Let $p=\pi((x_0,y_0))$ and $q=\pi((x_1,y_1))\in\mathbb P^1$. We then have
$$
\begin{align*}
\rho_v(\phi(p),\phi(q))&=\frac{\Phi(x_0,y_0)\Phi(x_1,y_1)-\Phi(x_1,y_1)\Phi(x_0,y_0)}{\|\Phi(x_0,y_0)\|_v\|\Phi(x_1,y_1)\|_v}\\
&\leq \dfrac{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}{\|\Phi(x_0,y_0)\|_v\|\Phi(x_1,y_1)\|_v}\dfrac{|H(x_0,y_0,x_1,y_1)|_v|x_0y_1-x_1y_0|_v}{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}\\
&\leq \frac{1}{c^2}M_\phi\frac{|x_0y_1-x_1y_0|_v}{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}\leq \frac{2}{c^2}M_\phi\rho(p,q)
\end{align*}
$$