Let count pairs of people in who know each other. I need to verify QI: for . Consider any pair with . Count how many times it appears on each side:
Both in : counted twice on both sides. Equal.
One in , one in : left has 1 (via ), right has 2 (via both). Right wins.
One in , one in : left has 0, right has 1 (via ). Right wins. Every pair contributes at least as much to the right side. QI holds.