The angle formula for geodesic hinges in the space of metric measure spaces

Let $(\boldsymbol{X}_\bullet,\boldsymbol{X}'_\bullet)$ be a geodesic hinge with endpoint distances $R$ and $R'$, and let ${\sf f}_0,{\sf f}_1,{\sf f}'_1$ be the associated functions on $X^2$. Let $\bar{\mu}$ be the coupling associated with the hinge, and write $\langle\cdot,\cdot\rangle_{L^2(X^2,\bar{\mu}^2)}$ for the corresponding inner product. The angle formula. For each geodesic hinge as above,

$$\cos\measuredangle\Big(\mathcal{X}_\bullet,\mathcal{X}'_\bullet\Big)=\frac1{RR'}\big\langle {\sf{f}}_1-{\sf{f}}_0,{\sf{f}}'_1-{\sf{f}}_0\big\rangle_{L^2(X^2,\bar{\mu}^2)}.$$

This identifies the angle between geodesics with the normalized $L^2$ inner product of their initial displacement functions. The supplied text does not indicate whether this claim has been established beyond the displayed proof, so its status is left open.