Theorem 1

Every connected group $G$ which is locally isomorphic to $\mathbf{R}^n$ is isomorphic to a group $\mathbf{R}^p \times \mathbf{T}^{n-p}$ ($0 \leqslant p \leqslant n$).

A suitably chosen local isomorphism $f$ of $\mathbf{R}^n$ with $G$ extends to a strict morphism of $\mathbf{R}^n$ onto an open subgroup of $G$ (Proposition 2, Corollary 2) and therefore onto $G$ itself, since $G$ is connected. It follows that $G$ is isomorphic to a quotient group $\mathbf{R}^n/H$ of $\mathbf{R}^n$: $H$ is discrete, otherwise there would exist points $x \neq o$ of $H$ arbitrarily close to $o$ and such that $f(x) = f(o)$, contrary to the hypothesis that $f$ is a local isomorphism. The theorem is therefore a consequence of Theorem 1 of § 1, no. 1.