
theorem Th16:
  for L being Field
  for p being non-zero Polynomial of L
  for a being Element of L
  for b being non zero Element of L st not -a/b in Roots(p)
  holds (canFS Roots(p))^<*-a/b*> is Enumeration of Roots(<%a,b%>*'p)
  proof
    let L be Field;
    let p be non-zero Polynomial of L;
    let a be Element of L;
    let b be non zero Element of L such that
A1: not -a/b in Roots(p);
    set C = canFS Roots(p);
A2: Roots(p) = rng C by FUNCT_2:def 3;
    then
A3: C^<*-a/b*> is one-to-one by A1,GRAPHSP:1;
A4: rng <*-a/b*> = {-a/b} by FINSEQ_1:38;
    Roots <%a,b%> = {-a/b} by Th10;
    then Roots(<%a,b%>*'p) = rng C \/ rng <*-a/b*> by A2,A4,UPROOTS:23
    .= rng(C^<*-a/b*>) by FINSEQ_1:31;
    hence thesis by A3,RLAFFIN3:def 1;
  end;
