Comparison Principle for Slope-Based Solutions of the Discounted Hopf-Lax Equation on a Complete Metric Space with Interpolation Points

In the setting of Slope-Based Viscosity Solutions on a Metric Space: Standing Notation, suppose that $(X,d)$ is complete and has interpolation points. Let $\rho$ be a positive real and let $f:X\to\mathbb{R}$ be uniformly continuous on $X$. Let $u,v:X\to\mathbb{R}$ be bounded above and below, and suppose that $u$ is an s-subsolution and $v$ is an s-supersolution of the discounted stationary Hopf--Lax equation $\rho\,u+\frac{1}{2}|\nabla u|^{2}=f$ in $X$ (the equation being taken with $\Omega=X$).

¶Then $u(x)\le v(x)$ for every $x\in X$.