Proposition 7 (Transitivity of quotient spaces)

Let $R$ and $S$ be two equivalence relations on a topological space $X$ such that $R$ implies $S$, and let $S/R$ be the quotient equivalence relation on the quotient space $X/R$ (Set Theory R, § 5, no. 9). Then the canonical bijection $(X/R)/(S/R) \to X/S$ is a homeomorphism.

This is a particular case of the transitivity of final topologies (§ 2, no. 4, Proposition 7. Cf. Set Theory, Chapter IV, § 2, no. 3, criterion CST 21).