
theorem ll:
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for h being F-fixing Monomorphism of E,K
for T being F-algebraic Subset of E holds h .: T is F-algebraic
proof
let F be Field, E be FieldExtension of F,
    K be F-extending FieldExtension of E;
let h be F-fixing Monomorphism of E,K, T be F-algebraic Subset of E;
now let a be Element of K;
  assume a in h.:T; 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 Element of E by A;
  consider p being non zero Polynomial of F such that
  B: Ext_eval(p,x) = 0.E by A,FIELD_6:43;
  p is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10; then
  Ext_eval(p,a) = h.(0.E) by A,B,fixeval .= 0.K by RING_2:6;
  hence a is F-algebraic by FIELD_6:43;
  end;
hence thesis by FIELD_7:def 12;
end;
