
theorem lift8:
for F1,F2 being Field
for p being Polynomial of F1 for q being Polynomial of F2
for a being Element of F1, b being Element of F2
st F1 == F2 & p = q & a = b holds eval(p,a) = eval(q,b)
proof
let F1,F2 be Field;
let p be Polynomial of F1; let q be Polynomial of F2;
let a be Element of F1, b be Element of F2;
assume AS: F1 == F2 & p = q & a = b; then
   F1 is Subfield of F2 by FIELD_7:def 2; then
A: F2 is FieldExtension of F1 by FIELD_4:7;
p is Element of the carrier of Polynom-Ring F1 &
q is Element of the carrier of Polynom-Ring F2 by POLYNOM3:def 10;
hence thesis by A,AS,FIELD_4:27;
end;
