Corollary 2

There exists an automorphism of $\mathbf{R}^n$ which maps G onto the group $G'$, isomorphic to $\mathbf{R}^p \times \mathbf{Z}^{r-p}$, which is the direct sum of the vector subspace generated by $e_1, e_2, \ldots, e_p$ and the (discrete) additive subgroup generated by $e_{p+1}, e_{p+2}, \ldots, e_r$.

This is an immediate consequence of Corollary 1. Corollary 2 of Theorem 2 shows that a closed subgroup G of $\mathbf{R}^n$ is completely determined up to isomorphism by two integers $\geq 0$: its rank, which we denote by $r(G)$, and the dimension of the largest vector subspace contained in G, which we call the dimension of G and denote by $d(G)$.

The only conditions which these integers have to satisfy are the inequalities $0 \leq d(G) \leq r(G) \leq n$.