# Unable to think about a part of lemma in Chapter Fields and Galois Theory from Algebra by Hungerford

I am self studying Fields and Galois Theory from Algebra by Hungerford and I am unable to think about how to prove (iii) of lemma 2.6 given on Page 246 of the book.

Defination that could be useful -> H' is fixed field.

I tried firstly by simply using the definition and then assuming some x belongs to L and then trying to prove x belongs to L" .

But I am not able to successfully use these.

I think you may have not understood who $$L''$$ and $$H''$$ are. Let's look at $$L''$$ for example:
$$L$$ is an intermediate field, so $$L'=\{\sigma \in Aut_KF: \ \sigma(v)=v \ \forall v \in L\}$$ is now a subgroup of $$Aut_{K}F$$, and therefore $$L''=(L')'$$ is now of the form of $$H'$$ in part $$(i)$$ of Theorem 2.3 (and not of the form of $$E'$$ in part $$(ii)$$). Therefore:
$$L''=\{v \in F: \ \sigma(v)=v \ \forall \sigma \in L'\}$$
Now let $$l\in L$$. By definition, $$\sigma(l)=l \ \forall \sigma \in L'$$, so we conclude that $$l\in L''$$.