on $\n$. By hypothesis, $u=\sum_{j=1}^{K}
U_j$ where each $U_j$ is a CR distribution that extends holomorphically to a wedge $\W_j\subset
\C^{k+m}$ with edge $\n$. We thus apply proposition \[Prop1\] to each $U_j$ to obtain a holomorphic extension to wedges $\W'_j$ all in the direction $v$. Let $\W=\cap_{j=1}^K \W'_j$ we conclude that the function $\sum_{j=1}^{K}
U_j$ is holomorphic in $\W$ and $\sum_{j=1}^{K}U_j=u$ on $\n$. This concludes the proof of theorem \[Th2\]. $\blacksquare$

The remainder of this section will be devoted to the proof of proposition \[Prop1\].

Local Coordinates
-----------------

We begin with a choice of local coordinates on $\n$. $\n$ is a generic manifold in $\C^{k+m}$. We introduce local coordinates near $p_0$. We may choose a local embedding so that $p_0=0$ and $\n$ is parameterized in $\C^{k+m}=\C^k_z \times \C^m_{w'}$ by

$$\label{e1} 
\n=\{(z,w')\in \C^k \times \C^m:Im(w')=a(z,
Re(w')),~~ a(0)=da(0)=0\}.$$

We will denote by $s=Re(w') \in \R^m$, we thus have

$$\label{e2} 
\n=\{(z,s+ia(z,s)\}\subset \C^k \times \C^m,~~~ T_0\n=\C^k\times \R^m.$$

Define $\C T_p\n=T_p\n \otimes \C$ and $T^{0,1}_p\n=T^{0,1}_p\C^{k+m} \cap \C T_p\n$. We say that $\n$ is a CR manifold if dim$_{\C}T^{0,1}_p\n$ does not depend on $p$. The CR vector fields of $\n$ are vector fields $L$ on $\n$ such that for any $p\in \n$ we have $L_p \in 