Definition 1

Tag 0AP2, permanent. Cite it as [Bourbaki, Tag 0AP2].

Lie Groups and Lie Algebras, chapter V, § 1. Hyperplanes, chambers and facets

A facet of E relative to $\mathscr{H}$ is an equivalence class of the equivalence relation defined above.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Nothing.