Lossless join decomposition

In database design, a lossless join decomposition is a decomposition of a relation formula_1 into relations formula_2 such that a natural join of the two smaller relations yields back the original relation. This is central in removing redundancy safely from databases while preserving the original data.
Criteria.
Can also be called nonadditive.
If formula_1 is split into formula_4 and formula_5, for this decomposition to be lossless (i.e., formula_6) then at least one of the two following criteria should be met.
Check 1: Verify join explicitly.
Projecting on formula_4 and formula_5, and joining them back, results in the relation you started with.
Check 2: Via functional dependencies.
Let formula_1 be a relation schema.
Let be a set of functional dependencies on formula_1.
Let formula_4 and formula_5 form a decomposition of formula_1 .
The decomposition is a lossless-join decomposition of formula_1 if at least one of the following functional dependencies are in + (where + stands for the closure for every attribute or attribute sets in ):