
theorem
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for T being finite F-algebraic Subset of K
st T c= the carrier of E holds FAdj(F,T) is Subfield of E
proof
let F be Field, E be FieldExtension of F,
    K be E-extending FieldExtension of F;
let T be finite F-algebraic Subset of K;
assume T c= the carrier of E; then
reconsider T1 = T as finite Subset of E;
now let a being Element of E;
  assume A: a in T1; then
  reconsider b = a as Element of K;
  consider p being non zero Polynomial of F such that
  B: Ext_eval(p,b) = 0.K by A,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,b) by FIELD_6:11
               .= 0.E by B,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,T) by lemh1;
hence thesis;
end;
