Let $R$ be a commutative Noetherian ring with non-zero identity, fa$ and fb$ ideals of $R$ with fb subseteq fa , and M a finitely generated $R$-module. In this paper, for a non-negative integer n , we show that prod_{i+j=n}(0:_R T{Ext}^i_R(R/ fa,H^j_ fb(M))) subseteq (0:_R T{Ext}^n_R(R/ fa,M)), where H^j_ fb(M) is the j th local cohomology module of M with respect to fb . This implies that there exists a positive integer ell , depending on M and fb , such that fb^ ell H^i_ fa(M)=0 for all i<f_ fb(M) and all ideals fa of R with fb subseteq fa where f_ fb(M) is the finiteness dimension of M relative to fb . Also, we obtain a new characterization of the concept of M -grade of an ideal of R .

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