\mu}_j$ still has $2$ eigenvalues $>0$ at every point.

In the next section, while applying the holomorphic extension procedure of [@mp2006b], we shall have to insure that the extensional domains attached to $M$ from either the outside or the inside [*cannot go beyond $\{ \rho \leqslant C\}$*]{}. So we have to prepare in advance the curvature of the limit hypersurface $\{ \rho =
C\} \cap X_{\rm reg}$.

Enlarging $C$ of an arbitrarily small increment if necessary, we can assume (thanks to Sard’s theorem) that $C$ is a regular value of $\rho
\big \vert_{ X_{\rm reg}}$, so that $$\Lambda
:=
\{\rho=C\}
\cap 
X_{\rm reg}$$ is a $\mathcal{ C }^\infty$ real hypersurface of $X_{\rm reg}$.

Lowering again $\varepsilon >0$ if necessary, the following holds[:]{}

-   [*At every point $q$ of the $\mathcal{ C}^\infty$ real hypersurface $\Lambda = \{ \rho = C \} \cap X_{\rm reg}$, one can find a complex line $E_q \subset T_q^c \Lambda$ on which the Levi-forms of both $\rho$ and $\mu$ are positive*]{}.

Here, $q \mapsto E_q$ might well be discontinuous, but this shall not cause any trouble in the sequel.

Each $p \in \{ \rho = C\}$ is contained in some $U_{ j ( p)}$, whence $\rho$ is 