
theorem Th17:
  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
  for E being Enumeration of Roots(<%a,b%>*'p) st E = (canFS Roots(p))^<*-a/b*>
  holds len E = 1 + card Roots(p) &
  E.(1+card Roots(p)) = -a/b &
  for n being Nat st 1 <= n <= card Roots(p) holds E.n = (canFS Roots(p)).n
  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;
    set C = canFS Roots(p);
    let E be Enumeration of Roots(<%a,b%>*'p) such that
A1: E = C^<*-a/b*>;
A2: len C = card Roots(p) by FINSEQ_1:93;
    len <*-a/b*> = 1 by FINSEQ_1:39;
    hence thesis by A1,A2,FINSEQ_1:22,64;
  end;
