Proposition 3
Tag 0BHI, permanent. Cite it as [Bourbaki, Tag 0BHI].
General Topology, chapter III, § 1. Topologies on groups
Let G and G' be two topological groups, and let f be a homeomorphism of a neighbourhood V of the identity element of G onto a neighbourhood V' of the identity element of G', which satisfies condition 1) of Definition 2. Then f is an extension of a local isomorphism of G with G'.
For it is not hard to see that if W is a neighbourhood of the identity element of G such that W.W ⊂ V, then the restriction of f to W is a local isomorphism of G with G'.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.