reserve n for Nat;

theorem
for F being Field,
    q being Polynomial of F
for p being non zero Polynomial of F
for b being non zero Element of F holds q divides p iff q divides (b*p)
proof
let F be Field, q be Polynomial of F;
let p be non zero Polynomial of F; let b be non zero Element of F;
X: b <> 0.F;
now assume q divides (b*p);
  then consider r being Polynomial of F such that
  A: b * p = q *' r by RING_4:1;
  q *' (b" * r)  = b" * (q *' r) by HURWITZ:19
               .= (b" * b) * p by A,HURWITZ:14
               .= 1.F * p by X,VECTSP_1:def 10
               .= p;
  hence q divides p by RING_4:1;
  end;
hence thesis by divi1;
end;
