Proposition 24

Let G be a Lie group germ, X a manifold of class C^r and (X_i)_{i \in I} a locally finite open covering of X. For all i \in I, let $\psi_i$ be a law chunk of left operation of class C^r of G on X_i. Suppose that the underlying topological space of X is normal and that, 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). There exists a law chunk $\psi$ of left operation of class C^r of G on X such that, for all i \in I and all x \in X_i, $\psi_i$ and $\psi$ coincide on a neighbourhood of (e, x).

For all (i, j) \in I \times I and all x \in X_i \cap X_j, choose an open neighbourhood V_{ij}(x) of x in X_i \cap X_j such that $\psi_i$ and $\psi_j$ are defined and equal on a neighbourhood of {e} \times V_{ij}(x) in G \times X. For all x \in X choose an open neighbourhood V(x) of x in X such that conditions (a) and (b) of Lemma 3 are fulfilled. Let I_x be the set of i \in I such that x \in X_i. This is a finite set. Let U_x be the set of (g, y) \in G \times V(x) such that the $\psi_i$ for i \in I_x are defined and coincide on a neighbourhood of (g, y). Then U_x is open and (e, x) \in U_x. The $\psi_i$ for i \in I_x all have the same restriction to U_x. Let x, y be in X. If U_x and U_y meet, V(x) and V(y) meet and hence there exists i \in I such that

Then i \in I_x, i \in I_y, $\psi_i|U_x = \psi_x, \psi_i|U_y = \psi_y$ and hence

The $\psi_x$ therefore define a mapping $\psi$ of $U = \bigcup_{x \in X} U_x$ into X and U is an open neighbourhood of {e} \times X in G \times X. Clearly $\psi$ is of class C^r and $\psi(e, x) = x$ for all x \in X. For all i \in I and all x \in X_i, $\psi$ coincides with $\psi_x$ and hence with $\psi_i$ in a neighbourhood of (e, x) and hence $\psi$ satisfies condition (iii) of Definition 6.