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Let $k{\\ge}-1$ be an integer and $ r=\\depth_k(I,N)$ the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536). For a subset $S\\subseteq \\Spec R$ we set $S_{{\\ge}k}={\\p\\in S\\mid\\dim(R/\\p){\\ge}k}$. We first prove in this paper that $\\Ass_R(H^j_I(N))_{\\ge k}$ is a finite set for all $j{\\le}r$}. 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