Remark 2

Let $(g, x) \mapsto gx$ be a law of left operation of class $C^r$ of G on X. Let $t \in U_s(G)$ with $s \leq r$, $\Omega$ be an open subset of X and $f : \Omega \to F$ a function of class $C^r$. $t * f$ can also be defined by formula (4) or (5); it is a function defined on $\Omega$ with values in F, of class $C^{r-s}$ if $s < \infty$. The results of this no. extend in an obvious way to this situation.