
theorem alg0:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F st K is F-finite
holds E is F-finite & deg(E,F) <= deg(K,F) &
      K is E-finite & deg(K,E) <= deg(K,F)
proof
let F be Field, E be FieldExtension of F, K be E-extending FieldExtension of F;
assume AS: K is F-finite;
hence E is F-finite & deg(E,F) <= deg(K,F) by FIELD_5:15;
set BF = the Basis of VecSp(K,F);
reconsider BF as finite Subset of VecSp(K,F) by AS;
H0: the carrier of VecSp(K,E)
      = the carrier of K by FIELD_4:def 6
     .= the carrier of VecSp(K,F) by FIELD_4:def 6; then
reconsider BE = BF as finite Subset of VecSp(K,E);
Lin BE = VecSp(K,E)
  proof
  H1: the carrier of Lin BE c= the carrier of VecSp(K,E) by VECTSP_4:def 2;
  the carrier of VecSp(K,E) c= the carrier of Lin BE
     proof
     now let o be object;
       assume o in the carrier of VecSp(K,E); then
       o in Lin BF by H0,VECTSP_7:def 3; then
       consider l being Linear_Combination of BF such that
       H2: o = Sum l by VECTSP_7:7;
       reconsider l1 = l as Linear_Combination of BE by sp1;
       o = Sum l1 by H2,sp2; then
       o in Lin BE by VECTSP_7:7;
       hence o in the carrier of Lin BE;
       end;
     hence thesis;
     end;
  hence thesis by H1,TARSKI:2,VECTSP_4:31;
  end; then
consider I being Subset of VecSp(K,E) such that
A: I c= BE & I is linearly-independent & Lin I = VecSp(K,E) by VECTSP_7:18;
reconsider I as finite Subset of VecSp(K,E) by A;
B: I is Basis of VecSp(K,E) by A,VECTSP_7:def 3;
D: VecSp(K,E) is finite-dimensional by A,VECTSP_7:def 3,MATRLIN:def 1;
hence K is E-finite by FIELD_4:def 8;
VecSp(K,F) is finite-dimensional by AS,FIELD_4:def 8; then
F: dim VecSp(K,E) = card I & dim VecSp(K,F) = card BE by B,D,VECTSP_9:def 1;
deg(K,E) = dim VecSp(K,E) & deg(K,F) = dim VecSp(K,F) by AS,D,FIELD_4:def 7;
hence deg(K,E) <= deg(K,F) by F,A,NAT_1:43;
end;
