Corollary 3

Let G be a simply connected topological group and let V be a connected neighbourhood of the identity element of G. One defines a presentation of G by taking as generator set the set V and as set $\mathbf{r}$ of relations the family of the $xyz^{-1}$, where $(x, y, z)$ runs through the triples of elements of V such that $xy=z$.

Let $F(V,\mathbf{r})$ be the quotient group of the free group F(V) by the smallest normal subgroup containing the elements $xyz^{-1}$, where $(x, y, z)\in V\times$ $V\times V$ and $xy=z$ (A, I, p. 86). Let us denote by $f: V\rightarrow F(V,\mathbf{r})$ and $g: F(V,\mathbf{r})\rightarrow G$ the canonical mappings. By construction, the mapping $f$ is a local homomorphism of G into $F(V,\mathbf{r})$. By corollary 1, there exists a unique group homomorphism $\overline{f}: G\rightarrow F(V,\mathbf{r})$ extending $f$. Since the group $F(V,\mathbf{r})$ is generated by $f(V)$, the homomorphism $\overline{f}$ is surjective. For $x\in V$, one has $g(\overline{f}(x)) =g(f(x)) =x$; since V generates G$,g\circ \overline{f}=$ Id$_G$, which proves that $\overline{f}$ is injective. It is therefore an isomorphism.