Let $M$ be a matroid on a finite ground set $E$, with rank function $r: 2^E \to \mathbb{N} \cup \{0\}$. Let $M^*$ denote the dual matroid of $M$, and let $r^*: 2^E \to \mathbb{N} \cup \{0\}$ be the rank function of $M^*$. Then, for every subset $A \subset E$,