in Lemma 15.1 Munkre shows:
Let $ K'$ be a subdivision of the complex $ K $. For each vertex $ w\in K'$ there is a vertex $ v\in K $, such that $ St (w, K')\subseteq St (v, K) $.
If $\sigma $ is a simplex in $ K $, such that $ w\in Int\sigma $, then the inclusion holds, iff $ v $ is a vertex of $\sigma $.
In the proof it is claimed that it suffices to show $|K|\setminus St (v, K)\subseteq |K| \setminus St (w, K') $ in order to prove the $\Leftarrow $ direction. Why does it suffices to show this?
The whole other part of the proof is clear.