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

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


 * $(6)\quad $every open cover of $T$ has an open $\sigma$-locally finite refinement
 * $(1)\quad T$ is paracompact

Proof

 * $(6)\quad $every open cover of $T$ has an open $\sigma$-locally finite refinement
 * $\ldots$
 * $(1)\quad T$ is paracompact