Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant $N^{-1/2}$

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used (the letter $P$ below denotes a probability measure on a configuration space), let $q,N\in\mathbb{N}$, with tensor powers $\rho^{\otimes N}$ of The Tensor Power of a Probability Measure on Euclidean Space §tensor and one-particle marginals $P^{[1]}$ of The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal; tensor powers and one-particle marginals of measures with finite second moment have finite second moment by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments, and $W_{2}$ is the Wasserstein distance on $\mathcal{P}_{2}(\mathbb{R}^{q})$ and on $\mathcal{P}_{2}(\mathbb{R}^{qN})$.

1. (One-particle marginals)¶ For $P,P'\in\mathcal{P}_{2}(\mathbb{R}^{qN})$, $N\,W_{2}(P^{[1]},P'^{[1]})^{2}\le W_{2}(P,P')^{2}$.

2. (Tensor powers)¶ For $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{q})$, $W_{2}(\rho^{\otimes N},\rho'^{\otimes N})^{2}=N\,W_{2}(\rho,\rho')^{2}$.