Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 2

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ Space.

If $T$ is paracompact then:
 * every open cover of $T$ has a locally finite refinement

Proof
Let $T$ be paracompact.

By definition of paracompact:
 * every open cover of $S$ has an open refinement which is locally finite.

By definition of open refinement:
 * every open refinement of a cover is a refinement of a cover.

It follows that:
 * every open cover of $T$ has a locally finite refinement.