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This paper contains the following sharper version of Marstrand's theorem. Let $V \\subset \\mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. Then, for $\\mathcal{H}^{1}$ almost every $e \\in S^{2} \\cap V$, the projection $\\pi_{e}(K)$ has Hausdorff dimension $\\min\\{\\dim_{\\mathrm{H}} K,1\\}$. 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