The need for a certain property in a proof of a basic fact about Lie derivatives

Question

Below is a proof from Lee's Smooth Manifolds, 2nd ed.

I don't see where the fact that the set of regular points is open is used in the proof. It seems never to be invoked.

I'd appreciate it if someone could either confirm this or point to where that fact is used.

UPDATE: I don't believe that the answer is the first sentence of Case 1, as this is what Theorem 9.22 says.

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Answer 1

In the first sentence of case 1: a neighbourhood of $p$ such that... This neighbourhood must be a subset of $\mathcal{R} (V)$.