Intersection properties of maximal directed cuts in digraphs

If $D$ is a finite digraph, a directed cut is a subset of arcs in $D$ having tail in some subset $X?V\left(D\right)$ and head in $V\left(D\right)?X$. In this paper we prove two general results concerning intersections between maximal paths, cycles and maximal directed cuts in $D$. As a direct consequence of these results, we deduce that there is a path, or a cycle, in $D$ that crosses each maximal directed cut.