The maximum sum of sizes of cross-intersecting families of subsets of a set

A set of sets is called a family. Two families and B are said to be cross-intersecting if each member intersects B. For any two integers n k with 1≤k≤n, let ([n]≤k) denote the family subsets [n]={1,…,n} that have at most elements. We show non-empty subfamily ([n]≤r), ([n]≤s), r≤s, cross-intersecting, then|A|+|B|≤1+∑i=1s((ni)−(n−ri)), equality holds A={[r]} in ([n]≤s) intersect [r].