
theorem Th15:
  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 card Roots(<%a,b%>*'p) = 1 + card Roots(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;
A1: Roots(<%a,b%>*'p) = Roots(<%a,b%>) \/ Roots(p) by UPROOTS:23;
    Roots <%a,b%> = {-a/b} by Th10;
    hence thesis by A1,CARD_2:41;
  end;
