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Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module. For a non-negative integer n it is shown that, if the sets $As{s}_R\left(Ex{t}_R^n(R/I,M)\right)$ and $Sup{p}_R\left(Ex{t}_R^i(R/I,H_{I,J}^j\left(M\right))\right)$ are finite for all i ≤ n+1 and all j < n, then so is $As{s}_R\left(Ho{m}_R(R/I,H_{I,J}^n\left(M\right))\right)$. We also study the finiteness of $As{s}_R\left(Ex{t}_R^i(R/I,H_{I,J}^n\left(M\right))\right)$ for i = 1,2.