Characterization of Paracompactness in T3 Space/Lemma 12

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

For each $V \in \VV$, let:
 * $U_V \in \UU : V \subseteq U_V$

Let:
 * $\UU^* = \set{V^* \cap U_V : V \in \VV}$

Then:
 * $\forall A \in \AA : \set{U^* \in \UU : U^* \cap A \ne \O}$ is finite

Lemma 9
Let $A \in \AA$.

From Lemma 4:
 * $\set{V \in \VV : V \cap A_0 \ne \O}$ is finite

Consider the surjection:
 * $f: \set{V \in \VV : V \cap A \ne \O} \to \set{V^* \cap U_V : V \in \VV : V \cap A \ne \O}$ defined by:
 * $\forall V \in \VV: \map f V = V^* \cap U_V$

From Cardinality of Surjection:
 * $\set{V^* \cap U_V : V \in \VV : V \cap A \ne \O}$ is finite

From Lemma 9:
 * $\set{V^* \cap U_V : V \in \VV : V^* \cap A \ne \O} \subseteq \set{V^* \cap U_V : V \in \VV : V \cap A \ne \O}$

From Subsets of Disjoint Sets are Disjoint:
 * $\set{V^* \cap U_V : V \in \VV : V^* \cap U_V \cap A \ne \O} \subseteq \set{V^* \cap U_V : V \in \VV : V^* \cap A \ne \O}$

From Subset of Finite Set is Finite:
 * $\set{V^* \cap U_V : V \in \VV : V^* \cap U_V \cap A \ne \O}$ is finite

That is:
 * $\set{U^* \in \UU : U^* \cap A \ne \O}$ is finite

Since $A$ was arbitrary, it follows that:
 * $\forall A \in \AA : \set{U^* \in \UU : U^* \cap A \ne \O}$ is finite