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We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\\geq m+f([G:\\bigcap_{i=1}^kG_i])$, where $f(\\prod_{t=1}^r p_t^{\\alpha_t})=\\sum_{t=1}^r\\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\\alpha_1,...,\\alpha_r$ are nonnegative integers. 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Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\\geq m+f([G:\\bigcap_{i=1}^kG_i])$, where $f(\\prod_{t=1}^r p_t^{\\alpha_t})=\\sum_{t=1}^r\\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\\alpha_1,...,\\alpha_r$ are nonnegative integers. 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