
theorem lift7:
for F1,F2 being Field,
    E1 being FieldExtension of F1, E2 being FieldExtension of F2
for p being Polynomial of F1 for q being Polynomial of F2
for a being Element of E1, b being Element of E2
st F1 == F2 & E1 == E2 & p = q & a = b holds Ext_eval(p,a) = Ext_eval(q,b)
proof
let F1,F2 be Field,
    E1 be FieldExtension of F1, E2 be FieldExtension of F2;
let p be Polynomial of F1; let q be Polynomial of F2;
let a be Element of E1, b be Element of E2;
assume AS: F1 == F2 & E1 == E2 & p = q & a = b;
B: p is Element of the carrier of Polynom-Ring F1 &
   q is Element of the carrier of Polynom-Ring F2 by POLYNOM3:def 10;
p is Polynomial of E1 by FIELD_4:8; then
reconsider p1 = p as Element of the carrier of Polynom-Ring E1
   by POLYNOM3:def 10;
q is Polynomial of E2 by FIELD_4:8; then
reconsider q1 = q as Element of the carrier of Polynom-Ring E2
   by POLYNOM3:def 10;
thus Ext_eval(p,a) = eval(p1,a) by B,FIELD_4:26
                  .= eval(q1,b) by AS,lift8
                  .= Ext_eval(q,b) by B,FIELD_4:26;
end;
