Proving that $\mathrm{Gal}(E/FL) = \mathrm{Gal}(E/F) \cap \mathrm{Gal}(E/L)$

Question

I am trying to apply a proof related to this discussion: $\text{Gal}(FL|L) \simeq \text{Gal}(F|F\cap L)$ I have managed to prove the following:
$ \mathrm{Gal}(E/(F \cap L)) = \mathrm{Gal}(E/F) \cdot \mathrm{Gal}(E/L). $ Here is a sketch of the proof:

1. Let $\sigma \in \mathrm{Gal}(E/F) \cdot \mathrm{Gal}(E/L)$. Then $\sigma = \sigma_1 \sigma_2$, where $\sigma_1 \in \mathrm{Gal}(E/F)$ and $\sigma_2 \in \mathrm{Gal}(E/L)$.
2. For any $x \in F \cap L$, we have $\sigma_1(x) = x$ (since $\sigma_1$ fixes $F$) and $\sigma_2(x) = x$ (since $\sigma_2$ fixes $L$). Hence, $\sigma(x) = \sigma_1(\sigma_2(x)) = x$, which implies $\sigma \in \mathrm{Gal}(E/(F \cap L))$.
3. This shows $\mathrm{Gal}(E/F) \cdot \mathrm{Gal}(E/L) \subseteq \mathrm{Gal}(E/(F \cap L))$. Using the Fundamental Theorem of Galois Theory and some intermediate steps, I proved the reverse inclusion.

However, I am struggling to prove the second part:
$ \mathrm{Gal}(E/FL) = \mathrm{Gal}(E/F) \cap \mathrm{Gal}(E/L). $

Any help on this would be greatly appreciated. Additionally, as a beginner in Galois theory, I would love recommendations for textbooks or resources that make the subject approachable, preferably ones that would help me prepare for my exam.

Thank you in advance.