The concept of a lossless join is among the most important and fundamental concepts of database design. The paper by Bernstein et al. [1] presents a necessary and sufficient condition for a lossless join, when functional dependencies are given. While the basic concepts of this paper are simple, the proof is delicate and complex.

Since the time this paper was written, the area has made significant progress, using general classes of dependencies and powerful methods. The present paper shows how the result above can be very simply proved, once these are mastered. A significant part of this short paper is devoted to a brief exposition of the concepts which are used in the proof. For anybody with sufficient acquaintance with the state-of-the-art theory of dependencies, the proof of the theorem itself is indeed almost trivial. The exposition in the paper assumes such an acquaintance; it cannot serve as an introduction to those who are not familiar with the theory.

In the proof of the second direction of the theorem, the last step is missing. This is the application of Lemma 3(2) to prove the claim.