Suppose $\{X_n\}$ has Markov property. Show that for any $n,r \in \Bbb N,i \in S,A \subset S^n, B \subset S^r$
$P[(X_{n+1}, \cdots , X_{n+r}) \in B\ \mid X_n=i,(X_0,\cdots,X_{n-1}) \in A]=P[(X_{n+1},\cdots,X_{n+r}) \in B\ \mid X_n=i]$.
If further $C \subset S$ then,
$P[(X_{n+1}, \cdots , X_{n+r}) \in B\ \mid X_n \in C,(X_0,\cdots,X_{n-1}) \in A]=P[(X_{n+1},\cdots,X_{n+r}) \in B\ \mid X_n \in C]$.
How do I proceed using Markov property? Please help me in this regard.
Thank you very much.