
theorem lems:
for F being Field
for p being non zero Polynomial of F
for c being non zero Element of F
for a being Element of F holds multiplicity(c*p,a) = multiplicity(p,a)
proof
let F be Field, p be non zero Polynomial of F;
let c be non zero Element of F, a be Element of F;
set n = multiplicity(p,a);
(X-a)`^n divides p & not (X-a)`^(n+1) divides p by FIELD_14:67; then
(X-a)`^n divides (c*p) & not (X-a)`^(n+1) divides (c*p) by RING_5:15;
hence thesis by FIELD_14:67;
end;
