Corollary
Tag 0BLI, permanent. Cite it as [Bourbaki, Tag 0BLI].
General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators
Let G be a Hausdorff topological group operating properly on a topological space X. If K is any compact subset of X and if F is any closed subset of G, then F.K is closed in X.
This follows from Proposition 12 and from Chapter I, § 10, no. 1, Proposition 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.