1)"Theorem:Given that the system is in State s at time n, the probability of making the transition from State i at time n + k to State j at time n + k + 1 is

P(X_{n+k} = i | X_n =s) * P(X_{n+k+1}=j | X_{n+k} =i)."

I believe the probability in question is

P(X_{n+k+1} =j ∩ X_{n+k}=i | X_n =s).

The source of the theorem did not include a proof, so I don't actually understand why this theorem must be true. How can we formally prove it? Where do we need to use Markov's property?

2) For a discrete time Markov Chain, is it always true that

P(X_{n+2}=j | X_n =i ∩ X_{n-1}=s) = P(X_{n+2}=j | X_n =i) ??

I know that Markov's property implies that the above equality would always be true if the part in red is replaced by 1. But in this case, when we have 2 (instead of 1), would the equality still be true? If so, how can we formally prove it?

Thanks in advance!