An isolated puncture is removable in complex dimension at least two

By [L1], choose radii ${\rho }_1>\rho >0$ and $\tau >0$ with ${\mathrm{\Delta }}_{{\rho }_1}(a_1)\times {\mathrm{\Delta }}_{\tau }(a^{\mathrm{\prime }})\subseteq \mathrm{\Omega }$. For $(z_1,z^{\mathrm{\prime }})$ with $\mathrm{\mid }{z}_1-a_1\mathrm{\mid }<\rho$ and $\mathrm{\mid }{z}^{\mathrm{\prime }}-a^{\mathrm{\prime }}\mathrm{\mid }<\tau$, define $F(z_1,z^{\mathrm{\prime }}):=\frac{1}{2\pi i}{\int }_{\mathrm{\mid }\zeta -a_1\mathrm{\mid }=\rho }\frac{f(\zeta ,z^{\mathrm{\prime }})}{\zeta -z_1}d\zeta \mathrm{.}$ When $\mathrm{\mid }\zeta -a_1\mathrm{\mid }=\rho$ and $\mathrm{\mid }{z}^{\mathrm{\prime }}-a^{\mathrm{\prime }}\mathrm{\mid }<\tau$, the point $(\zeta ,z^{\mathrm{\prime }})$ lies in ${\mathrm{\Delta }}_{{\rho }_1}(a_1)\times {\mathrm{\Delta }}_{\tau }(a^{\mathrm{\prime }})\setminus \{a\}$, so the integrand is well defined and continuous there.

Fix all variables except one. If the free variable is $z_1$, the Cauchy kernel makes $z_1\mapsto F(z_1,z^{\mathrm{\prime }})$ holomorphic on $\mathrm{\mid }{z}_1-a_1\mathrm{\mid }<\rho$. If the free variable is one coordinate of $z^{\mathrm{\prime }}$, then [L2] applies to that single complex parameter. So $F$ is separately holomorphic on ${\mathrm{\Delta }}_{\rho }(a_1)\times {\mathrm{\Delta }}_{\tau }(a^{\mathrm{\prime }})$. The same integral formula gives local bounds on compact subsets, so [L5] upgrades $F$ to a jointly holomorphic function there.

If $z^{\mathrm{\prime }}\mathrm{\ne }{a}^{\mathrm{\prime }}$, then the slice $\zeta \mapsto f(\zeta ,z^{\mathrm{\prime }})$ is holomorphic on the full disc $\mathrm{\mid }\zeta -a_1\mathrm{\mid }<\rho$, because the deleted point $a$ does not lie on that slice. Hence [L3] gives $F(z_1,z^{\mathrm{\prime }})=f(z_1,z^{\mathrm{\prime }})$ for every $\mathrm{\mid }{z}_1-a_1\mathrm{\mid }<\rho$.

The set $\{(z_1,z^{\mathrm{\prime }})\in P:z^{\mathrm{\prime }}\mathrm{\ne }{a}^{\mathrm{\prime }}\}$ is a nonempty open subset of $P\setminus \{a\}$, and step 2.2 shows that $F$ and $f$ agree there. Both are holomorphic on the connected punctured polydisc $P\setminus \{a\}$, so [L4] forces $F=f$ on all of $P\setminus \{a\}$.

Thus $F$ is a holomorphic extension of $f$ across $a$ on the neighborhood ${\mathrm{\Delta }}_{\rho }(a_1)\times {\mathrm{\Delta }}_{\tau }(a^{\mathrm{\prime }})$. Repeating the same construction at each puncture point gives a local extension, and uniqueness on overlaps again follows from [L4]. Therefore the local extensions glue to a unique holomorphic function on all of $\mathrm{\Omega }$.