Proposition 23

Let $(G, e, \theta, m)$ be a Lie group germ, $X$ a manifold of class $C^r$, $x_0$ a point of $X$, $\Omega$ an open neighbourhood of $(e, x_0)$ in $G \times X$ and $\psi$ a mapping of $\Omega$ into $X$ with the following properties: (i) $\psi$ is of class $C^r$; (ii) $\psi(e, x)$ is equal to $x$ for $x$ sufficiently close to $x_0$; (iii) $\psi(m(g, g'), x) = \psi(g, \psi(g', x))$ for $(g, g', x)$ sufficiently close to $(e, e, x_0)$. Then there exist an open neighbourhood $X'$ of $x_0$ in $X$ and an open subset $\Omega'$ of $\Omega \cap (G \times X')$ such that $\psi|_{\Omega'}$ is a law chunk of left operation of class $C^r$ of $G$ on $X'$. There exist an open neighbourhood $X'$ of $x_0$ in $X$ and an open neighbourhood $G'$ of $e$ in $G$ such that $\psi(e, x) = x$ for all $x \in X$, and

for $(g, g', x) \in G' \times G' \times X'$. Let $\Omega'$ be the set of $(g, x) \in \Omega \cap (G' \times X')$ such that $\psi(g, x) \in X'$. Then $\Omega'$ is open in $G \times X'$ and $X'$, $\Omega'$ have the properties of the proposition.