
theorem lemma2d:
for F being Field,
    E being FieldExtension of F
for a being non zero Element of F,
    b being non zero Element of E
for p being Ppoly of F
for q being Ppoly of E st a * p = b * q holds a = b & p = q
proof
let F be Field, E be FieldExtension of F;
let a be non zero Element of F, b be non zero Element of E;
let p be Ppoly of F; let q be Ppoly of E;
assume AS: a * p = b * q;
thus T: b = b * 1.E
         .= b * LC(q) by RING_5:50
         .= LC(b * q) by RATFUNC1:18
         .= LC(a * p) by AS,lemma2e
         .= a * LC(p) by RATFUNC1:18
         .= a * 1.F by RING_5:50
         .= a;
F is Subfield of E by FIELD_4:7; then
the carrier of F c= the carrier of E by EC_PF_1:def 1; then
reconsider a1 = a as Element of E;
reconsider q1 = p as Ppoly of E by lemmapp;
a1|E = a|F by FIELD_6:23; then
(a1|E) *' q1 = (a|F) *' p by FIELD_4:17 .= a * p by poly1; then
(a1|E) *' q1 = (b|E) *' q by AS,poly1;
hence thesis by T,RATFUNC1:7;
end;
