ILL 1984 Problem 29

Let $S_n = \{1, \cdots, n\}$ and let $f$ be a function that maps every subset of $S_n$ into a positive real number and satisfies the following condition: For all $A \subseteq S_n$ and $x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)$ . Prove that for all $A,B \subseteq S_n$ the following inequality holds: $$f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)$$