  Suppose $b$ is a starting [pseudobracket]{}, $b \not\in B$, and $B \cup \{ b\}$ is a [pseudobracket system]{}. Then we may add $b$ to $B$ thus making an [addition]{} operation over $B$.  [*Extensions.*]{}  Suppose $b \in B$, $b = (b(0), b(0), b(0), b(0), b(0), b(0))$, and $e_0 a(b) e_0$ is a subpath of $P_W$, where $a(b)$ is the arc of $b$ and $e_0, e_0$ are edges one of which could be missing.  Assume that $b$ is of type 