Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5

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

If every open cover of $T$ is even then:
 * every open cover of $T$ has an open $\sigma$-discrete refinement

Proof
Let every open cover of $T$ be even.


 * $\cdots$

It follows that:
 * every open cover of $T$ has an open $\sigma$-discrete refinement