Corollary 2

If there is a continuous mapping $u : X \to A$ such that $u(x)$ is congruent to $x$ mod R for each $x \in X$, then $f(A) = X/R$ and the canonical mapping $h : A/R_A \to X/R$ is a homeomorphism.

Since each equivalence class mod R meets A, the canonical image of $A/R_A$ in $X/R$ is the whole of $X/R$; on the other hand, if U is open in A and is saturated with respect to $R_A$, it follows from the hypothesis that $\bar{u}^1(U)$ is the set obtained by saturating U with respect to R; since $u$ is continuous, $\bar{u}^1(U)$ is open in X (§ 2, no. 1, Theorem 1). The corollary follows from this fact by virtue of Proposition 10.