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With some conditions on a regular local ring $(R,\\m)$ containing a field, and an ideal $I$ of $R$ with analytic spread $\\ell$ and a minimal reduction $J$, we prove that for all $w \\geq -1$, $ \\bar{I^{\\ell+w}} \\subseteq J^{w+1} \\mathfrak{a} (I,J),$ where $\\mathfrak{a}(I,J)$ is the coefficient ideal of $I$ relative to $J$, i.e. the largest ideal $\\mathfrak{b}$ such that $I\\mathfrak{b}=J\\mathfrak{b}$. 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