Corollary 3
Tag 0AEP, permanent. Cite it as [Bourbaki, Tag 0AEP].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let G be a Lie group and H_1 and H_2 integral subgroups of G whose topology admits a countable base. If H_1 and H_2 have the same underlying set, the Lie group structures on H_1 and H_2 are the same.
This follows from Corollary 2.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.