We will prove the statement for lattice functions. The statement for rational piecewise affine functions is proved with the same argument. If $f$ is an H-lattice function, we can write $f=g-h$, where $g$ and $h$ are H-lattice concave functions. We obtain $\Pi$ as any common refinement of $\Pi(g)$ and $\Pi(h)$ to a polyhedral complex. Then the statement follows from the definition of H-lattice concave functions. The converse is an easy consequence of Corollary 3.84. ∎