I have been looking for a proof of the following theorem without much success. Does someone know the proof outline for this theorem?Let $r$ be a relation with attribute set $R$. Let $R_{1}$ and $R_{2}$ be subsets of $R$ such that $R_{1} \cup R_{2}  = R$. Then$ r = \pi_{R_{1}}(r) \bowtie \pi_{R_{2}}(r)$if and only if at least one of the following functional dependencies holds:$ R_{1} \cap R_{2} \rightarrow R_{1}$$ R_{1} \cap R_{2} \rightarrow R_{2}$I'm trying to prove both directions of this theorem, but am not sure where to start or where I may find an accessible proof. My book (DBMS by Ramakrishnan) states the theorem without proof.
Lossless join decomposition theorem proof
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