
theorem lems1:
for F being Field
for E being FieldExtension of F
for p being non zero Polynomial of F
for c being non zero Element of F
for a being Element of E holds multiplicity(c*p,a) = multiplicity(p,a)
proof
let F be Field, E be FieldExtension of F, p be non zero Polynomial of F;
let c be non zero Element of F, a be Element of E;
set n = multiplicity(p,a);
reconsider q = p as Polynomial of E by FIELD_4:8;
p <> 0_.(F); then
p <> 0_.(E) by FIELD_4:12; then
reconsider q as non zero Polynomial of E by UPROOTS:def 5;
F is Subfield of E & c <> 0.F by FIELD_4:7; then
H: c <> 0.E &  @(c,E) = c by EC_PF_1:def 1,FIELD_7:def 4; then
K: @(c,E) is non zero; then
reconsider cq = @(c,E) * q as non zero Polynomial of E;
I: @(c,E) * q = c * p by H,ll;
multiplicity(q,a) = n by sepsep1; then
(X-a)`^n divides q & not (X-a)`^(n+1) divides q by FIELD_14:67; then
(X-a)`^n divides (@(c,E)*q) & not (X-a)`^(n+1) divides (@(c,E)*q)
   by K,RING_5:15;
then multiplicity(cq,a) = n by FIELD_14:67;
hence thesis by I,sepsep1;
end;
