Let $V_1$, $V_2$ be two projective subspaces of $\mathbb{P}^n$ such that $V_1\cap V_2 \neq \emptyset$, and let $\Pi$ be an hyperplane of $\mathbb{P}^n$.
Is it true that the affine correspondence of $V_1$ and $V_2$ in $\mathbb{P}^n$ \ $\Pi$ are parallel iff $V_1\cap V_2 \subset$ $\Pi$?