Corollary 2

Let $f$ be a local isomorphism of $\mathbf{R}^n$ with a topological group $G$. Then there exists a unique strict morphism of $\mathbf{R}^n$ onto an open subgroup of $G$ which agrees with $f$ at all the points of some neighbourhood of $o$.

Let $\overline{f}$ be the continuous homomorphism of $\mathbf{R}^n$ into $G$ which agrees with $f$ at all points of a neighbourhood of $o$; $\overline{f}(\mathbf{R}^n)$ contains, by hypothesis, a neighbourhood of the identity element of $G$ and therefore (Chapter III, § 2, no. 1, Corollary to Proposition 4) is an open subgroup of $G$; moreover $\overline{f}$ is a strict morphism of $\mathbf{R}^n$ onto $\overline{f}(\mathbf{R}^n)$, by Chapter III, § 2, no. 8, Proposition 24.