w)$ is in some open neighborhood $\B_{3\epsilon}\times
U_1$ of $(z_0,  w_0)$. Let $  w_1 \in \U_1$, $  w_1 \not =  w_0$. Consider then $P=(z_0,  w_1)$. By the above reasoning, we find the existence of neighborhoods $\U_2$ and $U_2$ of $  w_1 $ and a function $  F_2$ defined on $\B_{3\epsilon} \times
U_2$, holomorphic as a function of $  w$ such that $$S(f)=F_2|_{\B_{3\epsilon} \times
\U_2}.$$

[**Claim:**]{} $ 
F_1=  F_2$ on $\B_{3\epsilon} \times \left [U_1 \cap U_2\right ]$.

[**Proof of the Claim.**]{} For $z$ fixed, $(\U_1 \cap \U_2) $ can be viewed as a totally real submanifold of $\C^{m+n}$. The function $G=
  F_1-  F_2$ extends holomorphically to $(U_1 \cap U_2)$ as a function of $  w$ and is null on $(\U_1 \cap \U_2)$, and is thus null where it is defined.

Using the claim, we conclude that the function $S(f)$ extends holomorphically as a function of $  w$ to a neighborhood $\UU$ of $\Omega$ in $\C^{m+n}$.

[**(B) Holomorphy in $z$**]{}

Let $  L_j \in   \L$. Since $S(f)$ extends holomorphically as a function of $  w$, we have $$\label{e4.2}
  L_j(S(f))=\left [{{\partial} \over {\partial \overline {z_j}}} 
S(f) \right ]_{\Omega}.$$

By proposition \[p4\], $  L_j(S(f))=S(  L_j(f))$ and since $f$ is CR and $S(0)=0$, we have 