Help to prove fact about Galois Theory?

Question

If E is extension field over L, and L is field extension over F how to prove Gal(E/L) is a subgroup of Gal(E/F)? Why is this?

Answer 1

To show that $G(E/L)$ is a subgroup of $G(E/F)$, note that any $L$-automorphism of $E$ is also an $F$-automorphism of $E$, since $F \subset L$.

Answer 2

Let $\sigma\in Gal(E/L)$. Then $\sigma$ is an automorphism of $E$ fixing the elements of $L$. Since $F\subset L$, $\sigma$ fixes the elements of $F$ hence $\sigma \in Gal(E/F)$.