 $b'$ 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) $${\varphi}({\partial}{\Delta}_b) \equiv {\varphi}(a(b))   , \quad | {\Delta}_b(0) | = b(0) .$$ Here and below we use the notation of the definition of a