In example 2 to Theorem X.70 in Reed-Simon we want to construct a propagator for the heat equation with generator $$A(t) = - \Delta + q(x,t) + M + 1$$ on $C_{\infty}$ where $q$ is a bounded (with bound $M$), real-valued continuously differentiable function with $\partial q(x,t)/ \partial t$ bounded. In order to fulfill the assumptions of Theorem X.70 we have to check that for all $\varphi \in C_{\infty}$, $$ (t-s)^{-1} C(t,s) \varphi$$ where $C(t,s) = A(t)A(s)^{-1} - I$, is uniformly strongly continuous (however I'm not sure what that means since it is a function and not an operator) and uniformly bounded in $t,s$ for $t\neq s$ on any fixed compact subinterval of $I \ni t,s$. Moreover, we have to check that for all $\varphi \in C_{\infty}$, $\lim_{t \uparrow s} (t-s)^{-1} C(t,s) \varphi$ exists uniformly for $t$ in each compact subinterval of $I$ and it is bounded and strongly continuous (again, what does that mean) in $t$.
Now since $A(t)$ generates a contraction semigroup with common domain $D(A(t)) = D(-\Delta)$ for all $t$, we know that $A(t)A(s)^{-1}$ is bounded, so it follows that $A(t)A(s)^{-1} - I = C(t,s)$ is bounded. Then $|(t-s)^{-1} C(t,s) \varphi| \leq K(t-s)^{-1} |\varphi|$ which is bounded and continuous for $t\neq s$. Is this reasoning correct so far?
However I have no idea how to tackle the 2ns requirement. Any help appreciated!