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In this paper we prove that if ${\\mathcal B}$ is a non-separable family of balls of radii $r_1, r_2,\\ldots , r_n$ ($n\\geq 2$) with respect to an arbitrary norm in ${\\mathbb R}^d$ ($d\\geq 2$), then $\\bigcup {\\mathcal B}$ can be covered by a ball of radius $\\sum_{i=1}^n r_i$. 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