Amida

Can you prove that for every Almida framework, no two paths ever end up at the foot of the same upright? We have to show that the system described is a permutation (re-arrangement) of the numbers $1$ to $n$ which occur at the top of the uprights.

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.

This is a good way of recording the sequence transpositions
$12345$
$21345$
$21354$
$23154$
$23514$
$32514$
$32541$
$35241$
$35214$
$35124$
$53124$