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Specifically, we prove the following strengthening of a result from \\cite{C}: For an exponentially closed ordinal $\\alpha$, we have $L_{\\alpha}\\models$ZF$^{-}$ if and only if COMP$^{\\text{ITRM}}_{\\alpha}=L_{\\alpha+1}\\cap\\mathfrak{P}(\\alpha)$, i.e. if and only if the set of $\\alpha$-ITRM-computable subsets of $\\alpha$ coincides with the set of subsets of $\\alpha$ in $L_{\\alpha+1}$. 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