
theorem np1:
for F being Field,
    p,q being Element of the carrier of Polynom-Ring F
holds q divides p iff NormPolynomial(q) divides p
proof
let F be Field, p,q being Element of the carrier of Polynom-Ring F;
per cases;
suppose q = 0_.(F);
  hence thesis by npl0;
end;
suppose AS: q <> 0_.(F);
  then AS1: q 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 q)";
     reconsider qq = NormPolynomial(q) as Polynomial of F;
     q is non zero by AS,UPROOTS:def 5; then
     LC q <> 0.F; then
     (LC q)" * (LC q) = 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"  * (q *' r))) by A1,poly3
      .= (a * (q  *' (a" * r))) by poly2
      .= (a * q) *' (a" * r) by poly2
      .= qq *' (a" * r) by AS1,npl;
     hence NormPolynomial(q) divides p by T2;
     end;
  now assume A0: NormPolynomial(q) divides p;
    reconsider qq = NormPolynomial(q) as Polynomial of F;
    consider r being Polynomial of F such that
    A1: p = qq  *' r by A0,T2;
    p = ((LC q)" * q) *' r by AS1,A1,npl
     .= (LC q)" * (q *' r) by poly2
     .= q *' ((LC q)" * r) by poly2;
    hence q divides p by T2;
    end;
  hence thesis by A;
  end;
end;
