
theorem sepsep1:
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 a being Element of E 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 a be Element of E;
reconsider K = E as E-extending FieldExtension of F by FIELD_4:6;
multiplicity(q,@(a,K)) = multiplicity(p,@(a,K)) by AS,sepsep
                      .= multiplicity(p,a) by multi3K;
hence thesis by multi3;
end;
