Theorem 5

Let G be a simply connected Lie group, X a compact manifold of class C^r (r \geq 2) and a \mapsto D_a a law of left (resp. right) infinitesimal operation of class C^{r-1} of L(G) on X. There exists one and only one law of left (resp. right) operation of class C^{r-1} of G on X such that the associated law of infinitesimal operation is a \mapsto D_a.

The uniqueness follows from Proposition 16. By § 4, no. 7, Corollary 1 to Theorem 6, there exist a neighbourhood V of \{e\} \times X in G \times X and a law chunk of left (resp. right) operation of class C^{r-1} of G on X, defined on V, such that the associated law of infinitesimal operation is a \mapsto D_a. As X is compact, V can be assumed to be of the form U × X, where U is an open neighbourhood of e in G. It then suffices to apply Lemma 5.