Using the H-representation of polyhedra, one verifies that, if $\Lambda_{1}$ and $\Lambda_{2}$ are polyhedra with non-empty intersection, then any face of $\Lambda_{1}\cap\Lambda_{2}$ is the intersection of a face of $\Lambda_{1}$ with a face of $\Lambda_{2}$. This implies that $\Pi_{1}\cdot\Pi_{2}$ is a polyhedral complex.

Now suppose that $\Pi_{1}$ and $\Pi_{2}$ are complete. Let $\sigma\in\operatorname{rec}(\Pi_{1}\cdot\Pi_{2})$. This means that $\sigma=\operatorname{rec}(\Lambda)$ and $\Lambda=\Lambda_{1}\cap\Lambda_{2}$ with $\Lambda_{i}\in\Pi_{i}$. It is easy to verify that $\Lambda\not=\emptyset$ implies $\operatorname{rec}(\Lambda)=\operatorname{rec}(\Lambda_{1})\cap\operatorname{rec}(\Lambda_{2})$. Therefore $\sigma\in\operatorname{rec}(\Pi_{1})\cdot\operatorname{rec}(\Pi_{2})$. This shows

Since both complexes are complete, they agree. ∎