
theorem Th53:
  for L being domRing, p, q being non-zero Polynomial of L holds
  BRoots(p*'q) = BRoots(p) + BRoots(q)
proof
  let L be domRing, p, q being non-zero Polynomial of L;
  now
    let i be object;
    assume i in the carrier of L;
    then reconsider x = i as Element of L;
    thus (BRoots(p*'q)).i = multiplicity(p*'q, x) by Def8
      .= multiplicity(p,x) + multiplicity(q,x) by Th52
      .= (BRoots p).i + multiplicity(q,x) by Def8
      .= (BRoots p).i + (BRoots q).i by Def8
      .= (BRoots(p) + BRoots(q)).i by PRE_POLY:def 5;
  end;
  hence thesis by PBOOLE:3;
end;
