A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121–138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150–153]. Let 2⩽t+1⩽k⩽2t+1 and n⩾(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |⋂F∈FF|