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.
Adding it's image->
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 $l\in L''$.
