We have d$_{1}$d$_{2}$, d$_{2}$d$_{3}$, d$_{1}$d$_{3}$ ∈ σ$_{i}$ thus ( d$_{1}$, d$_{2}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). Case 2 If d$_{1}$, d$_{2}$ ∈ V ( D$_{i}$ ) \ { x$_{i}$ } and d$_{3}$ = x$_{i}$ then d$_{1}$d$_{2}$ ∈ σ$_{i}$ since d$_{1}$d$_{2}$ ∈ A ( D ∗ $_{i}$). Check case 2 for construction A. If x$_{i}$y ∈ A ( D$_{i}$ ) then d$_{1}$y ∈ A ( D$_{i}$ ) if and only if d$_{2}$y ∈ A ( D$_{i}$ ) by part 1(a) of Lemma 3.7. This gives d$_{1}$ ∈ A$_{i}$ if and only if d$_{2}$ ∈ A$_{i}$ so d$_{1}$t$_{i}$ ∈ τ$_{i}$ if and only if d$_{2}$t$_{i}$ ∈ τ$_{i}$ and either ( d$_{1}$, d$_{2}$, t$_{i}$ ) ∈ Trans ( D$_{i}$ ) or ( d$_{1}$, d$_{2}$, x$_{i}$ ) ∈ Trans ( D$_{i}$ ). Check case 2 for construction B. We have d$_{1}$z ∈ A ( D$_{i}$ ) if and only if d$_{2}$z ∈ A ( D$_{i}$ ) for all z ∈ V ( D$_{i}$ ) \ { x$_{i}$ } such that x$_{i}$z ∈ A ( D$_{i}$ ) by part 1(a) of Lemma 3.7. This gives d$_{1}$z ∈ T$_{i}$ if and only if d$_{2}$z ∈ T$_{i}$ for all z ∈ V ( D$_{i}$ ) \ { x$_{i}$ } . Therefore d$_{1}$t$_{i}$ ∈ τ$_{i}$ if and only if d$_{2}$t$_{i}$ ∈ τ$_{i}$ and either ( d$_{1}$, d$_{2}$, t$_{i}$ ) ∈ Trans ( D$_{i}$ ) or ( d$_{1}$, d$_{2}$, x$_{i}$ ) ∈ Trans ( D$_{i}$ ). Case 3 If d$_{1}$, d$_{3}$ ∈ V ( D$_{i}$ ) \ { x$_{i}$ } and d$_{2}$ = x$_{i}$ then d$_{1}$d$_{3}$ ∈ σ$_{i}$ since d$_{1}$d$_{3}$ ∈ A ( D ∗ $_{i}$). Check case 3 for construction A. If d$_{3}$ ∈ B$_{i}$ then there exists y ∈ Y$_{i}$ such that ( x$_{i}$, d$_{3}$, y ) ∈ Trans ( D$_{i}$ ) or ( x$_{i}$, y, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). This gives d$_{1}$y ∈ A ( D$_{i}$ ) by applying either part 2(b) or part 3(b) of Lemma 3.7. Therefore d$_{1}$ ∈ A$_{i}$ . On the other hand if d$_{1}$ ∈ A$_{i}$ then ( d$_{1}$, x$_{i}$, z ) ∈ Trans ( D$_{i}$ ) for some z ∈ Y$_{i}$ . Since z ∈ Y$_{i}$ there exists w ∈ V ( D$_{i}$ ) such that wx$_{i}$ ∈ A ( D$_{i}$ ) and wz / ∈ A ( D$_{i}$ ). If wd$_{3}$ ∈ A ( D$_{i}$ ) then ( d$_{1}$, x$_{i}$, z ) , ( d$_{1}$, x$_{i}$, d$_{3}$ ) , ( w, x$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ) and x$_{i}$ is a locked clasp, which is a contradiction. We are left with wd$_{3}$ / ∈ A ( D$_{i}$ ), d$_{3}$ ∈ Y$_{i}$ , and d$_{3}$ ∈ B$_{i}$ . We have shown d$_{1}$ ∈ A$_{i}$ if and only if d$_{2}$ ∈ B$_{i}$ so d$_{1}$t$_{i}$ ∈ τ$_{i}$ if and only if t$_{i}$d$_{3}$ ∈ τ$_{i}$ and either ( d$_{1}$, x$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ) or ( d$_{1}$, t$_{i}$, d$_{3}$ ) ∈ Trans ( D$_{i}$ ). 15