Corollary
Tag 0A8I, permanent. Cite it as [Bourbaki, Tag 0A8I].
Lie Groups and Lie Algebras, chapter III, § 1. Lie groups
Preserving the notation of Proposition 22, let G be the subgroup of H generated by U. Then G is an open subgroup of N. There exists one and only one Lie group structure on G such that U is an open submanifold of G and the canonical injection of G into H is an immersion.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.