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 $T$

$\VV^*$ is a Set of Open Subsets
Let $V^* \in \VV^*$ for some $V \in \VV$.

Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.

By definition of subset:
 * $\AA_V \subseteq \AA$

From Subset of Locally Finite Set of Subsets is Locally Finite:
 * $\AA_V$ is closed locally finite

From Union of Closed Locally Finite Set of Subsets is Closed:
 * $\bigcup \set{A \in \AA | A \cap V = \O}$ is closed in $T$

By definition of closed set:
 * $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O} \in \tau$

$\VV^*$ is a Cover
Let $x \in X$.

By definiton of a cover:
 * $\exists V \in \VV : x \in V$

From Subset of Set Difference iff Disjoint Set:
 * $\forall A \in \AA_V : V \subseteq X \setminus A$

We have:

By definition of subset:
 * $x \in V^*$

Since $x$ was arbitrary, it follows that $\VV^*$ is a cover by definition.