to construct analytic vectors of the vector fields $\{R_1,...,R_{n+m}\}$ and then apply proposition \[p4\] to these vectors to obtain a holomorphic extension in the variables $  w$.

Proof of Proposition \[Prop1\]
------------------------------

We have $$\label{e'5} 
T_0N=\C^k\times \R^m \times \{0\}\subset \C^k\times \C^m\times\C^n.$$ If $v$ is complex transversal to $N$ at the origin, then $v\in\{0\}
\times \{0\}  \times\C^n$. Therefore after a linear change of variables in $\C^{k+m+n}$ (viewed as a linear change of variables in $\C^n$ being the identity on $\C^{k+m}$ so that \[e4\] is unaffected) we may assume that $v$ has the following form $$\label{cpx} 
v=(0,0,v'),~~v'=(1,0,...,0) \in \R^n \times\{0\}\subset \C^n.$$

We first show that $f$ extends holomorphically to a neighborhood of $\Omega$.

[**Notation**]{} Given $P=(z,  w)$ a point in $\Omega$, $\U$ will denote a neighborhood of $  w$ in $\Omega$ and $U$ will denote a neighborhood of $  w$ in $\C^{m+n}$.

[**(A) Holomorphy in $  w$**]{}

By construction, we have $\Delta_{\n\times\R^n} S(f)=0$ on $\Omega$. Let $Q=(z_0,  w_0) \in \Omega$. By proposition \[p5\], $S(f)$ is an analytic vector for the vector fields $R_j$ on some open neighborhood $\U_1$ of $  w_0$. By proposition \[p4\], $S(f)$ extends holomorphically as a function of $  w$ to a function denoted by $  F_1$ when $(z,  