Uniqueness of a Bounded Continuous Viscosity Solution on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$, since every open ball of $(H,d_{H})$ is a subset of $H$; accordingly $V\cap H=V$ in the notation of Hilbert Triples: Standing Notation and Background §open-sets. Let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$ that is locally strictly proper, satisfies the first-order structure condition and satisfies the shift-continuity condition.

Let $C\in\mathbb{R}$ and let $u_{1},u_{2}:H\to\mathbb{R}$ be viscosity solutions of $F$ on $H$ that are continuous on $H$ and satisfy

¶ Then $u_{1}(x)=u_{2}(x)$ for every $x\in H$.