Assume to the Contrary. Then There Exists a Feasible U-fragment with the Blocks of Overlapping B Pq

pair of interlacing bights, A 1 and A 2. Let A 1 be between a 1 and a 2 and A 2 , between b 1 and b 2 , where a 1 , a 2 , b 1 and b 2 are four distinct vertices on J such that they are external vertices of attachment on J or one of s and t. Further assume that U has the least number of blocks of overlapping B PQ-bridges. Assume that B 1 , B 2 ,. . ., B l are so so ordered that if i < j (for 1 i, j l) then the vertices of attachment of B i on P and Q are to the left of those of B j on P and Q, respectively. It then follows that of the four vertices a 1 , a 2 , b 1 and b 2 , at least two lie on the subpath J]t P (B 1); t Q (B 1) and at least two lie on the subpath J]s P (B l); s Q (B l) and that l = 2. But then it is easy to see that in this case either U B1 or U B2 has a pair of interlacing bights; this contradicts the assumption.