Corollary 1

*Let $G$ be a Lie group germ and $X$ a paracompact manifold of class $C^r$ ($r \geq 2$). Let $a \mapsto D_a$ be a law of left infinitesimal operation of class $C^{r-1}$ of $L(G)$ on $X$.

(i) There exists a law chunk of left operation of class $C^{r-1}$ of $G$ on $X$ such that the associated law of infinitesimal operation is $a \mapsto D_a$.

(ii) Two laws of left operation of class $C^{r-1}$ of $G$ on $X$ which admit $a \mapsto D_a$ as associated law of infinitesimal operation coincide on a neighbourhood of $\{e\} \times X$.

Assertion (ii) follows from the Corollary to Proposition 11. By Theorem 6 (i), there exist an open covering $(X_i)_{i \in I}$ of $X$ and, for all $i \in I$, a law chunk of left operation $\psi_i$ of Class $C^{r-1}$ of $G$ on $X_i$ such that the associated law of infinitesimal operation is $a \mapsto D_a|_{X_i}$. As $X$ is paracompact, the covering $(X_i)_{i \in I}$ can be assumed to be locally finite. For all $(i, j) \in I \times I$ and all $x \in X_i \cap X_j$, $\psi_i$ and $\psi_j$ coincide on a neighbourhood of $(e, x)$ (Corollary to Proposition 11). As $X$ is normal, we can apply Proposition 24 of § 1, no. 11, which proves (i).