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. ∎