A Triangle Inequality for Discrepancies Along a Composite Coupling

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $\nu,\rho,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\pi_{12}\in\Pi(\nu,\rho)$ and $\pi_{23}\in\Pi(\rho,\mu)$, let $\sigma$ be a gluing of $\pi_{12}$ and $\pi_{23}$, and let $\pi_{13}=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\sigma$, which belongs to $\Pi(\nu,\mu)$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling (iii). Let $q\in L^{2}(\nu;\mathbb{R}^{d})$, $\eta\in L^{2}(\rho;\mathbb{R}^{d})$ and $\theta\in L^{2}(\mu;\mathbb{R}^{d})$; the three discrepancies below are the nonnegative real numbers of The Intrinsic Calculus on the Wasserstein Space: Standing Notation (i).

(Triangle inequality along the composite coupling)(i)

$$\sqrt{\int_{\mathbb{R}^{d+d}}\lVert q(x)-\theta(y)\rVert^{2}\,\pi_{13}(dz)}\ \le\ \sqrt{\int_{\mathbb{R}^{d+d}}\lVert q(x)-\eta(y)\rVert^{2}\,\pi_{12}(dz)}+\sqrt{\int_{\mathbb{R}^{d+d}}\lVert\eta(x)-\theta(y)\rVert^{2}\,\pi_{23}(dz)} .$$