
theorem np0:
for R being domRing,
    p,q being monic Element of the carrier of Polynom-Ring R
holds p is_associated_to q iff p = q
proof
let R be domRing,
    p,q be monic Element of the carrier of Polynom-Ring R;
K1: LC p = 1.R & LC q = 1.R by RATFUNC1:def 7;
now assume AS: p is_associated_to q;
  then consider c being Element of Polynom-Ring R such that
  K2: q = p * c by GCD_1:def 1;
  reconsider r = c as Element of the carrier of Polynom-Ring R;
  K3: q = p *' r by K2,POLYNOM3:def 10;
  r <> 0_.(R) by K3,POLYNOM3:34;
  then deg q = deg p + deg r by K3,HURWITZ:23
            .= deg q + deg r by AS,np00;
  then r is constant;
  then consider a being Element of R such that K5: r = a|R by T11;
  X: deg p = deg q by AS,np00;
  r = a * 1_.(R) by K5,LX1;
  then q = a * (p *' 1_.(R)) by K3,poly2 .= a * p by poly1;
  then q.(len q-'1) = a * 1.R by X,K1,POLYNOM5:def 4
                   .= a;
  then r = 1_.(R) by K5,K1;
  hence p = q by K3,poly1;
  end;
hence thesis;
end;
