
theorem fixp:
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 h being F-fixing Homomorphism of E,K holds (PolyHom h).p = p
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,
    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;
set g = (PolyHom h).p1;
H: dom g = NAT by FUNCT_2:def 1 .= dom p by FUNCT_2:def 1;
now let o be object;
   assume o in dom g; then
   reconsider i = o as Nat;
   g.i = h.(p1.i) by FIELD_1:def 2 .= p.i by FIELD_8:def 2;
   hence g.o = p.o;
   end;
hence thesis by H,FUNCT_1:2;
end;
