Suppose that $X_n$ is a Markov Chain.Then for $m,n \in N$ such that $m<n$
$Pr[X_n=j_n|X_m=j_m,X_{m-1}=j_{m-1},...=X_0=j_0]=Pr[X_n=j_n|X_m=j_m]$
When proving for n=3,m=1 case we have to show that $Pr[X_3=j_3|X_1=j_1,X_0=j_0]=Pr[X_3=j_3|X_1=j_1] $
I don't understand the following step.
$Pr[X_3=j_3|X_1=j_1 and X_0=j_0]=
\sum_{j_2}Pr[(X_3=j_3)\cap (X_2=j_2)|(X_1=j_1)\cap(X_0=j_0)]$
I don't understand how this $j_2$ terms are brought in and why?