KöMaL 2008 A451

Let $\mathcal{U}\subset P(S)$ be an upset and let $\mathcal{D}\subset P(S)$ be a downset on a set $S$ of $n$ elements, i.e. ($a$) for every $X\in\mathcal{U}$ we have $Y\in\mathcal{U}$ for all $X \subset Y \subset S$; ($b$) for every $X\in\mathcal{D}$ and $Y \subset X$, $Y\in\mathcal{D}$ . Prove that $|\mathcal{U}| \cdot |\mathcal{D}| \ge 2^n \cdot |\mathcal{U}\cap\mathcal{D}|$.