Lemma 5

Let G be a simply connected topological group, X a Hausdorff topological space, U an open neighbourhood of e in G and \psi a continuous mapping of U \times X into X such that \psi(e, x) = x and \psi(s, \psi(t, x)) = \psi(st, x) for all x \in X and s, t in U such that st \in U. Let W be a symmetric connected open neighbourhood of e such that W^3 \subset U. There exists one and only one law of continuous left operation \psi' of G on X such that \psi' and \psi coincide on W \times X. If G is a Lie group and X a manifold of class C^r and \psi is of class C^r, then \psi' is of class C^r.

The uniqueness on \psi follows from Lemma 4. Let P be the permutation group of X. For u \in W^3, the mapping x \mapsto \psi(u, x) is an element f(u) of P and

for u_1, u_2, u_3 in W. Applying Lemma 1 of no. 1, we obtain a morphism f' of G into P which extends f|W. Let \psi'(g, x) = f'(g)(x) for all (g, x) \in G \times X. Then \psi' is a law of left operation of G on X which coincides with \psi on W \times X. As \psi'(g, \psi'(g', x)) = \psi'(gg', x) for (g, g', x) \in G \times G \times X, the continuity of \psi on W \times X implies the continuity of \psi' on gW \times X for all g \in G. Hence \psi' is continuous. If \psi is of class C^r, it is seen similarly that \psi' is of class C^r.