Corollary

Tag 0A31, permanent. Cite it as [Bourbaki, Tag 0A31].

Lie Groups and Lie Algebras, chapter II, § 2. Free Lie algebras

For every family $(S_i)_{i \in I}$ of subsets of X,

L\left(\bigcap_{i \in I} S_i\right) = \bigcap_{i \in I} L(S_i).

This follows from Proposition 4 and the obvious formula

\mathbf{N}^{(S)} = \bigcap_{i \in I} \mathbf{N}^{(S_i)}

where we have written

S = \bigcap_{i \in I} S_i

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

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