
theorem multi1:
for F being Field,
    p,q being Polynomial of F
for n,k being Element of NAT st q`^n divides p & k <= n holds q`^k divides p
proof
let F be Field, p,q be Polynomial of F; let n,k be Element of NAT;
assume A: q`^n divides p & k <= n;
then consider r being Polynomial of F such that
B: p = (q`^n) *' r by RING_4:1;
consider j being Nat such that C: n = k + j by A,NAT_1:10;
reconsider j as Element of NAT by ORDINAL1:def 12;
(q`^n) *' r = ((q`^k) *' (q`^j)) *' r by C,multi0
  .= (q`^k) *' ((q`^j) *' r) by POLYNOM3:33;
hence thesis by B,RING_4:1;
end;
