According to Proposition 3.2 we can choose $s$ big enough so that the $\mathrm{Log}_{s}$-image of the affine part of every hypersurface $H_{s}^{w}(a)$ lies in the $\epsilon$-neighborhood of the Minkowski sum $V^{\delta}_{w}+\operatorname{cone}(w)$. Also, recall from Lemma 3.3 that $V^{\delta}_{w}+\operatorname{cone}(w)$ lies in the domain $Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon)$. Thus, we can assume that the affine parts of all hypersurfaces $H_{s}^{w}(a)$ lie in $\mathrm{Log}_{s}^{-1}(Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon))$.

We choose a basis $\{e_{i}\}$ of $(\mathbb{Z}^{d})^{*}$ such that

Then the affine coordinate functions $y_{i}=x^{e_{i}}$ can be extended (by allowing zero values for $y_{1}$) to the open part of the toric divisor $Z_{w}$ corresponding to the facet $F_{w}\subset\partial{\Delta_{\nu}}$. Moreover, in these coordinates the preimage of each flow line $\mathcal{F}_{q}$ in $(\mathbb{C}\backslash\{0\})^{d}$ is defined by fixing the values of $|y_{2}|,\dots,|y_{d}|$, so that its closure $X_{s}(q)$ is defined by the same equations, but allowing the zero value for $y_{1}$. Hence, we can use $\{y_{i}\}$ as global coordinates on $X_{s}(V^{\delta}_{w})$.

Multiplying the affine equation of $H_{s}^{w}(a)$ by $y_{1}$ we note that the Laurent polynomial

has only positive powers of $y_{1}$, where as its first part $P_{1}(y)=y_{1}\sum\limits_{m\in G_{w}}s^{\lambda(m)}x^{m}$ is independent of $y_{1}$ at all. Thus, we get the global equation for the family in $X_{s}(V^{\delta}_{w})$.

Now we can repeat the estimates for the monomials and their log-derivatives. Whenever $\mathrm{Log}_{s}(x)\in Q^{\lambda}_{(\{0\}\mid w^{\perp})}(\epsilon)$, we have

or, equivalently,

When written in the $y$-coordinates these estimates extends by continuity from the affine part to the entire $X_{s}(V^{\delta}_{w})$.

To see that $H_{s}^{w}(a)$ has no critical points we differentiate its defining equation with respect to $y_{1}$:

for large enough $s$.

Finally, Lemma 3.6 asserts that the vector field $\mathfrak{X}$ in $\bigcup_{q\in V^{\delta}_{w}}\mathcal{F}_{q}$ is constant and equal to $w$. It means that $\frac{\partial}{\partial y_{1}}$ is a tangent vector to $X_{s}(q)$, $q\in V^{\delta}_{w}$, and it is transversal to $H_{s}^{w}(a)$ by the above calculation. ∎