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

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

Let $\UU$ be an open cover of $T$.

Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$.

From Sigma-Locally Finite Cover has Locally Finite Refinement:
 * there exists a locally finite refinement $\AA$ of $\VV$

From Refinement of a Refinement is Refinement of Cover:
 * $\AA$ is a locally finite refinement of $\UU$

Since $\UU$ was arbitrary, it follows that:
 * every open cover of $T$ has a locally finite refinement