'Show that Dijkstra on the potential-modified graph is same as A*

Section 4 of this paper states that it is easy to see that A* is equivalent to dijkstra used on a modified graph where edge weights add the potential from the outgoing node and subtract the potential from the ingoing node. I can see that the priority queue would have the same order between the two algorithms since the additional terms telescope away, but I don't think that constitute a proof of the equivalence. Can someone enlighten me?

algorithmshortest-pathdijkstrapath-findinga-star


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