The proof is quite simple. First we glue positive fibrations over sufficiently small neighborhoods of positive vertices of $\Delta$ using Proposition 4.17. Now given a negative vertex $p^{-}\in\mathcal{N}$, we have that a neighborhood of $p^{-}$ is affine isomorphic to a neighborhood $U$ of zero in the local model Example 3.12. Consider a negative Lagrangian fibration $\mathcal{F}^{-}=(X^{-},\omega^{-},f^{-},B^{-})$ (cf. Definition 7.1), which we have constructed in Theorem 7.3. The discriminant locus $\Delta^{-}$ of $f^{-}$ has the shape of an amoeba with thin legs and there is a disc $D$ containing the codimension $1$ part of $\Delta^{-}$ such that $f^{-}$ is smooth except at points of $(f^{-})^{-1}(D)$ (cf. part (i) and (ii) of Definition 7.1). Moreover we may assume that $B^{-}-(\Delta^{-}\cup D)$ is affine isomorphic to $(U^{\prime}-(D^{\prime}\cup\Delta_{\tau}),\mathscr{A}_{\tau})$, where $U^{\prime}$ is a neighborhood of $0$ in the affine manifold with singularities of Example 3.13 and $D^{\prime}\subset\{x_{1}=0\}$ contains $0$ and is homeomorphic to a disc (cf. point (iii) of Definition 7.1). It may happen that $U^{\prime}$ is too big for us to glue the Lagrangian negative fibration as it is. However, if we replace $\omega^{-}$ with $\epsilon\,\omega^{-}$ for a sufficiently small $\epsilon>0$, this has the effect of scaling the affine coordinates on the base by a factor of $\epsilon$ (i.e. of making the amoeba as small as we please). Therefore we may assume that $U^{\prime}\subset U$. Moreover, we may also assume that the legs of $\Delta^{-}$ (in affine coordinates) are straight towards their ends, i.e. they coincide with the legs of $\Delta$ outside an open subset $U^{\prime\prime}$ such that $D^{\prime}\subset\bar{U}^{\prime\prime}\subset U^{\prime}$. The localized thickening $\Delta_{\blacklozenge}$ of $\Delta$ around $p^{-}$ consists in replacing $U^{\prime}\cup\Delta$ with $\Delta^{-}$ and defining $D_{p^{-}}=D^{\prime}$. The affine structure $\mathscr{A}_{\blacklozenge}$ is inherited from $\mathscr{A}$. This can be done at every negative vertex $p^{-}$. Tautologically, we have that $X(U^{\prime}-(D_{p^{-}}\cup\Delta_{\blacklozenge}),\mathscr{A}_{\blacklozenge})$ is symplectically conjugate to $(f^{-})^{-1}(B^{-}-(\Delta^{-}\cup D))$ and therefore we can glue $X^{-}$ to $X(B_{\blacklozenge},\mathscr{A}_{\blacklozenge})$.

Finally, now that singular fibres have been glued on top of all vertices, it only remains to glue generic-singular fibres along the edges. This can be easily done by applying directly Proposition 4.18, notice in fact that Lagrangian negative fibrations are smooth and generic-singular towards the ends of the legs. ∎