Characterization of Paracompactness in T3 Space/Lemma 3

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

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

Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denoted the closure of $V$ in $T$.

Let $\AA$ be a locally finite refinement of $\VV$.

Let $\BB = \set{A^- : A \in \AA}$ be locally finite.

Then:
 * $\BB$ is a refinement of $\UU$

Proof
Let $B \in \BB$.

By definition of $\BB$:
 * $\exists A \in \AA : A^- = B$

By definition of refinement:
 * $\exists V \in \VV : A \subseteq V$

From Set Closure Preserves Set Inclusion:
 * $B = A^- \subseteq V^-$

By definition of $\VV$:
 * $\exists U \in \UU : V^- \subseteq U$

From Subset Relation is Transitive:
 * $B \subseteq U$

It follows that $\BB$ is a refinement of $\UU$ by definition.