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Let $(R,\m)$ be a Noetherian local ring, two ideals of , and an Artinian -module. Let be an integer and $r=\Width_{k}(I,A)$ the supremum of lengths of -cosequences in dimension in defined by Nhan-Hoang \cite{NhHo}. It is first shown that for each and each sequence which is an -cosequence in dimension , the set $$\big (\overset t{\bigcup\limits_{i=0}} \Att_R(0:_A(x_1^{n_1},\ldots, x_i^{n_i}))\big )_{\ge k}$$ is independent of the choice of . Let be the eventual value of $\Width_{k}(0:_AJ^n)$. Then our second result says that for each the set $(\overset t{\bigcup\limits_{i=0}}\Att_R(\Tor_i^R(R/I, (0:_AJ^n))))_{\ge k}$ is stable for large .