Proposition 11

— Suppose the set $\mathscr{P}^*$ is a lattice, and let $(a_i)$, $(b_j)$ be two finite families of elements of A. Then every gcd of $\prod_i a_i$ and $\prod_j b_j$ divides the product $\prod_{i,j} \gcd(a_i, b_j)$.