yBu\,d(yB-g^2) - (yB-g^2 + a+ \delta)\,d(yBu)&=&0\end{aligned}$$ We see easily that the curve of tangencies is given by $ABau^2{\phi}{\phi^{\prime}}y + y^2(...)=0$, so that the component different from $C=\{y=0\}$ crosses $C$ at $p$ transversaly.

We proceed now to examine the points of tangency between $\mathcal I^*$ and $\mathcal L^*$ that possibly appear at ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. If we denote their number as $tang_1 (\mathcal I^*,\mathcal L^*)$, we have seen that $$tang_1 (\mathcal I^*,\mathcal L^*) = Z_1(\mathcal I^*,C)+ Z_1(\mathcal L^*,C).$$ In fact, we have seen that out of ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$ tangency curves correspond to each other when restricted to $C$ and $C_0$.

[**We claim that this equality holds at each point of**]{} ${\mathcal Z}_1(\mathcal I^*,C)\cup {\mathcal Z}_1(\mathcal L^*,C)$. Let us consider some point $p\in {\mathcal Z}_1(\mathcal I^*,C)$; since $\Phi_{F,G}$ is not a local biholomorphism, we have as explained before the possibilities $\bf A1$, $\bf A2$ and $\bf B$, the first two occuring when $\phi(p)\in tang({\mathcal F}_0,{\mathcal G}_0 \cap C_0$. If $p$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,G}$, then $p$ is $\bf B$ for $\Phi_{F,H}$ (in the same way, when $q\in {\mathcal Z}_1(\mathcal L^*,C)$ is $\bf A1$ or $\bf A2$ for $\Phi_{F,H}$, then $q$ is 