
theorem Th49:
  for L being non degenerated comRing, p being non-zero (
Polynomial of L), x being Element of L holds x is_a_root_of p iff multiplicity(
  p,x) >= 1
proof
  let L be non degenerated comRing, p being non-zero (Polynomial of L), x
  being Element of L;
  set r = <%-x, 1.L%>;
  set m = multiplicity(p,x);
  consider F being finite non empty Subset of NAT such that
A1: F = {k where k is Element of NAT : ex q being Polynomial of L st p =
  (r`^k) *' q} and
A2: m = max F by Def7;
  m in F by A2,XXREAL_2:def 8;
  then consider k being Element of NAT such that
A3: m = k and
A4: ex q being Polynomial of L st p = (r`^k) *' q by A1;
  hereby
    assume x is_a_root_of p;
    then
A5: p = r*'poly_quotient(p,x) by Th47;
    r`^1 = r by POLYNOM5:16;
    then 1 in F by A1,A5;
    hence multiplicity(p,x) >= 1 by A2,XXREAL_2:def 8;
  end;
  consider q being Polynomial of L such that
A6: p = (r`^k) *' q by A4;
  assume multiplicity(p,x) >= 1;
  then consider k1 being Nat such that
A7: k = k1+1 by A3,NAT_1:6;
  reconsider k1 as Element of NAT by ORDINAL1:def 12;
  p = r *' (r`^k1) *' q by A6,A7,POLYNOM5:19
    .= r *' ((r`^k1) *' q) by POLYNOM3:33;
  hence thesis by Th46;
end;
