Proposition 1
Tag 0C5H, permanent. Cite it as [Bourbaki, Tag 0C5H].
General Topology, chapter IX, § 4. Normal spaces
A compact space is normal.
A compact space satisfies axiom , by Proposition 2 of Chapter I, § 9, no. 2.
As to locally compact spaces, every point of such a space has a compact neighbourhood, which is a normal subspace; but there are examples of locally compact spaces which are not normal (cf. Exercises 9 and 26, and § 5, Exercise 15).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.