CWMO 2002 Problem 4

Let $n$ be a positive integer, let the sets $A_{1},A_{2},\cdots,A_{n + 1}$ be non-empty subsets of the set $\{1,2,\cdots,n\}.$ prove that there exist two disjoint non-empty subsets of the set $\{1,2,\cdots,n + 1\}$ : $\{i_{1},i_{2},\cdots,i_{k}\}$ and $\{j_{1},j_{2},\cdots,j_{m}\}$ such that $A_{i_{1}}\cup A_{i_{2}}\cup\cdots\cup A_{i_{k}} = A_{j_{1}}\cup A_{j_{2}}\cup\cdots\cup A_{j_{m}}$ .