom-Ring F;
assume AS: q divides p;
set s = NormPolynomial q;
B: s divides p by AS,RING_4:25;
   q = q *' 1_.(F); then
C: s divides q by RING_4:25,RING_4:1;
now let r be Polynomial of F;
   assume D: r divides p & r divides q;
   r is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
   hence r divides s by D,RING_4:26;
   end;
hence thesis by C,B,RING_4:53;
end;
