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Then adic reduction to the diagonal holds: $A\\otimes^{L}_{ A\\hat{\\otimes}_{K} A } ( M\\hat{\\otimes}^{L}_{K} N ) \\cong M \\otimes^{L}_A N$. A similar result is given in the case where $M,N$ are not necessarily finitely generated. 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Then adic reduction to the diagonal holds: $A\\otimes^{L}_{ A\\hat{\\otimes}_{K} A } ( M\\hat{\\otimes}^{L}_{K} N ) \\cong M \\otimes^{L}_A N$. A similar result is given in the case where $M,N$ are not necessarily finitely generated. 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