Remark
Tag 0C9C, permanent. Cite it as [Bourbaki, Tag 0C9C].
General Topology, chapter IX, § 6. Polish spaces, Souslin spaces, Borel sets
In particular a closed equivalence relation R satisfies the hypotheses of Theorem 4. When X is a compact metrizable space, Theorem 4 therefore applies to any Hausdorff equivalence relation R, since a Hausdorff equivalence relation on a compact space is closed (Chapter I, § 10, no. 4, Proposition 8).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.