reserve n for Nat;

theorem llll:
for R being domRing,
    p being non zero Polynomial of R
for b being non zero Element of R holds BRoots(b * p) = BRoots p
proof
let F be domRing, p be non zero Polynomial of F;
let b be non zero Element of F;
now let a be Element of F;
   multiplicity(p,a) = multiplicity(b*p,a) by multip1d;
   hence (BRoots(b*p)).a = multiplicity(p,a) by UPROOTS:def 9
                        .= (BRoots p).a by UPROOTS:def 9;
   end;
then for o be object holds o in the carrier of F implies
     (BRoots(b*p)).o = (BRoots p).o;
hence thesis by PBOOLE:3;
end;
