Proposition 1

Let $G$ be a discrete subgroup of $\mathbf{R}^n$ of rank $p$, let $(a_i)_{1 \leq i \leq p}$ be a free system of $p$ points of $G$, and let $P$ be the closed parallelo-tope with centre $o$ and basis vectors $a_i$ (Chapter VI, § 1, no. 3). Then the set $G \cap P$ is finite and generates $G$, and every point of $G$ is a linear combination of the $a_i$ with rational coefficients.

$G \cap P$ is compact and discrete, hence finite. Let $x$ be any point of $G$; it is equal to a linear combination $\sum_{i=1}^p t_i a_i$ of the $a_i$ with real coefficients. For each integer $m > 0$, consider the point

it belongs to $G$, and since $0 \leq mt_i - [mt_i] < 1$, it lies in $P$. Hence, first, $x = z_1 + \sum_{i=1}^p [t_i] a_i$, so that $G$ is generated by $G \cap P$; and secondly, since $G \cap P$ is finite, there exist two distinct integers $h, k$ such that $z_h = z_k$, so that $(h - k)t_i = [ht_i] - [kt_i]$ less ($1 \leq i \leq p$), and therefore the $t_i$ are rational numbers.