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.

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

Lemma 7

 * there exists a $\sigma$-discrete refinement $\AA$ of $\UU$

Lemma 8

 * there exists an open $\sigma$-discrete cover $\VV$ of $X$ such that $\AA$ is a precise refinement of $\VV$

From [User:Leigh.Samphier/Topology/Common Refinement Condition for Open Sigma-Discrete Refinement of Open Cover]:
 * there exists an open $\sigma$-discrete cover $\WW$ of $\UU$

Since $\UU$ was arbitrary, it follows that:
 * every open cover of $T$ has an open $\sigma$-discrete refinement