This is proposition 2.9 from Ethier and Kurtz' Markov Processes. The proof simply states that this result is a consequence of the next proposition 2.10. However, I can't figure out how 2.10 leads to 2.9. I would greatly appreciate any help.
Prop 2.9. Let $\{T(t)\}$ and $\{S(t)\}$ be strongly continuous contraction semigroups on $L$ with generators $A$ and $B$, respectively. If $A=B$, then $T(t)=S(t)$ for all $t\ge 0$.
Fix a point $x_0 \in X$, and define $u(t) := (S(t)-T(t))x_0$ for $t \geq 0$. Note that $u(t) \in \mathcal D(A)$ for all $t>0$, since $S(t)$ and $T(t)$ map $X$ into $\mathcal D(A)$ for any $t > 0$.
Since $\frac{d}{dt}S(t)x_0 = AS(t)x_0$ and similarly for $T$, it follows that $u$ satisfies the criteria of proposition 2.10 and $u(0)=0$, hence $u(t)=0$ for all $t \geq 0$. Thus $S(t)x_0 = T(t)x_0$ for all $t \geq 0$.