The proof follows the same lines of Lemma 7.6. Assume that $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$ has been constructed with Theorem 6.19. In particular the wall $\Gamma$ consists of the union of three disjoint sets, denoted $\Gamma_{c}$, $\Gamma_{d}$ and $\Gamma_{e}$. The corresponding components of the seam are $Z_{c}=f^{-1}(\Gamma_{c})$, $Z_{d}=f^{-1}(\Gamma_{c})$ and $Z_{e}=f^{-1}(\Gamma_{e})$ with corresponding quotients denoted by $\bar{Z}_{c}$, $\bar{Z}_{d}$ and $\bar{Z}_{e}$. The invariants of $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$ are given by sequences $\ell^{c}$, $\ell^{d}$ and $\ell^{e}$. In particular the first order invariants satisfy the integral conditions (60) with $m_{1}=-1$ and $m_{2}=1$.

Over the same wall $\Gamma$ and seam $Z$, we could define another triple of invariants as follows. Define $(\ell^{c})^{\prime}$ to be the zero sequence, while $(\ell^{d})^{\prime}$ and $(\ell^{e})^{\prime}$ to be sequences whose only non-zero terms are the first order ones, which we define to be

As we saw in Example 6.21, these choices of invariants give rise to a fake stitched fibration $\mathcal{F}^{\prime}$ which is topologically conjugate to $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$.

Using Theorem 6.19 we now construct a new stitched fibration with the same wall $\Gamma$ and seam $Z$ as $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$, but whose invariants interpolate between those of $\mathcal{F}^{\prime}$ and those of $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$. Let $A^{\prime}$ be a small tubular neighborhood of $\Delta$ and denote $\bar{A}^{\prime}=A^{\prime}\cap\{b_{1}=0\}$. Assume that $\bar{A}^{\prime}$ is entirely contained in the region in Figure 15 (a) delimited by the dotted lines. In particular we want the ends of $\bar{A}^{\prime}$ to be contained in the white region where $\mathcal{F}$ is smooth. Let $A\subset A^{\prime}$ be a smaller open neighborhood of $\Delta$ and denote $\bar{A}=A\cap\{b_{1}=0\}$. Let $\rho\in C^{\infty}(\Gamma)$ be a cut-off function which is 1 on $\bar{A}$ and $0$ on $\Gamma-\bar{A}^{\prime}$. Define $\tilde{\ell}_{k}^{c}=(1-\rho)(\ell_{k}^{c})^{\prime}+\rho\,\ell_{k}^{c}$ and similarly define $\tilde{\ell}_{k}^{d}$ and $\tilde{\ell}_{k}^{e}$. It follows from Theorem 6.19 that the sequences $\tilde{\ell}_{c}$, $\tilde{\ell}_{d}$ and $\tilde{\ell}_{e}$ give rise to a stitched Lagrangian fibration $\tilde{\mathcal{F}}^{o}$ which is topologically conjugate to $\mathcal{F}|_{\mathbb{R}^{3}-\Delta}$. Moreover $\tilde{\mathcal{F}}^{o}|_{A-\Delta}$ and $\mathcal{F}|_{A-\Delta}$ are symplectically conjugate so we can glue $\mathcal{F}|_{A}$ to $\tilde{\mathcal{F}}^{o}|_{A-\Delta}$ along $\mathcal{F}|_{A-\Delta}$. This produces a piecewise smooth Lagrangian fibration $\tilde{\mathcal{F}}$ which is topologically conjugate to $\mathcal{F}$, moreover the chosen invariants guarantee that after a change of coordinates on the base $\tilde{\mathcal{F}}$ satisfies the smoothness condition $(ii)$. ∎

It only remains to show that $\mathcal{\tilde{F}}$ satisfies property (iii) of Definition 7.1, but this immediately follows from the construction. In fact, $\tilde{\mathcal{F}}|_{\mathbb{R}^{3}-A^{\prime}}$ coincides with the fibration described in Example 6.21 restricted to a suitable neighborhood of the vertex. We observed that the latter fibration induces an affine structure on the base which is affine isomorphic to a negative vertex of Example 3.12 (or of Example 3.13). This concludes the proof. ∎