Here you have to prove that, for all frameworks of the type described in this question, no two paths from different starting points ever end up at the foot of the same upright. In other words we have to show that the system described always produces a permutation (a re-ordering) of the numbers 1 to n which occur at the top of the uprights. In this case the permutation is

Here is a hint to help you to prove this: imagine moving numbered counters down the paths at the same rate and every time a rung is encountered the two counters on adjacent uprights change places; this is called a transposition.