Corollary 1

Let $\Omega$ be an extension of a field $K$ and $L$ a subextension of $\Omega$ which is regular over $K$. Every subextension $M$ of $\Omega$ which is algebraically disjoint from $L$ over $K$ is linearly disjoint.

The ring $L \otimes_K M$ is an integral domain by definition of regular extension and it suffices now to apply Prop. 10.