so that the Levi extension theorem with analytic discs ([*cf.*]{} the survey [@mp2006a]) applies at each point of $\{ \mu = c\}$. In Section 3 of [@mp2006b], we defined $(n-a)$-concave Hartogs figures for $1 \leqslant a \leqslant
n-1$, but we used only $1$-concave ones, because the Levi-form of exterior of spheres $\{ \vert \!  \vert z \vert \! \vert < r\}$ in ${\mathbb{C}}^n$ had $(n-1)$ negative eigenvalues. Here, we start from $(n-1)$-concave Hartogs figures, we modify them similarly as in Section 3 of [@mp2006b] (details are skipped) and we call them [*$(n-1)$-concave Levi-Hartogs figures*]{}.

Next, we use a finite number of these figures, [*via*]{} some local charts of $X_{\rm reg}$, to cover $\{\mu = c\}$ and to show that holomorphic[^8] (or meromorphic) functions in $\{ \mu > c\}$ extend to a slightly deeper super-level set $\{ \mu > c - \eta\}$ (provided no critical point of $\mu$ or of $\mu \big \vert_M$ is encountered in the shell $\{ c 
\geqslant \mu > c - \eta \}$), for some $\eta > 0$ which depends on $X$, on $n$, on $\mu$, but not on $c$.

[**2)**]{} Contrary to the ${\mathbb{C}}^n$ case treated in [@mp2006b], $\mu$ may have critical points on $X_{\rm reg}$. 