Let $D\subseteq \mathbb{C}^{n}$ be a domain.
We say that $D$ is a domain of holomorphy if, for every domain $V\subseteq \mathbb{C}^{n}$ for which $V\cap \partial D\not=\emptyset$ and for every connected component $W$ of $D\cap V$, there exists $f\in\mathcal{O}(D)$ such that $f|_{W}$ cannot be extended to a holomorphic function on $V$.
Proposition: $D$ is a domain of holomorphy if and only if the following holds: if $\emptyset\not=V\subseteq U\subseteq \mathbb{C}^{n}$ are domains with the properties
[H1] $V\subseteq D$,
[H2] For every $f\in\mathcal{O}(D)$ there exists $F\in\mathcal{O}(U)$ such that $\left.f\right|_V =\left.F\right|_V$,
then $U\subseteq D$.
Will anyone have any idea to show this equivalence with respect to domains of holomorphy?
Thanks