In this paper we prove that there holds SuppHo(K.)⊃…⊃SuppHn(K.), where Hi(K.) is the homology of the Koszul complex:K.=K.(x,M), for a sequence ot elements x=x1,…,xn of a commutative noetherian ring, and a finitely generated module M. This is a generalization of Vasconcelos' Lemma. Further, we give a proof of the non-negativity of partial Euler characteristics of the Koszul complex as an application of Vasconcelos' lemma.