Let $L_1/K$ and $L_2/K$ be Galois extensions contained in a (bigger) field $M$ and let $F$ be the smallest subfield of $M$ containing both $L_1$ and $L_2$ (in other words, $F$ is the field generated by products of elements in $L_1$ with $L_2$). Prove that if $L_1 \cap L_2 = K$, then $|F:K| = |L_1:K||L_2:K|$.
If you wish, you can assume that $F$ is Galois and that "$\leq$" holds (I know how to prove both). In general I tried to use the Fundamental Theorem of Galois Theory in the setting that $L_1 \cap L_2 = K$ gives that the corresponding groups of $L_1, L_2$ generate $\Gamma(F:K)$, but this does not give anything about cardinalities.
Any help appreciated!