Corollary 3

If $(f_i)$ is a family of proper mappings, then the product mapping $(x_i) \to (f_i(x_i))$ is proper.

In view of Theorem 1, this is just Lemma 2 above.

If we apply this corollary to the family of mappings $X_i \to P$ and use Corollary 1, we have Tychonoff’s theorem (§ 9, no 5, Theorem 3).