Let $X : Ω \to S^{Z_+}$ be a random process for the discrete state space $S$. Show that $X$ is a time homogeneous Markov chain if and only if there exists a function $f(x,y)$ such that for all $x_0, x_1,. . ., x_{n−1}, x,y ∈ S$, we have $$P(X_{n+1} = y| X_0 = x_0,X_1 = x_1,. . .,X_{n−1} = x_{n−1},X_n = x) = f(x,y)$$
I could prove it just one way $\implies$
Considering $X \to$ homogeneous markov chain so, the whole above equation will come down to $$P(X_{n+1} = y | X_n = x) = f(x,y) = p_{xy}$$ where $p_{xy} = $ the transition probability from state $x \to y$ so it will be some $f(x,y)$.
But how to prove the other way ?