Proposition 2

Let $X_1, X_2$ be two sets, let $Y$ be a uniform space and let $\mathscr{S}_i$ be a set of subsets of $X_i$ ($i = 1, 2$) and $\mathscr{S}_1 \times \mathscr{S}_2$ the set of subsets of $X_1 \times X_2$ of the form $A_1 \times A_2$, where $A_i \in \mathscr{S}_i, i = 1, 2$. Then the canonical bijection

(Set Theory, R, § 4, no. 14) is an isomorphism of the uniform space onto $\mathscr{F}_{\mathscr{S}_1}(X_1; \mathscr{F}_{\mathscr{S}_2}(X_2; Y))$.

Let $V$ be an entourage of $Y$ and let $A_i \in \mathscr{S}_i (i = 1, 2)$; then it follows immediately from the definitions that $W(A_1 \times A_2, V)$ is identified with $W(A_1, W(A_2, V))$ by the canonical bijection, and the result is immediate.