
theorem
for F being Field
for E being FieldExtension of F
for K being E-extending FieldExtension of F
st K is F-algebraic holds K is E-algebraic & E is F-algebraic
proof
let F be Field, E be FieldExtension of F, K be E-extending FieldExtension of F;
H0: E is Subring of K & F is Subring of E by FIELD_4:def 1; then
H1: 0.K = 0.E by C0SP1:def 3;
assume AS: K is F-algebraic;
now let a be Element of K;
  consider p being non zero Polynomial of F such that
  A: Ext_eval(p,a) = 0.K by AS,FIELD_6:43;
  reconsider p1 = p as Polynomial of E by H0,FIELD_4:9;
  0_.(E) = 0_.(F) & p <> 0_.(F) by FIELD_4:12; then
  reconsider p1 as non zero Polynomial of E by UPROOTS:def 5;
  p is Element of the carrier of Polynom-Ring F &
  p1 is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
  Ext_eval(p1,a) = Ext_eval(p,a) by lemma7b .= 0.E by A,H0,C0SP1:def 3;
  hence a is E-algebraic by H1,FIELD_6:43;
  end;
hence K is E-algebraic;
now let a be Element of E;
 :: @(a,K) is F-algebraic by AS; then
  consider p being non zero Polynomial of F such that
  A: Ext_eval(p,@(a,K)) = 0.K by AS,FIELD_6:43;
  p is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10; then
  Ext_eval(p,a) = Ext_eval(p,@(a,K)) by FIELD_6:11 .= 0.E by A,H0,C0SP1:def 3;
  hence a is F-algebraic by FIELD_6:43;
  end;
hence thesis;
end;
