PomeProposition 17.10.83.3A33DCAAE061 Let $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$ and $\ell \leq \ell '$. Then
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17.10.84
\begin{equation} d_1(\ell ,\ell ')=\mathbf{E}^{\phi }(\ell ')-\mathbf{E}^{\phi }(\ell ). \end{equation}
Proposition 17.10.833A33DCAAE061
Proof. By proposition 17.10.65, the energy is linear along finite energy rays. Hence
\[ E_{\theta }^{\phi }(\ell _ t)=tE_{\theta }^{\phi }(\ell _1),\quad E_{\theta }^{\phi }(\ell '_ t)=tE_{\theta }^{\phi }(\ell '_1). \]
Since $\ell _ t\leq \ell '_ t$ for all $t\geq 0$, (17.10.42) gives
\[ \frac{1}{t}d_1(\ell _ t,\ell '_ t)=\frac{1}{t}\left(E_{\theta }^{\phi }(\ell '_ t)-E_{\theta }^{\phi }(\ell _ t)\right). \]
Letting $t\to \infty $ proves the assertion. $\square$
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