Partitioning the Cartesian product of an arbitrarily partitionable graph and a complete graph

A graph G is arbitrarily partitionable (AP for short) if for every sequence (n_1, ..., n_p) of positive integers summing up to |V(G)| there exists a partition (V_1, ..., V_p) of V(G) such that G[V_i] is a connected graph on n_i vertices for every i in {1,...,p}. We show that the Cartesian product G x K_l is AP whenever G is AP and K_l is a complete graph on l >= 1 vertices.