Characterization of Paracompactness in T3 Space/Statement 1 implies Statement 6

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

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

Proof
Let $T$ be paracompact.

By definition of paracompact:
 * every open cover of $T$ has an open locally finite refinement

From Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets
 * every open cover of $T$ has an open $\sigma$-locally finite refinement