Characterization of Paracompactness in T3 Space/Lemma 6

Theorem
Let $T = \struct{X, \tau}$ be a topological Space such that every open cover of $T$ is even.

Let $T \times T = \struct{X \times X, \tau_{X \times X}}$ be the product space of $T$ with itself.

Let $U$ be a neighborhood of the diagonal $\Delta_S$ of $X \times X$ in $T \times T$.

Then there exists a neighborhood $V$ of the diagonal $\Delta_S$:
 * $(1) \quad V$ is symmetric, as a relation
 * $(2) \quad $ the composite $V \circ V$ is a subset of $U$