  In \cite{Covolo:2016}, \cite{Covolo:2012} and \cite{Poncin:2016}, we
introduced the category of colored supermanifolds
($\mathbb{Z}_2^n$-super\-ma\-ni\-folds or just $\mathbb{Z}_2^n$-manifolds
($\mathbb{Z}_2^n=\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2$ ($n$ times))),
explicitly described the corresponding $\mathbb{Z}_2^n$-Berezinian and gave
first insights into $\mathbb{Z}_2^n$-integration theory. The present paper
contains a detailed account of parts of the $\mathbb{Z}_2^n$-differential
calculus and of the $\mathbb{Z}_2^n$-variants of the trilogy of local theorems,
which consists of the inverse function theorem, the implicit function theorem
and the constant rank theorem.
