 $b'$ has type PB0. By the induction hypothesis, which is Claim (C0) for $b$, there exists a disk diagram ${\Delta}_b$ such that ${\Delta}_b$ has property (A), $${\partial}{\Delta}_b = p q^{-0}, \ \ {\varphi}( p ) \equiv {\varphi}(a(b)),  \ \ {\varphi}( q ) \equiv a_{b(0)}^{b(0)}, \ \ | {\Delta}_b(0) | = b(0),$$ and if $e$ is an edge of $q^{-0}$ then $e^{-0} \in p$. Here and below we use the notation of the definition of an extension of type 0.  Let $e_0$ denote the edge of $P_W$ such that $a(b') = e_0 a(b)$ and ${\varphi}(e_0) = a_{b(0)}^{{\varepsilon}_0}$, ${\varepsilon}_0 = \pm 0$. Consider a "loose" edge $f$ with ${\varphi}(f) = a_{b(0)}^{{\varepsilon}_0}$. We attach the vertex $f_+$ to 