Theorem 2

Every continuous mapping $f$ of a compact space $X$ into a uniform space $X'$ is uniformly continuous.

Let $g = f \times f$, then $g$ is continuous on $X \times X$ (Chapter I, § 4, no. 1, Proposition 1, Corollary 1); hence for each open entourage $V'$ of $X'$, $\overline{g^{-1}}(V')$ is an open subset of $X \times X$, which evidently contains the diagonal. The theorem thus follows from Theorem 1, since the open entourages of $X'$ form a fundamental system of entourages (§ 1, no. 2, Proposition 2, Corollary 2).

Under the hypotheses of Theorem 2, the restriction of $f$ to any subspace $A$ of $X$ is uniformly continuous; hence (§ 3, no. 6, Theorem 2):