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:
 * $\map {V_A} x = \set {y \in X : \tuple {x, y} \in V_A}$

where:
 * $V_A$ is seen as a relation on $X \times X$
 * $\map {V_A} x$ denotes the image of $x$ under $V_A$.

Then:
 * $\forall A \in \AA, x \in A : \map {V_A} x = U_A$

Proof
We have: