Lemma 5.4.2. Let $(X, \fU)$ be a uniform space, $V \in \fU$, and $A, B \subset X$ such that:
$A, B$ are $V$-small.
$A \cap B \ne \emptyset$.
then $A \cup B$ is $V \circ V$-small.
Proof. Since $V \circ V \supset V$, it is sufficient to consider the case where $x \in A$ and $y \in B$. By assumption (2), there exists $z \in A \cap B$. By assumption (1), $(x, z), (z, y) \in V$, so $(x, y) \in V \circ V$.$\square$