Bourbaki

Lemma 1

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

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

Any two non-parallel hyperplanes have non-empty intersection.

Let H and H′H' be two non-parallel hyperplanes, a∈Ha \in H and a′∈H′a' \in H'; there exist two hyperplanes M and M′M' of the vector space T such that H=M+aH = M + a and H′=M′+a′H' = M' + a'; since H and H′H' are not parallel, we have M≠M′M \neq M' and hence T=M+M′T = M + M'; hence there exist u∈Mu \in M and u′∈M′u' \in M' such that a′−a=u−u′a' - a = u - u', and the point u+a=u′+a′u + a = u' + a' belongs to H∩H′H \cap H'.

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

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Cites

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Transcribed from the printed volume. Not translated.