field theory - Composite of Galois extensions is Galois, a particular proof.

This has been asked over and over again on math.stackexchange and I will ask it again.

Let $L_1/K$ and $L_2/K$ be finite Galois extensions of $K$ inside a common field, then $L_1L_2/K$ is a finite Galois extension.

I'm interested in one common proof of that fact. It goes like this:

$L_1L_2/K$ is finite so it suffices to prove that $L_1L_2$ is the splitting field of a separable polynomial over $K$. $L_i$ is the splitting field of a separable polynomial $f_i$ over $K$. Then $L_1L_2$ is the splitting field for the product of $f_1$ and $f_2$ with common factors only used once.

However, to me it seems that this only works when (the product of) common factors belong to $K[X]$ and I cannot think of a reason why this would be guaranteed (except e.g. when $L_1\cap L_2 = K$).What am I missing, or is this a bogus proof?