A proof of the Multiplicative 1-2-3 Conjecture

We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kazi{\'o}w in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1,2,3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.