Corollary 3
Tag 0BKX, permanent. Cite it as [Bourbaki, Tag 0BKX].
General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators
Let K be a quasi-compact normal subgroup of a topological group G, and let φ be the canonical mapping G → G/K. Then for each closed subgroup A of G the canonical bijection of A/A ∩ K onto φ(A) is an isomorphism of topological groups.
Since x⁻¹y ∈ K is a closed equivalence relation (Corollary 2), the Corollary follows from Chapter I, § 5, no. 2, Proposition 4.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.