Corollary 2

Two locally connected and simply connected topological groups which are locally isomorphic are isomorphic.

Let G and $G'$ be locally connected topological groups, let V and $V'$ be connected neighbourhoods of the identity element of G and of $G'$ respectively, and let $f: V\rightarrow V'$ be a homeomorphism which is a local isomorphism of G onto $G'$. By corollary 1, there exists a unique continuous group homomorphism $\varphi : G\rightarrow G'$ extending $f$ and a unique continuous group homomorphism $\varphi ': G'\rightarrow$ G extending $f^{-1}$. The homomorphisms $\varphi '\circ \varphi$ and Id$_G$ coincide on a neighbourhood of $e$ in G, hence are equal, since G is connected. Analogously, $\varphi \circ \varphi '=$ Id$_{G'}$, whence the corollary.