Note that if all $a_{i}=1$ in family $H_{s}^{v}(a)$, then we have the original equation of $H_{s}$. On the other hand, if all $a_{i}=0$, then the family $H_{s}^{v}(a)$ degenerates to the hyperbola:

Because $X_{s}(\mathcal{F}_{q})$, $q\in U^{\epsilon}_{v}$, intersect every $H_{s}^{v}(a)$ transversally, the corresponding fibers $F_{q}:=H_{s}\cap X_{s}(\mathcal{F}_{q})$ and $F_{q}^{v}:=H_{s}^{v}(0)\cap X_{s}(\mathcal{F}_{q})$ are diffeomorphic.

If $\theta=\{\theta_{i}\}$ denote the coordinates of the torus $\mathbb{T}$ and $\theta_{s}$ is the phase of $s$, then the fiber $F_{q}^{v}$ of $H^{v}_{s}(0)$ is the torus

which, for a fixed $s$, can be identified with the torus $\mathbb{T}_{v}$ (though, see the remark below about monodromy as $\theta_{s}\mapsto\theta_{s}+2\pi$).

Similarly, the fibers $F_{q}=H_{s}\cap X_{s}(\mathcal{F}_{q})$ and $F_{q}^{w}:=H_{s}^{w}(0)\cap X_{s}(\mathcal{F}_{q})$ for $q\in V^{\delta}_{w}$ are diffeomorphic. But $F_{q}^{w}$ can be naturally identified with the torus $\mathbb{T}/w$, which follows from writing the equation for $H_{s}^{w}(0)$ in the local coordinates $\{y_{i}\}$ from Lemma 3.9:

where $P_{1}(y_{2},\dots,y_{d})$ is a Laurent polynomial independent of $y_{1}$. Restricting to the fiber $X_{s}(q)$ means fixing absolute values of $y_{i},\ i=2,\dots,d$. A point on the torus $\mathbb{T}/w$ determines the phases of $y_{i},\ i=2,\dots,d$. Once $y_{i},\ i=2,\dots,d$, are fixed, there is a unique solution to the equation of $H_{s}^{w}(0)$.

Thus, $f_{s}:H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ is a torus fibration. The only thing left to check is that it has the correct monodromy.

Note that all diffeomorphisms $F_{q}\cong F^{v}_{q}$, $q\in U^{\epsilon}_{v}$, and $F_{q}\cong F^{w}_{q}$, $q\in V^{\delta}_{w}$, are deformation diffeomorphisms. Hence, the transitions maps between $\mathbb{T}_{v}$ and $\mathbb{T}/w$, for $q\in U_{v}\cap V_{w}$, are homotopic to the map $f_{vw}:\mathbb{T}_{v}\to\mathbb{T}/w$. But monodromy is a homotopy invariant, hence, it has to be equal to the one given by the maps $f_{vw}$. This completes the proof. ∎