
theorem lemNor3e:
for F1 being Field,
    F2 being F1-isomorphic F1-homomorphic Field
for h being Isomorphism of F1,F2
for E1 being FieldExtension of F1, E2 be FieldExtension of F2
for a being Element of E1, b be Element of E2
for p being irreducible Element of the carrier of Polynom-Ring F1
st Ext_eval(p,a) = 0.E1 & Ext_eval((PolyHom h).p,b) = 0.E2
holds Psi(a,b,h,p).a = b
proof
let F1 be Field,
    F2 be F1-isomorphic F1-homomorphic Field;
let h be Isomorphism of F1,F2;
let E1 be FieldExtension of F1, E2 be FieldExtension of F2;
let a be Element of E1, b be Element of E2;
let p be irreducible Element of the carrier of Polynom-Ring F1;
assume AS: Ext_eval(p,a) = 0.E1 & Ext_eval((PolyHom h).p,b) = 0.E2;
   F1 is Subfield of E1 by FIELD_4:7; then
I1: 0.E1 = 0.F1 by EC_PF_1:def 1;
H1: X-(0.F1) = rpoly(1,0.F1) by FIELD_9:def 2
            .= rpoly(1,0.E1) by I1,FIELD_4:21
            .= X-(0.E1) by FIELD_9:def 2;
   F2 is Subfield of E2 by FIELD_4:7; then
I2: 0.E2 = 0.F2 by EC_PF_1:def 1;
H2: X-(0.F2) = rpoly(1,0.F2) by FIELD_9:def 2
            .= rpoly(1,0.E2) by I2,FIELD_4:21
            .= X-(0.E2) by FIELD_9:def 2;
A: (PolyHom h).(X-(0.F1)) = X-(h.(0.F1)) by lemNor3d .= X-(0.F2) by RING_2:6;
B: Ext_eval(X-(0.F1),a)
     = eval(X-(0.E1),a) by H1,FIELD_4:26
    .= eval(rpoly(1,0.E1),a) by FIELD_9:def 2
    .= a - 0.E1 by HURWITZ:29;
Psi(a,b,h,p).Ext_eval(X-(0.F1),a)
     = Ext_eval(X-(0.F2),b) by A,AS,FIELD_8:def 10
    .= eval(X-(0.E2),b) by H2,FIELD_4:26
    .= eval(rpoly(1,0.E2),b) by FIELD_9:def 2
    .= b - 0.E2 by HURWITZ:29;
hence thesis by B;
end;
