Theorem 1 (Kurosh-Ore)   Let be a modular lattice and suppose that has two irredundant decompositions of joins of join-irreducible elements:

Then

1. , and
2. every can be replaced by some , so that
   

There is also a dual statement of the above theorem in terms of meets.

Remark. Additionally, if is a distributive lattice, then the second property above (known the replacement property) can be strengthened: each is equal to some . In other words, except for the re-ordering of elements in the decomposition, the above join is unique.