Prove lemma 3.1 geometrically by showing that the following construc- tion establishes a one-to-one correspondence between the two classes of paths: Given a path to (2n, 0) denote its leftmost minimum point by M = (k, m). Reflect the section from the origin to M on the vertical line = k and slide the reflected section to the endpoint (2n, 0). If M is taken as origin of a new coordinate system the new path leads from the origin to (2n, 2m) and has all vertices strictly above or on the axis. (This construction is due to E. Nelson.)