
theorem
for F being Field
for E being FieldExtension of F
for K being FieldExtension of E holds F_Alg K is FieldExtension of F_Alg E
proof
let F be Field, E be FieldExtension of F, K be FieldExtension of E;
set FE = F_Alg E, FK = F_Alg K;
H0: F is Subring of E & E is Subring of K by FIELD_4:def 1;
H1: the carrier of FE = Alg_El E by d
                     .= the set of all a where a is F-algebraic Element of E;
H5: the carrier of FK = Alg_El K by d
                     .= the set of all a where a is E-algebraic Element of K;
    the carrier of FE c= the carrier of E
    proof
    now let o be object;
      assume o in the carrier of FE;
      then consider a being F-algebraic Element of E such that B: a = o by H1;
      thus o in the carrier of E by B;
      end;
    hence thesis;
    end; then
H4: [:the carrier of FE,the carrier of FE:] c=
         [:the carrier of E,the carrier of E:] by ZFMISC_1:96;
now let o be object;
  assume o in the carrier of FE; then
  consider a being F-algebraic Element of E such that B: a = o by H1;
  @(a,K) is E-algebraic;
  hence o in the carrier of FK by B,H5;
  end; then
A: the carrier of FE c= the carrier of FK; then
H3: [:the carrier of FE,the carrier of FE:] c=
         [:the carrier of FK,the carrier of FK:] by ZFMISC_1:96;
B: (the addF of FK)||the carrier of FE
 = ((the addF of K)||the carrier of FK)||the carrier of FE by d
.= (the addF of K)||the carrier of FE by H3,FUNCT_1:51
.= ((the addF of K)||the carrier of E)||the carrier of FE by H4,FUNCT_1:51
.= (the addF of E)||the carrier of FE by H0,C0SP1:def 3
.= the addF of FE by d;
C: (the multF of FK)||the carrier of FE
 = ((the multF of K)||the carrier of FK)||the carrier of FE by d
.= (the multF of K)||the carrier of FE by H3,FUNCT_1:51
.= ((the multF of K)||the carrier of E)||the carrier of FE by H4,FUNCT_1:51
.= (the multF of E)||the carrier of FE by H0,C0SP1:def 3
.= the multF of FE by d;
D: 1.FK = 1.K by d .= 1.E by H0,C0SP1:def 3 .= 1.FE by d;
   0.FK = 0.K by d .= 0.E by H0,C0SP1:def 3 .= 0.FE by d;
then FE is Subring of FK by A,B,C,D,C0SP1:def 3;
hence thesis by FIELD_4:def 1;
end;
