
theorem lemh:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for h being F-fixing Homomorphism of E,(K qua FieldExtension of E)
for T being finite F-algebraic Subset of E
st h.:T c= the carrier of E holds FAdj(F,h.:T) is Subfield of E
proof
let F be Field, E be FieldExtension of F,
    K be E-extending FieldExtension of F;
let h be F-fixing Homomorphism of E,(K qua FieldExtension of E);
let T be finite F-algebraic Subset of E;
assume AS: h.:T c= the carrier of E;
reconsider T1 = h.:T as finite Subset of E by AS;
now let a be Element of E;
  assume a in T1; then
  consider x being object such that
  A: x in dom h & x in T & a = h.x by FUNCT_1:def 6;
  reconsider x as F-algebraic Element of E by A;
  consider p being non zero Polynomial of F such that
  B: Ext_eval(p,x) = 0.E by FIELD_6:43;
  reconsider p as non zero Element of the carrier of Polynom-Ring F
     by POLYNOM3:def 10;
  C: E is Subfield of K by FIELD_4:7;
  Ext_eval(p,a) = Ext_eval(p,h.x) by A,FIELD_6:11
               .= h.(0.E) by B,fixeval
               .= 0.K by RING_2:6
               .= 0.E by C,EC_PF_1:def 1;
  hence a is F-algebraic by FIELD_6:43;
  end; then
reconsider T1 as finite F-algebraic Subset of E by FIELD_7:def 12;
FAdj(F,T1) = FAdj(F,h.:T) by lemh1;
hence thesis;
end;
