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In this article, by using the theory of Gorenstein dimensions, it is shown that whenever $R$ is a homomorphic image of a Noetherian Gorenstein ring, then the invariants $\\inf\\{i\\in\\nat_0|\\, {\\dim\\Supp}(\\fb^tH_{\\fa}^i(M))\\geq n\\text{for all} t\\in\\nat_0\\}$ and $\\inf\\{\\lambda_{\\fa R_{\\p}}^{\\fb R_{\\p}}(M_{\\p})|\\,\\p\\in {\\rm Spec} \\, R \\text{and} \\dim R/ \\p\\geq n\\}$ are equal, for every finitely generated $R$-module $M$ and for every ideals $\\frak a, \\frak b$ of $R$ with $\\frak b\\subseteq \\frak a$. 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