Proof of hall's marriage theorem by Halmos

Question

This is one of the various proofs for Hall's theorem.

I have a question in Case 2 explanation. Let T $\subseteq$S. Then how can we conclude $N_{G}(S\cup T)$ $\subseteq$ $N_{G}(S)$$\cup$ $N_{Q}(T)$ What I dont understand is that some of the vertices in T can have corresponding vertices in N(S) and not necessarily only in Q.