Check case 3 for construction B. If d$_{1}$y$_{i}$ / ∈ A ( D$_{i}$ ) then d$_{1}$d$_{3}$ ∈ T$_{i}$ and ( d$_{1}$, t$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). If d$_{1}$y$_{i}$ ∈ A ( D$_{i}$ ) then d$_{1}$x$_{i}$ ∈ σ$_{i}$ and we must show x$_{i}$d$_{3}$ ∈ σ$_{i}$ . Note that ( d$_{1}$, x$_{i}$, d$_{3}$ ) , ( d$_{1}$, x$_{i}$, y$_{i}$ ) , ( b$_{i}$, x$_{i}$, y$_{i}$ ) ∈ Trans ( D$_{i}$ ) so b$_{i}$d$_{3}$ ∈ A ( D$_{i}$ ) since x$_{i}$ is an un- locked clasp. If x$_{i}$d$_{3}$ / ∈ σ$_{i}$ then cd$_{3}$ ∈ T$_{i}$ for some c ∈ V ( D$_{i}$ ) such that cy$_{i}$ / ∈ A ( D$_{i}$ ). This gives ( b$_{i}$, x$_{i}$, y$_{i}$ ) , ( b$_{i}$, x$_{i}$, d$_{3}$ ) , ( c, x$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ) with cy$_{i}$ / ∈ A ( D$_{i}$ ) and x$_{i}$ is a locked clasp. This is a contradiction. We are left with d$_{1}$x$_{i}$, x$_{i}$d$_{3}$ ∈ σ$_{i}$ and ( d$_{1}$, x$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). Case 4 If d$_{2}$, d$_{3}$ ∈ V ( D$_{i}$ ) \ { x$_{i}$ } , d$_{1}$ = x$_{i}$ then d$_{2}$d$_{3}$ ∈ σ$_{i}$ since d$_{2}$d$_{3}$ ∈ A ( D ∗ $_{i}$). Check case 4 for construction A. Suppose d$_{j}$ ∈ B$_{i}$ for j = 2 or j = 3 and let k ∈ { 2 , 3 } be such that k = j . If d$_{j}$ ∈ B$_{i}$ then ( x$_{i}$, d$_{j}$, y ) ∈ Trans ( D$_{i}$ ) or ( x$_{i}$, y, d$_{j}$ ) ∈ Trans ( D$_{i}$ ) for some y ∈ Y$_{i}$ . There exists w ∈ V ( D$_{i}$ ) \ { x$_{i}$ } such that wx$_{i}$ ∈ A ( D$_{i}$ ) and wy / ∈ A ( D$_{i}$ ) since y ∈ Y$_{i}$ . Then wd$_{2}$, wd$_{3}$ / ∈ A ( D$_{i}$ ) by two applications of part 3(b) of Lemma 3.7. Thus d$_{2}$, d$_{3}$ ∈ Y$_{i}$ and ( t$_{i}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). We have shown d$_{2}$ ∈ B$_{i}$ or d$_{3}$ ∈ B$_{i}$ imply ( t$_{i}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). On the other hand if d$_{2}$ / ∈ B$_{i}$ and d$_{3}$ / ∈ B$_{i}$ then x$_{i}$d$_{2}$, x$_{i}$d$_{3}$ ∈ σ$_{i}$ and ( x$_{i}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). Check case 4 for construction B. We have cd$_{2}$ ∈ A ( D$_{i}$ ) if and only if cd$_{3}$ ∈ A ( D$_{i}$ ) for all c ∈ V ( D$_{i}$ ) such that cx$_{i}$ ∈ A ( D$_{i}$ ) and cy$_{i}$ / ∈ A ( D$_{i}$ ) by part 3(b) of Lemma 3.7. This gives cd$_{2}$ ∈ T$_{i}$ if and only if cd$_{3}$ ∈ T$_{i}$ for all c ∈ V ( D$_{i}$ ) \ { x$_{i}$ } . Therefore t$_{i}$d$_{2}$ ∈ τ$_{i}$ if and only if t$_{i}$d$_{3}$ ∈ τ$_{i}$ and either ( x$_{i}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ) or ( t$_{i}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). 16