reserve n for Nat;

theorem
for R being domRing,
    p being non zero Polynomial of R holds card(Roots p) <= card(BRoots p)
proof
let R be domRing, p be non zero Polynomial of R;
per cases;
suppose p is with_roots;
  then reconsider p1 = p as non zero with_roots Polynomial of R;
  consider q being (Ppoly of R,BRoots p1),
           r being non with_roots Polynomial of R such that
  A: p1 = q *' r & Roots q = Roots p1 by acf;
  deg q = card(BRoots q) by lemacf5 .= card(BRoots p1) by pf2;
  hence thesis by A,degpoly;
  end;
suppose A: p is non with_roots;
  then card(Roots p) = 0 .= card(EmptyBag(the carrier of R)) by UPROOTS:11
                    .= card(BRoots p) by A,lemacf1;
  hence thesis;
  end;
end;
