Background
Let $\left(X_t\right)_{t \in I}$ ($I\subseteq\mathbb R$) be an $E$-valued stochastic process ($E$ being a Polish space with the Borel $\sigma$-algebra $\mathcal{B}\left(E\right)$) equipped with the filtration generated by $X$, $\left(\mathcal F_t\right)_{t\in I}=\left(\sigma\left(X_s\space:\space s\leq t\right)\right)_{t\in I}$. Suppose $E$ is countable.
Question
Why is it the case (as claimed in Klenke, Remark 17.2) that if for all $n\in\mathbb N$, all $s_1<\cdots
$\mathbb{P}\left[\left.X_t=i\space\right|\space X_{s_1}=i_1,\dots,X_{s_n}=i_n\right]=\mathbb{P}\left[\left.X_t=i\space\right|\space X_{s_n}=i_n\right]$
then the Markov property applies, namely
$\forall s\leq t\in I\bullet\mathbb{P}\left[\left.X_t\in A\space\right|\space\mathcal{F}_s\right]=\mathbb{P}\left[\left.X_t\in A\space\right|\space X_s\right]$