  In the present paper we give some explicit proofs for folklore theorems on
holomorphic functions in several variables with values in a locally complete
locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on
vector-valued holomorphic functions is either devoted to the case of one
variable or to infinitely many variables whereas the case of (finitely many)
several variables is only touched or is subject to stronger restrictions on the
completeness of $E$ like sequential completeness. The main tool we use is
Cauchy's integral formula for derivatives for an $E$-valued holomorphic
function in several variables which we derive via Pettis-integration. This
allows us to generalise the known integral formula, where usually a
Riemann-integral is used, from sequentially complete $E$ to locally complete
$E$. Among the classical theorems for holomorphic functions in several
variables with values in a locally complete space $E$ we prove are the identity
theorem, Liouville's theorem, Riemann's removable singularities theorem and the
density of the polynomials in the $E$-valued polydisc algebra.
