Corollary 1

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

Algebra, chapter IX, § 4. Sous-espaces totalement isotropes. Théorème de Witt

If F is a totally isotropic (resp. totally singular) subspace of dimension r, there exists a totally isotropic (resp. totally singular) subspace F' of dimension r such that $F \cap F' = \{0\}$ and $F + F'$ is non-isotropic.

It suffices to take $G = \{0\}$ in prop. 2, b).

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

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