Consider one of the fibrations $\mathcal{F}_{h,M}$, $\mathcal{F}_{v,M}$ or $\mathcal{F}_{d,M}$ as above (whenever necessary, we allow ourselves to restrict to smaller neighborhoods of $\Delta_{h,M}$, $\Delta_{v,M}$ or $\Delta_{d,M}$). To keep the notation simple we temporarily drop the subindices and denote it by $\mathcal{F}$.

Since $\mathcal{F}$ satisfies Assumption 6.22, it follows from Proposition 6.28 that we can associate to $\mathcal{F}$ a normal form of cylindrical type $\mathcal{F}_{u,H}$ together with its invariants given by a triple $(Z^{\#}_{H},\ell,H_{\Delta})$ which, in view of Theorem 6.29, uniquely determine $\mathcal{F}$ as a germ around $\Gamma=B\cap\{b_{1}=0\}$. By slight abuse of notation we will denote by the same letter $\Gamma$ both $B\cap\{b_{1}=0\}$ and $B_{u}\cap\{b_{1}=0\}$, where $B_{u}$ is the base of $\mathcal{F}_{u,H}$. For the duration of this proof $H$ will remain unchanged, so we drop the subindex $H$ and denote $\mathcal{F}_{u}:=\mathcal{F}_{u,H}$ for short.

The proof consists in suitably deforming the sequence $\ell$. Let $\bar{A}\subset\Gamma$ and $\bar{A}^{\prime}\subset\bar{A}$ be (planar) regions as depicted in Figure 14. Given a cut-off function $\rho\in C^{\infty}(\Gamma)$ such that $\rho$ is 1 on $\Gamma-\bar{A}$ and $0$ on $\bar{A}^{\prime}$, define a new (fibrewise closed) sequence $\tilde{\ell}$ whose elements are $\tilde{\ell}_{k}=(\rho\circ\bar{\pi}^{\#})\,\ell_{k}$ for each $k\in\mathbb{N}$. We obtain a triple $(Z^{\#}_{H},\tilde{\ell},H_{\Delta})$, such that $\ell|_{(\bar{\pi}^{\#})^{-1}(\Gamma-\bar{A})}=\tilde{\ell}|_{(\bar{\pi}^{\#})^{-1}(\Gamma-\bar{A})}$ and $\tilde{\ell}|_{(\bar{\pi}^{\#})^{-1}(\bar{A}^{\prime})}=0$.

In view of Proposition 6.30, $(Z^{\#}_{H},\tilde{\ell},H_{\Delta})$ gives rise to a normal form of cylindrical type $\mathcal{F}_{\tilde{u}}$ defined over a neighborhood of $\Gamma$. By construction and by Theorem 6.29, $\mathcal{F}_{u}$ and $\mathcal{F}_{\tilde{u}}$ define the same germ around $\Gamma-\bar{A}$, i.e. there are open neighborhoods $U$ and $\tilde{U}$ of $\Gamma-\bar{A}$ (satisfying $U\cap\{b_{1}=0\}=\tilde{U}\cap\{b_{1}=0\}=\Gamma-\bar{A}$) such that $\mathcal{F}_{u}|_{U}$ and $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ are symplectically conjugate. Moreover $\mathcal{F}_{\tilde{u}}$ is smooth when restricted to any open neighborhood $A^{\prime}$ of $\bar{A}^{\prime}$ such that $A^{\prime}\cap\{b_{1}=0\}=\bar{A}^{\prime}$. Now recall that $\mathcal{F}_{u}$ is symplectically conjugate to $\mathcal{F}$, so we have that $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ is symplectically conjugate $\mathcal{F}|_{U}$.

Let us summarize the result using our original notation for the horizontal leg. For $\Gamma_{h,M}=B_{h,M}\cap\{b_{1}=0\}$, we have found sets $\bar{A}^{\prime}\subset\bar{A}\subset\Gamma_{h,M}$ (as in Figure 14) and a normal form of cylindrical type $\mathcal{F}_{\tilde{u}}$, defined over a neighborhood of $\Gamma_{h,M}$, smooth over $\bar{A}^{\prime}$ and such that $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ is symplectically conjugate to $\mathcal{F}_{h,M}|_{U}$, where $U$ and $\tilde{U}$ are neighborhoods of $\Gamma_{h,M}-\bar{A}$ (satisfying $U\cap\{b_{1}=0\}=\tilde{U}\cap\{b_{1}=0\}=\Gamma_{h,M}-\bar{A}$).

If we go back denoting by $\mathcal{F}$ the fibration of Lemma 7.4, we can form a new fibration $\tilde{\mathcal{F}}$ in the following way. Let $\mathcal{F}^{\prime}=\mathcal{F}|_{\mathbb{R}^{3}-(\mathbb{R}\times\bar{A})}$ and symplectically glue $\mathcal{F}_{\tilde{u}}$ to $\mathcal{F}^{\prime}$ using the conjugation between $\mathcal{F}_{\tilde{u}}|_{\tilde{U}}$ and $\mathcal{F}^{\prime}|_{U}=\mathcal{F}_{h,M}|_{U}$. The fibration $\tilde{\mathcal{F}}$ is the result of this gluing. Notice that $\tilde{\mathcal{F}}$, due to the properties of $\mathcal{F}_{\tilde{u}}$, is such that for some $m>M$ (depending on $\bar{A}^{\prime}$) and a suitable neighborhood of $B_{h,m}$ of $\Delta_{h,m}$, the restriction $\tilde{\mathcal{F}}_{h,m}$ is smooth. Notice that $\bar{A}^{\prime}$ can be chosen so that the latter holds for any $m>M$.

The above method applied to all legs, produces the required result. ∎