
theorem
for F being Field,
    E being FieldExtension of F
for K being E-extending FieldExtension of F
holds K is F-infinite or (E is F-finite & deg(E,F) <= deg(K,F))
proof
let F be Field, E be FieldExtension of F;
let K be E-extending FieldExtension of F;
set VK = VecSp(K,F), VE = VecSp(E,F);
now assume K is F-finite; then
  reconsider VK as finite-dimensional VectSp of F by FIELD_4:def 8;
  A: deg(K,F) = dim VK by FIELD_4:def 7;
  B: VE is Subspace of VK by sp; then
  C: dim VE <= dim VK by VECTSP_9:25;
  thus E is F-finite by B,FIELD_4:def 8;
  thus deg(E,F) <= deg(K,F) by A,C,FIELD_4:def 7;
  end;
hence thesis;
end;
