$(M\cap E)(M\cap F)= M$ for linearly disjoint fields $E$ and $F$?

Question

Let $E$ and $F$ be linearly disjoint fields over a base field $K$ (all contained in an algebraic closure $\overline K$). Suppose there is an extension $M/K$ contained in $EF$.
Is it true that $(M\cap E)(M\cap F)= M$?
I was not able to show it. Does anybode have an idea?

Answer 1

It is not true, see mihaild's comment.