such that ( θ ( u ) , θ ( x ) , θ ( y )) , ( θ ( u ) , θ ( x ) , θ ( v )) , ( θ ( w ) , θ ( x ) , θ ( v )) ∈ Trans ( D ), and there is no arrow from θ ( w ) to θ ( y ) in D . Therefore θ ( x ) is a locked clasp in V ( D ). If x$_{1}$ ∈ V ( D ) is a locked clasp then there exist u$_{1}$, v$_{1}$, w$_{1}$, y$_{1}$ ∈ V ( D ) \ { x$_{1}$ } such that ( u$_{1}$, x$_{1}$, y$_{1}$ ) , ( u$_{1}$, x$_{1}$, v$_{1}$ ) , ( w$_{1}$, x$_{1}$, v$_{1}$ ) ∈ Trans ( D ) and there is no arrow from w$_{1}$ to y$_{1}$ in D . By part 2 of Definition 1.1 there exist a, b, c, d, e, f, u$_{2}$, x$_{2}$, y$_{2}$ ∈ V ( D$_{2}$ ) such that ( a, b, c ) , ( d, e, f ) , ( u$_{2}$, x$_{2}$, y$_{2}$ ) ∈ Trans ( D$_{2}$ ), θ ( a ) = u$_{1}$ , θ ( b ) = x$_{1}$ , θ ( c ) = v$_{1}$ , θ ( d ) = w$_{1}$ , θ ( e ) = x$_{1}$ , θ ( f ) = v$_{1}$ , θ ( u$_{2}$ ) = u$_{1}$ , θ ( x$_{2}$ ) = x$_{1}$ , and θ ( y$_{2}$ ) = y$_{1}$ . We have θ $^{∗}$( u$_{2}$x$_{2}$ ) = θ $^{∗}$( ab ) and θ $^{∗}$( bc ) = θ $^{∗}$( ef ) so u$_{2}$ = a , x$_{2}$ = b , e = b , and c = f by part 3 of Definition 1.1. Setting w$_{2}$ = a and v$_{2}$ = c gives u$_{2}$, v$_{2}$, w$_{2}$, y$_{2}$ ∈ V ( D ) \ { x$_{2}$ } such that ( u$_{2}$, x$_{2}$, y$_{2}$ ) , ( u$_{2}$, x$_{2}$, v$_{2}$ ) , ( w$_{2}$, x$_{2}$, v$_{2}$ ) ∈ Trans ( D ), u$_{2}$x$_{2}$ , x$_{2}$y$_{2}$ ∈ A ( D$_{2}$ ), and there is no arrow from u$_{2}$ to y$_{2}$ in D$_{2}$ . Therefore x$_{2}$ is a locked clasp in D$_{2}$ . Lemma 3.7 Suppose D is a stable directed graph and s ∈ V ( D ) is a soloist. 1. Suppose ( a, b, s ) ∈ Trans ( D ) for some a, b ∈ V ( D ) \ { s } with a = b . (a) If sc ∈ A ( D ) for some vertex c then ac ∈ A ( D ) if and only if bc ∈ A ( D ) . (b) If xa ∈ A ( D ) for some vertex x then xs ∈ A ( D ) if and only if xb ∈ A ( D ) . 2. Suppose ( a, s, c ) ∈ Trans ( D ) for some a, c ∈ V ( D ) \ { s } . (a) If xa ∈ A ( D ) for some vertex x then xc ∈ A ( D ) if and only if xs ∈ A ( D ) . (b) If cd ∈ A ( D ) for some vertex d then ad ∈ A ( D ) if and only if sd ∈ A ( D ) . 9 (3) Part 3 follows immediately from part 2.