Characterization of Paracompactness in T3 Space/Lemma 6

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

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

Let $\VV$ be a closed locally finite refinement of $\UU$.

For all $x \in X$, let:
 * $W_x \in \tau: x \in W_x$ and $\set{V \in \VV : V \cap W \ne \O}$ is finite

Let $\WW = \set{W_x : x \in X}$ be an open cover of $T$.

Let $\AA$ be a closed locally finite refinement of $\WW$.

For each $V \in \VV$, let:
 * $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O}$

Let $\VV^* = \set{V^* : V \in \VV}$.

Then:
 * $\VV^*$ is an open locally finite cover of $T$