# How to prove $\operatorname{Gal}(K/LM)=\operatorname{Gal}(K/L) \cap \operatorname{Gal}K/M)$

If $K/F$ is splitting field, $M,L$ are intermediate fields, how to prove $\operatorname{Gal}(K/LM)=\operatorname{Gal}(K/L) \cap \operatorname{Gal}(K/M)?$

An automorphism is in $\newcommand{\Gal}{\text{Gal}}\Gal(K/LM)$ iff it fixes all elements of $LM$.
An automorphism is in $\Gal(K/L)\cap\Gal(K/M)$ iff it fixes all elements of $L$ and all elements of $M$.
Now why must an automorphism fixing all elements of $M$ and of $L$ also fix all elements of $LM$ and vice versa?