Proposition 12
Tag 0BIC, permanent. Cite it as [Bourbaki, Tag 0BIC].
General Topology, chapter III, § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups
The group G operates continuously on every homogeneous space .
Since the equivalence relation is open (no. 4, Lemma 2) this is a particular case of Proposition 11 of no. 4.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.