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Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic M\\\"{o}bius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. 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Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic M\\\"{o}bius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. 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