Unfortunately, the third version of the orthogonality principle as defined in the previous section is still missing something, and revisiting the light vs. heavy parts example shows what it is: It’s missing that business about restrictions. (In that example, the EQD wasn’t between database relvars as such, nor between projections of such relvars, but rather between certain restrictions of such relvars.) In other words, the third version of the principle failed to subsume the second version. By contrast, the following formulation takes care of both the restriction issue and the projection issue:
Definition (fourth attempt): Let relvars R1 and R2 be distinct, and let the JD
{X1,...,Xn} be irreducible with respect to R1. Let there exist some Xi (1 ≤ i ≤ n) and some possibly empty set of attribute renamings on the projection, R1X say, of R1 on Xi that maps R1X into R1Y, say, where R1Y has the same heading as some subset Y of the heading of R2. Further, let the projection of R2 on Y be R2Y. Then The Principle of Orthogonal Design is violated by R1 and R2 if and only if there exist restriction conditions c1 and c2, neither of which is identically false, such that the equality dependency (R1Y WHERE c1) = (R2Y WHERE c2) holds.