Let $\Lambda\in\Pi$ and $(m,l)\in M\times\mathbb{Z}$ such that $f(u)=\langle m,u\rangle+l$ for $u\in\Lambda$. Then, by the definition of $\operatorname{rec}(f)$, it is clear that $\operatorname{rec}(f)|_{\operatorname{rec}(\Lambda)}(u)=\langle m,u\rangle$. Hence, $\operatorname{rec}(f)$ is a conic H-lattice function on $\operatorname{rec}(\Pi)$. ∎