of $(s_0,t_0)$ in $ \omega$, $K$ a compact subset of $\U$. To every pair of positive constants $C_1, C_2$, there is another pair of positive constants $C_1',C_2'$ with $C_2'$ independent of $C_1$ such that for any $\CC^{\infty}$ function $f$ in an open neighborhood of the closure of $\U$, if we have, for all $k\in \NN$ and all $z\in
\B_{3\epsilon}$, $$\label{e3.1}
\|\Delta_{\n\times\R^n} ^k f\|_{L^{\infty}(\U)} \leq C_1 C_2^k (2k)!,$$ then for every $z \in \B_{3\epsilon}$ and every $\alpha \in \NN^{n+m}$ $$\label{e3.2}
\|R^{\alpha} f\|_{L^{\infty}(K)}\leq C_1'{C_2'}^{|\alpha|}|\alpha|!$$

\[p4\] (Proposition II.4.1 in [@[Tre]]). Let $p \in \Omega$, in order that $f\in \CC^0$ be an analytic vector of the system of vector fields $\{R_1,...,R_{n+m}\}$ it is necessary and sufficient that there exists an open neighborhood $\V$ of $p$ in $\C^k \times \C^{n+m}$ and a continuous function $
F(z,w)$ in $\V$ holomorphic with respect to $w$ and such that $f(z,s,t)=F(z,  w(z,s,t))$.

The main difficulty in the proof of proposition \[p4\] is to show that the function defined by $$F(z,  w,s,t)=\sum_{\alpha \in \NN^{m+n}}{{R^{\alpha}f(z,s,t)} \over
{\alpha!}}
\left (  w-  w(z,s,t) \right )^{\alpha}$$ is equal to $f$ for $  w$ near $  w(z,s,t)$ in $\Omega$ if $f$ is an analytic vector of the vector fields $\{R_1,...,R_{n+m}\}$.

We shall use proposition \[p5\] 