
theorem sepsep:
for F being Field,
    p being non zero Polynomial of F
for E being FieldExtension of F,
    q being non zero Polynomial of E st q = p
for K being E-extending FieldExtension of F
for a being Element of K holds multiplicity(q,a) = multiplicity(p,a)
proof
let F be Field, p be non zero Polynomial of F;
let E be FieldExtension of F, q be non zero Polynomial of E;
assume AS: q = p;
let K be E-extending FieldExtension of F, a be Element of K;
reconsider r = q as Polynomial of K by FIELD_4:8;
q <> 0_.(E); then
r <> 0_.(K) by FIELD_4:12; then
reconsider r as non zero Polynomial of K by UPROOTS:def 5;
thus multiplicity(q,a) = multiplicity(r,a) by FIELD_14:def 6
  .= multiplicity(p,a) by AS,FIELD_14:def 6;
end;
