Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 3

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

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

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

For each $A \in \AA$, let $U_A \in \UU$ such that $A \subseteq U_A$.

For each $A \in \AA$, let:
 * $V_A = \paren{U_A \times U_A} \cup \paren{\paren{X \setminus A} \times \paren{X \setminus A}}$

For each $x \in X, A \in \AA$, let:
 * $V_A \sqbrk x =\set{ y∈X : \tuple{x,y} ∈ V_A}$

Then:
 * $\forall A \in \AA, x \in A : V_A \sqbrk x = U_A$

Proof
We have: