Proposition 16
Tag 0AFT, permanent. Cite it as [Bourbaki, Tag 0AFT].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let G be a connected Lie group, X a Hausdorff manifold of class C^r and f_1, f_2 laws of left (resp. right) operation of class C^r of G on X. If the laws of infinitesimal operation associated with f_1 and f_2 are equal, then f_1 = f_2.
By § 4, no. 7, Corollary to Proposition 11, there exists a neighbourhood V of \{e\} \times X in G \times X such that f_1 and f_2 coincide on V. Hence f_1 = f_2 (Lemma 4).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.