
theorem np2:
for F being Field,
    p,q being Element of the carrier of Polynom-Ring F
holds q divides p iff q divides NormPolynomial(p)
proof
let F be Field, p,q being Element of the carrier of Polynom-Ring F;
per cases;
suppose p = 0_.(F);
  hence thesis by npl0;
end;
suppose AS: p <> 0_.(F);
  then AS1: p is non zero by UPROOTS:def 5;
  A: now assume q divides p;
     then consider r being Polynomial of F such that
     A1: p = q *' r by T2;
     set a = (LC p)";
     NormPolynomial p = a * (q *' r) by AS1,A1,npl
                     .= q *' (a * r) by poly2;
     hence q divides NormPolynomial(p) by T2;
     end;
  now assume q divides NormPolynomial(p);
    then consider r being Polynomial of F such that
    A1: NormPolynomial(p) = q *' r by T2;
    set a = (LC p)";
    p is non zero by AS,UPROOTS:def 5; then
    LC p <> 0.F; then
    (LC p)" * (LC p) = 1.F by VECTSP_1:def 10; then
    a <> 0.F; then
    a" * a = 1.F by VECTSP_1:def 10; then
    p = (a" * a) * p by poly2a
     .= a" * (a * p) by poly3
     .= a" * (q *' r) by AS1,A1,npl
     .= q *' (a" * r) by poly2;
    hence q divides p by T2;
    end;
  hence thesis by A;
  end;
end;
