
theorem lemA:
for F being Field
for E being F-algebraic FieldExtension of F
for K being E-extending FieldExtension of F
for a being Element of K st a is E-algebraic holds a is F-algebraic
proof
let F be Field, E be F-algebraic FieldExtension of F;
let K be E-extending FieldExtension of F, a be Element of K;
assume a is E-algebraic; then
consider p being non zero Polynomial of E such that
A: Ext_eval(p,a) = 0.K by FIELD_6:43;
reconsider T = Coeff p as finite Subset of E;
for a be Element of E st a in T holds a is F-algebraic; then
reconsider T as finite F-algebraic Subset of E by FIELD_7:def 12;
set B = FAdj(F,T);
  E is FieldExtension of B by FIELD_4:7; then
  reconsider K1 = K as FieldExtension of B;
  E: K1 is B-extending & E is B-extending by FIELD_4:7;
  reconsider a1 = a as Element of K1;
  T is Subset of B by FIELD_6:35; then
  reconsider p1 = p as non zero Polynomial of B by E,FIELD_7:12;
  p1 is Element of the carrier of Polynom-Ring B &
  p is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
  Ext_eval(p1,a) = 0.K by A,E,FIELD_7:15;then
  reconsider a1 = a as B-algebraic Element of K1 by FIELD_6:43;
  reconsider B1 = FAdj(B,{a1}) as FieldExtension of F;
  F: {a} is Subset of FAdj(B,{a1}) by FIELD_6:35;
  G: K is B1-extending by FIELD_4:7;
  a in {a} by TARSKI:def 1;
  then a in FAdj(B,{a1}) by F;
hence a is F-algebraic by G,FIELD_7:33;
end;
