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We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $\\Omega \\subset {\\mathbb{C}}^n \\times {\\mathbb{R}}$, $n \\geq 2$, be a bounded domain with a connected real-analytic boundary such that $\\partial \\Omega$ has only nondegenerate CR singularities. 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