
theorem Th14:
  for L being Field
  for a,x being Element of L
  for b being non zero Element of L st x <> -a/b holds
  multiplicity(<%a,b%>,x) = 0
  proof
    let L be Field;
    let a,x be Element of L;
    let b be non zero Element of L;
    assume
A1: x <> -a/b;
    set f = <%a,b%>;
    Roots(f) = {-a/b} by Th10;
    then not x in Roots(f) by A1,TARSKI:def 1;
    then not x is_a_root_of f by POLYNOM5:def 10;
    hence thesis by NAT_1:14,UPROOTS:52;
  end;
