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

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


 * $(4)\quad $every open cover of $T$ is even
 * $(5)\quad $every open cover of $T$ has an open $\sigma$-discrete refinement

Proof

 * $(4)\quad $every open cover of $T$ is even
 * $(5)\quad $every open cover of $T$ has an open $\sigma$-discrete refinement