
theorem ZZ1fB:
for F being Field
for p,q being non zero Polynomial of F holds BRoots(p) divides BRoots(p *' q)
proof
let F be Field, p,q being non zero Polynomial of F;
now let o be object;
  per cases;
  suppose o in support(BRoots p); then
    reconsider a = o as Element of the carrier of F;
    (BRoots p).a = multiplicity(p,a) &
    (BRoots(p *' q)).a = multiplicity(p*'q,a) by UPROOTS:def 9;
    hence (BRoots p).o <= (BRoots(p*'q)).o by ZZ7;
    end;
  suppose not o in support(BRoots p);
    hence (BRoots p).o <= (BRoots(p*'q)).o by PRE_POLY:def 7;
    end;
  end;
hence thesis by PRE_POLY:def 11;
end;
