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Suppose R is a relation schema and F is a set of functional dependencies on R. Further, suppose R1 and R2 form a decomposition of R. Then the decomposition is a lossless join decomposition of R provided that:

  1. R1∩R2→R1 is in F+
  2. R1∩R2→R2 is in F+
  3. both R1∩R2→R1 and R1∩R2→R2 functional dependencies are in F+
  4. at least one from R1∩R2→R1 and R1∩R2→R2 is in F+
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