Remark

Let G be a finite-dimensional Lie group, X a manifold of class C^r and (g, x) ↦ gx a law of left operation of class C^r of G on X. Then Proposition 14 remains valid. The only point which needs a different proof is the fact that G_x is a Lie subgroup. But if r ≠ ω, K = \mathbf{R}; as clearly G_x is closed, G_x is a Lie subgroup by § 8, Theorem 2.