Corollary 1

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

There exists a basis $(a_i)_{1 \leq i \leq n}$ of $\mathbf{R}^n$ , such that

a_i \in G \quad (1 \leq i \leq r), \qquad a_i \in V \quad (1 \leq i \leq p)

and such that G is the set of points

\sum_{i=1}^p t_i a_i + \sum_{j=p+1}^r n_j a_j

, where the

t_i

take all real values and the

n_j

take all integer values.

This follows from Theorem 2, and Theorem 1 of no. 1 applied to the discrete group $W \cap G$ .

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

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