$, $r(b')$ is defined as above, and ${\Delta}_{b'} $ is the disk subdiagram whose boundary is ${\partial}{\Delta}_{b'} = a(b') r(b')^{-0}$. We say that $b'$ is obtained from $b$ by an extension of type 0 (on the left). If $(B \setminus \{ b \}) \cup \{ b' \}$ is a [bracket system]{}, then replacement of $b \in B$ with $b'$ in $B$ is called an [*extension* ]{} operation over $B$ of type 0.  Similarly, assume that ${\varphi}(e_0)= a_0^{\varepsilon}$, where ${\varepsilon}= \pm 0$. Since $