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I'm sure there's a rule about this, but I couldn't remember nor find it.
However, it's quite simple; "That book is mine" = "Den bog er min."
There aren't really any words for 'mine', other than 'min/mit/mine'.
You can re-phrase the sentence; "Den bog er min." = "Det er min bog." ("That is my book.")
In most cases, I would say the latter sounds more pleasant and is the best option to use.
So, re-arrange the words in the sentence; "That book is mine." = "Det er min bog."
Btw - glad you caught the connection between 'en/et' and 'min/mit'.
