Mixed Volume Is Symmetric

Prove that mixed volume is symmetric: For any polytopes $P_1,P_2,\dots,P_n \subset \mathbb{R}^n$ and permutation $\sigma$ of $\{1,2,\dots,n\}$, $$\text{MV}(P_1,P_2,\dots,P_n) = \text{MV}(P_{\sigma(1)}, P_{\sigma(2)}, \dots, P_{\sigma(n)}).$$

Solution: