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Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID such that $A\oplus B\cong C\oplus D$ and $A\oplus D\cong C\oplus B$. Prove that $A\cong C$ and $B\cong D$.

The only tool I have is the theorem about finitely generated modules, but I don't quite see the connection. Please Help. Thanks.

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