
theorem UP55:
for F being Field
for E being FieldExtension of F
for p,q being non zero Polynomial of F
for a being Element of E
holds multiplicity(p*'q,a) = multiplicity(p,a) + multiplicity(q,a)
proof
let F be Field, E be FieldExtension of F;
let p,q be non zero Polynomial of F; let a be Element of E;
reconsider pE = p, qE = q as Polynomial of E by FIELD_4:8;
p <> 0_.(F); then
p <> 0_.(E) by FIELD_4:12; then
reconsider pE as non zero Polynomial of E by UPROOTS:def 5;
q <> 0_.(F); then
q <> 0_.(E) by FIELD_4:12; then
reconsider qE as non zero Polynomial of E by UPROOTS:def 5;
pE *' qE = p *' q by FIELD_4:17;
hence multiplicity(p*'q,a)
    = multiplicity(pE*'qE,a) by sepsep1
   .= multiplicity(pE,a) + multiplicity(qE,a) by UPROOTS:55
   .= multiplicity(p,a) + multiplicity(qE,a) by sepsep1
   .= multiplicity(p,a) + multiplicity(q,a) by sepsep1;
end;
