How to Invoke the Galois Correspondence in the Proof of the Abstract Primitive Element Theorem

I was going through the proof of the Abstract Primitive Element Theorem and had minor concerns about how the Galois correspondence is invoked.

Abstract Primitive Element Theorem: Let $K$ be an infinite field and let $L/K$ be a finite separable extension. Then there exists $\theta\in\ L$ such that $L=K(\theta)$.

Note that $L/K$ is not assumed to be normal. Anyways, in the proof, Stewart says, "...Therefore, $\Gamma$(L/K) has only a finite number of subgroups. By the Galois correspondence, there are only finitely many intermediate fields $M$ with $K\subsetneq M\subsetneq L$"

The invoked bijection only works for normal field extensions. Does he implicitly assume that we are corresponding with the normal closure? I guess it would have the same effect because intermediate fields are still intermediate fields. Does he not mention so because it is a basic thing to do?