Integer-weight shortest-path tie-breaking lemma ($w^{\prime }(e)=|V|w(e)+1$)

In a finite graph with positive integral edge weights, replacing $w(e)$ by $|V|w(e)+1$ makes every transformed shortest path minimise original weight first and edge count second. Positive weights imply that an optimal route is simple, with at most $|V|-1$ edges. Original integral path weights that differ have a gap of at least one; multiplication by $|V|$ outweighs every possible difference in simple-path edge counts. Among equal-weight paths, only the added edge count differs. Thus a single ordinary shortest path computation solves lexicographic weight-and-hop minimisation.