# Why is there “markov property” in proving the renewal equation for a renewal process?

When proving the renewal equation for a renewal process in Wikipedia

The renewal function satisfies $$m(t) = F_S(t) + \int_0^t m(t-s) f_S(s)\, ds$$ where $F_S$ is the cumulative distribution function of $S_1$ and $f_S$ is the corresponding probability density function.

Proof of the renewal equation

We may iterate the expectation about the first holding time: $$m(t) = \mathbb{E}[X_t] = \mathbb{E}[\mathbb{E}(X_t \mid S_1)]. \,$$ But by the Markov property $$\mathbb{E}(X_t \mid S_1=s) = \mathbb{I}_{\{t \geq s\}} \left( 1 + \mathbb{E}[X_{t-s}] \right). \,$$ ...

A renewal process is not Markovian. Why is there "by the Markov property" in the proof? Thanks!

The renewal process $(X_t)_{t\geqslant0}$ is not a Markov process but it satisfies the Markov property at time $S_1$. That is, for every $s$, the process $(X_{t+s})_{t\geqslant0}$ conditioned by the event $[S_1=s]$ is distributed like the process $(1+X_t)_{t\geqslant0}$.