
theorem ThsepsplB:
for F being Field
for p being non zero Polynomial of F
for a being Element of F
holds multiplicity(NormPolynomial p,a) = multiplicity(p,a)
proof
let F be Field, p be non zero Polynomial of F; let a be Element of F;
set n = multiplicity(p,a);
A: (X-a)`^n divides p & not (X-a)`^(n+1) divides p by FIELD_14:67;
   p is Element of the carrier of Polynom-Ring F &
   (X-a)`^n is Element of the carrier of Polynom-Ring F &
   (X-a)`^(n+1) is Element of the carrier of Polynom-Ring F &
   NormPolynomial p is Element of the carrier of Polynom-Ring F
   by POLYNOM3:def 10; then
(X-a)`^n divides (NormPolynomial p) &
   not (X-a)`^(n+1) divides (NormPolynomial p) by A,RING_4:26;
hence thesis by FIELD_14:67;
end;
