Corollary 3

Let $I$ be a finite set and for each $i \in I$, let $f_i : X \to Y_i$ be a proper mapping. If $X$ is Hausdorff, then the mapping $x \mapsto (f_i(x))$ of $X$ into $\prod_{i \in I} Y_i$ is proper.

This mapping is the composition of the product mapping $(x_i) \mapsto (f_i(x_i))$ of $X^I$ into $\prod_i Y_i$ and the diagonal mapping of $X$ into $X^I$; since the latter is proper (by Proposition 2 and § 8, no. 1, Proposition 1) the conclusion follows from Proposition 4 and Proposition 5 a).