On a finite discrete state space $S=\{s_1,\ldots,s_N\}$ the general Markov property, that for every $s\in S$, denoting $\mathcal{F}_n=\sigma(X_1,\ldots,X_n)$, $\mathbb{P}(X_{n+1}=s|\mathcal{F}_n)=\mathbb{P}(X_{n+1}=s|X_n)$, reduces to $\mathbb{P}(X_{n+1}=s|X_n=s_n,\ldots,X_1=s_1)=\mathbb{P}(X_{n+1}=s|X_n=s_n)$ for all $s_i\in S$. I have always used the second since I never worked in continuous time, where I think the first version is mostly used. However I am curious to know how one derives that the two statements are equivalent for $S$ finite.
My attempt is evaluating the conditional expectation on $\omega$, but I am not sure if I can say that, for example, if $\omega\in\{X_1=s_1,\ldots,X_n=s_n\}$ then $\mathbb{P}(X_{n+1}=s|\mathcal{F}_n)(\omega)= \mathbb{P}(X_{n+1}=s|X_n=s_n,\ldots,X_1=s_1)$. This would rely on $\mathcal{F}_n=\sigma(\{X_1=s_1,\ldots,X_n=s_n\}_{s_1\in S,\ldots,s_n\in S})$ which I believe it is true since as a discrete vector $(X_1,\ldots,X_n)^{-1}(s_1,\ldots,s_n)=(X_1^{-1}(s_1)\cap\ldots\cap X_n^{-1}(s_n))$ and all the singletons $(s_1,\ldots,s_n)$ generate the Borel $\sigma$-algebra on $S$ do to it being discrete. The property would then follow from the fact that $\{X_1=s_1,\ldots,X_n=s_n\}_{s_1\in S,\ldots,s_n\in S}$ is a partitioning of $\Omega$, and the properties of conditional expectation on partitioning.
Is this the right way to derive the property? Thank you in advance.