
theorem fixeval:
for F being Field,
    E being FieldExtension of F, K being FieldExtension of E
for p being Element of the carrier of Polynom-Ring F
for a being Element of E, h being F-fixing Homomorphism of E,K
holds h.(Ext_eval(p,a)) = Ext_eval(p,h.a)
proof
let F be Field, E be FieldExtension of F, K be FieldExtension of E;
let p be Element of the carrier of Polynom-Ring F;
let a be Element of E, h be F-fixing Homomorphism of E,K;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E by FIELD_4:10;
then reconsider p1 = p as Element of the carrier of Polynom-Ring E;
reconsider p2 = (PolyHom h).p as Element of the carrier of Polynom-Ring F
  by fixp;
thus h.(Ext_eval(p,a)) = h.eval(p1,a) by FIELD_4:26
                      .= eval((PolyHom h).p1,h.a) by FIELD_1:27
                      .= Ext_eval(p2,h.a) by FIELD_4:26
                      .= Ext_eval(p,h.a) by fixp;
end;
