I'm following definitions in Lang's graduate Algebra. Suppose that $E/L$ and $F/L$ are field extensions, assumed to exist in some common big field extension $\Omega.$ We say that $E$ is linearly disjoint from $F$ over $L$ iff any set of elements of $E$ which are linearly independent over $L$ are also linearly independent over $F$. Similarly, we say $E$ is free from $F$ over $L$ iff any set of elements of $E$ which is algebraically independent over $L$ is algebraically independent over $F$.
My question specifically relates to under what conditions linearly disjointness/freeness transfers to algebraic closures of the fields. Specifically, I've seen the following used: assuming $E/L$ and $F/L$ are free over $L$ and both extensions are regular, then $\overline{E}$ is linearly disjoint from $\overline{F}$ over $\overline{L}$. This seems reasonable but the proof eludes me and I cannot find it in the literature.
NB. I take $\overline{E}$ to mean algebraic closure or separable closure.