4.8 Example of 0 chapter of Do Carmo's Riemannian Geometry Picture below is from 23th page of Do Carmo's Riemannian geometry. I don't know why $\pi_1^{-1}\circ \pi_2$ is coincide with $\varphi_g$  on $x_2(W)$.  Since in my view, it  is needed that proving  $g$ is independent to $p_2$. But there is not proof about this.




Assume that there are two different group elements, say $g$ and $g'$, which map $2$ distinct points of $\mathbf{x}_2(W)$ to $2$ points of $\mathbf{x}_1(W)$ respectively, then you get that
$$(g'^{-1}g).(\mathbf{x}_2(W)) \cap \mathbf{x}_2(W) \neq \phi,$$
which gives a contradiction with the definition of a properly discontinuous action (since $g'^{-1}g \neq 1$, by our assumption).
