Characterization of Paracompactness in T3 Space/Statement 6 implies Statement 2

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

If every open cover of $T$ has an open $\sigma$-locally finite refinement then:
 * every open cover of $T$ has a locally finite refinement

Proof
Let every open cover of $T$ have an open $\sigma$-locally finite refinement.


 * $\cdots$

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