The arguments are more or less formal from Prop. 3.4.1 ; let us detail them anyway for the sake of the reader.

1)$\Rightarrow$2). Like any rational map, $\varphi$ has infinitely many preperiodic points in ${\mathbf{P}}^{1}(\overline{F})$, and they satisfy $h_{\varphi}(x)=0$. If $h_{\varphi}=h_{\psi}$, then they also satisfy $h_{\psi}(x)=0$. Under the assumptions of the proposition, they are preperiodic for $\psi$.

2)$\Rightarrow$3) is obvious, for common preperiodic points of $\varphi$ and $\psi$ satisfy $h_{\varphi}(x)+h_{\psi}(x)=0$/.

3)$\Rightarrow$4). By Prop. 3.4.1, the line bundle $\mathscr{O}(1)_{\varphi}-\mathscr{O}(1)_{\psi}$ has the constant metric at all places. In particular, the local measures $\mu_{\varphi}$ and $\mu_{\psi}$ coincide at all places.

4)$\Rightarrow$5). Let $s$ be a non zero global section of $\mathscr{O}(1)$. For any place $v$, $f_{v}=\log(\left\|{s}\right\|_{v,\varphi}/\left\|{s}\right\|_{v,\psi})$ ; one has $\mu_{v,\psi}-\mu_{v,\varphi}=\mathop{\mathrm{d}\mathrm{d}^{c}}f_{v}$, hence $\mathop{\mathrm{d}\mathrm{d}^{c}}f_{v}=0$. By the maximum principle of [51], $f_{v}$ is constant. Moreover,

5)$\Rightarrow$1). This is obvious. ∎