A direct proof of well-definedness for the polymatroid Tutte polynomial

For a polymatroid P over $[n]$, Bernardi, K\'{a}lm\'{a}n and Postnikov [Adv. Math. 402 (2022) 108355] introduced the polymatroid Tutte polynomial TP relying on the order 1<2<⋯<n of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that TP is equivalent to another polynomial that only depends on P. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.