does not intersect $\VV$,

2.  the neighbourhood $M_y \{ d^{- \zeta'' n} \}$ does not intersect $\VV_{\theta_n}$ for any $n \geq n_{z,w}(y)$.

Let us now prove proposition \[touch\]. We start with some notations. Let $H^+$ and $H^-$ be the (open) North and South hemispheres of $L$, these sets induce a partition $L = H^+ \sqcup E
\sqcup  H^-$. We denote by ${{\textsf {Leb}  }}$ the Lebesgue measure on $E$ and by $p_1$ (resp. $p_2$) the spherical projection from $z$ (resp. $w$) to $E$. For any $y \in E$ and $\delta > 0$, let $\II(y,\delta)$ be the interval in $E$ centered at $y$ with length $2\delta$. We also denote by $D(c,\delta) \subset L$ the disc with center $c$ and radius $\delta$. We define $p_\kappa(c):= p_1(c)$ if $c \in H^+ \cup E$ and $p_\kappa(c):= p_2(c)$ if $c \in H^-$. The same convention holds for the projection of $D(c,\delta)$ to $E$: we use $p_1$ or $p_2$ depending on $c \in H^+ \cup E$ or $c \in H^-$.\
Let $\{ c_i \, , \, 1 \leq i \leq
l_{\theta_n} \}:= \VV_{\theta_n} \cap L$, where $l_{\theta_n} \leq
{{\rm deg \, }}(\VV_{\theta_n}) \leq \tau_{\theta_n}$. Since the Fubini-Study metric induces ${{\textsf s }}$ on $L$, the set 