points of $\mathcal Z_1(\mathcal I^*,C)$ (we put $Z_1(\mathcal L^*,C)$ for the correspondent sum at points of $\mathcal Z_1(\mathcal L^*,C)$).

As for the foliations $\mathcal I^*$ and $\mathcal L^*$, we have that $$\begin{aligned}
tang(\mathcal I^*,\mathcal L^*)\cdot C&=& N_{\mathcal I^*} \cdot C + N_{\mathcal L^*}\cdot C + K_S \cdot C \\
&=& Z(\mathcal I^*,C) + Z(\mathcal L^*,C) + 2d^2-3d\\
&=& Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C) + 2d^2-d\end{aligned}$$ We conclude therefore that $$tang(\mathcal I^*,\mathcal L^*)\cdot C = Z_1(\mathcal I^*,C) + Z_1(\mathcal L^*,C)+ tang(\mathcal I,\mathcal L)\cdot C_0$$ Observe that curves $C$ and $C_0$ appear as components of the tangency locus in both sides of the last equation, thus we cancel $d$ from the equation and consider, from now, tangency loci besides $C$ and $C_0$. This formula suggests that $tang(\mathcal I^*,\mathcal L^*)\cap C$ may be also computed looking at the points of $\phi^{-1}(tang(\mathcal I, \mathcal L)\cap C) \cup  \mathcal Z_1(\mathcal I^*,C) \cup \mathcal Z_1(\mathcal L^*,C)$.

Our aim is to prove that $\Phi_{F,G}$ and $\Phi_{F,H}$ are everywhere local biholomorphisms. First of all we have to associate the tangencies between $\mathcal I$ and $\mathcal L$ to tangencies between $\mathcal I^*$ and $\mathcal L^*$. There is a little difficulty here because $\mathcal I^*$ and $\mathcal L^*$ are obtained 