Intersections of vertical lines with non-vertical lines

We answer negatively a question: if $\mathcal{F}$ is a family of $n⩾3$ non-vertical, pairwise non-parallel lines on the plane and $\bigcap \mathcal{F}=\mathrm{\varnothing }$, is there a vertical line $L$ such that $L\cap \bigcup \mathcal{F}$ has exactly $n-1$ or $n-2$ points?