Finite axiomatization of $\textbf{GL}\times\textbf{S5}$ and $\textbf{Grz}\times\textbf{S5}$

Abstract: We prove that $\mathbf{GL} \times \mathbf{S5}$ is product matching, and that $\mathbf{Grz} \times \mathbf{S5}$ is axiomatizable by adding to $[\mathbf{Grz},\mathbf{S5}]$ the Gödel translation of the monadic Casari formula. This settles the question of finite axiomatization these logics.