... (5), it follows thatr(t3[X - YZ],t4[X - YZ])2:aX-YZ.By (17) andX - YZ ~ R - Zthenr(t2[X - YZ],t4[X - YZ])2:aX-YZ.Therefore,r(tdX - YZ],t4[X - YZ])2:aX-YZ.(21)Combining (20) with ... thatr(tdZ - Y],t4[Z - Y])2:aZ-Y.By (18), it is easy to see thatr(tdZ - Y], t4[Z - Y])2:aZ-Y.Final, we show thatr(t2[W], t4[W])2:aw,whereW=R - X(Z - Y).SinceR - XYZ ~ R - Z,andR - XYZ ... by(F,G)+,is the set of all(a,,B)-ffds and(a,,B)-fmvds that canbe derived fromFUG by repeated application of rules in the set {FFDI-FFD6, MVDI-FMVD7,FFD- FMVD 1-FFD- FMVD3}.Theorem 4.2.LetS=(R,...