I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need that the following version of Markov property also holds: if $1 \leq t_1 < t_2 < ... < t_n$ and $s_1, ... , s_{n+1} \in S$ we have that
$$\mathbb{P}[X_{t_n+1}=s_{n+1} | X_{t_n}=s_n, X_{t_n-1}=s_{n-1},...,X_{t_1}=s_1 ]=\mathbb{P}[X_{t_n+1}=s_{n+1}| X_{t_n}=s_n]$$
It looks like an obvious fact but I have no idea how to prove it. I tried to start from the definition of a Markov chain but I don't see how to make the step towards this version. Could anyone help me? Thanks in advance!
Definition: Consider $X_1, X_2, ...$ to be a sequence of random variables taking values on $S$ defined on a probability space $(\Omega, F, \mathbb{P})$ then the Markov property holds if: $$\mathbb{P}[X_{n+1}=s_{n+1}|X_n=s_n,..., X_1=s_1]=\mathbb{P}[X_{n+1}=S_{n=1}|X_n=s_n]$$
Starting with the definition, $\newcommand{\prob}{\mathbb{P}}$ $$ \prob(X_n = x_n | X_{n-1} = x_{n-1},…,X_0 = x_0 ) = \prob(X_n = x_n | X_{n-1} = x_{n-1}) $$
Then let $j_1...j_N$ be a labelling of $\{0,1,2,3,...,t_n\}\setminus \{t_1,t_2,...,t_n\}$. You just need to sum over the values you don't specify in the conditioning:
\begin{align} &\prob(X_{t_n + 1} = s_{n+1} | X_{t_n} = s_{n},…,X_{t_0} = s_0 ) \\ = &\sum_{a_1∈ S} … \sum_{a_N∈ S} \prob(X_{t_n +1} = s_n | X_{j_i} = a_i ∀ i\ X_{t_i} = s_i ∀ i\ )\prob(X_{j_i} = a_i ∀ i\ ) \\ = &\sum_{a_1∈ S} … \sum_{a_N∈ S} \prob(X_{t_n +1} = s_n | X_k = x_k ∀ k<t_n+1 )\prob(X_{j_i} = a_i ∀ i\ ) \\ \overset{\star}{=} &\sum_{a_1∈ S} … \sum_{a_N∈ S} \prob(X_{t_n +1} = s_n | X_{t_n} = s_n )\prob(X_{j_i} = a_i ∀ i\ ) \\ =&\prob(X_{t_n +1} = s_n | X_{t_n} = s_n )\sum_{a_1∈ S} … \sum_{a_N∈ S} \prob(X_{j_i} = a_i ∀ i\ ) \\ =& \prob(X_{t_n +1} = s_n | X_{t_n} = s_n ) \end{align}
where the equality marked $\star$ is by the markov property.