how to reformulate general markov property in discrete case

Question

I read the wiki article on the markov property http://en.wikipedia.org/wiki/Markov_property#Definition and wondered how to work out this reformulation. It seems intuitively but I can not work it out.

I guess one would need to show two implications to verify the equivalence of both conditions, the general and the special case one:

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From the general Markov property one simply obtains $$P(X_n=x_n | X_{n-1},...,X_0 ) = P(X_n=x_n | X_{n-1})$$

But how to continue from there?

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From the special discrete case Markov property I do not even know any way to start rigorously..

Answer 1

For every $n\geqslant0$, $(x_0,x_1,\ldots,x_n)$ such that $P(X_0=x_0,X_1=x_1,\ldots,X_{n}=x_{n} )\ne0$, and every $x$, $$ P(X_{n+1}=x | X_0=x_0,X_1=x_1,\ldots,X_{n}=x_{n} ) = P(X_{n+1}=x | X_{n}=x_{n}). $$