Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 2

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

Let $\VV = \ds \bigcup_{n = 0}^\infty \VV_n$ be an cover of $X$, where each $\VV_n$ is a set of subsets of $X$ for each $n \in \N$.

For each $n \in \N$, let:
 * $W_n = \ds \bigcup \VV_n$

Let:
 * $\WW = \set{W_n : n \in \N}$

For each $n \in \N$, let:
 * $A_n = W_n \setminus \ds \bigcup_{i < n} W_i$

Let:
 * $\AA = \set{A_n : n \in \N}$

Then:
 * $\AA$ is a locally finite refinement of $\WW$

$\AA$ is a Cover of $X$
Let $x \in X$.

By definition of cover of set:
 * $\exists n \in \N : x \in W_n$

From Well-Ordering Principle:
 * $\exists m \in \N : x \in W_m : \forall n < m : x \notin W_n$

By definition of set difference:
 * $\forall n < m : x \in W_m \setminus W_n$

By definition of set intersection:
 * $x \in \ds \bigcap_{i < m} \paren{W_m \setminus W_i}$

From De Morgan's Laws for Set Difference:
 * $\ds \bigcap_{i < m} \paren{W_m \setminus W_i} = W_m \setminus \paren{\bigcup_{i < m} W_i}$

Hence:
 * $x \in A_m$

It follows that $\AA$ is a cover of $X$.

$\AA$ is a Refinement of $\WW$
From Set Difference is Subset:
 * $\forall n \in \N : A_n \subseteq W_n$

It follows that $\AA$ is a refinement of $\WW$ by definition.

$\AA$ is Locally Finite
Let $x \in X$.

By definition of cover of set:
 * $\exists V \in \VV : x \in V$

By definition of union:
 * $\exists m \in \N : V \in \VV_m$

From Set is Subset of Union:
 * $V \subseteq W_m$

From Set is Subset of Union:
 * $\forall n > m : W_m \subseteq \ds \bigcup_{i < n} W_i$

From Set Difference with Subset is Superset of Set Difference:
 * $\forall n > m : A_n = W_n \setminus \ds \bigcup_{i < n} W_i \subseteq W_n \setminus W_m$

From Subset of Set Difference iff Disjoint Set:
 * $\forall n > m : A_n \cap W_m = \O$

From Subsets of Disjoint Sets are Disjoint:
 * $\forall n > m : A_n \cap V = \O$

Hence:
 * $\set{A_n \in \AA : A_n \cap V \ne \O} \subseteq \set{A_n \in \AA : n \le m}$

The set $\set{A_n \in \AA : n \le m}$ is finite.

From Subset of Finite Set is Finite:
 * $\set{A_n \in \AA : A_n \cap V \ne \O}$ is finite

It follows that $\AA$ is a locally finite refinement of $\WW$ by definition.