$H[X_1, .., X_{n-1}|X_n] = H[X_n , ..., X_2|X_1]$ for a stationary source

Question

I posted this question before but i want to give more context. I have the following theorem

For a stationary source $X_1,..,X_n$, the term $H(X_n|X_{n−1}, . . . ,X_1)$ is nonincreasing in n and has a limit $\lim_{n \rightarrow \infty} H[X_n|X_{n-1}, X_{n-2}, \ldots, X_1]$.

Proof

Be $X_1, X_2, \ldots$ a stationary source. Then

$H[X_{n+1}|X_1, X_2, \ldots, X_n] \leq H[X_{n+1}|X_n, \ldots, X_2]$

and because of the stationary of the source follows $H[X_{n+1}|X_n, \ldots, X_2] = H[X_n| X_{n-1}, \ldots, X_1]$.

I dont understand the last step.

I have the following definition:

A sequnece of random variables is called stationary source, if for all m,n $\in N$ the joint distributions of $X_1, ..., X_n$ and $X_{m+1}, .., X_{m+n}$ are identical.

Now i want to show that

$H[X_1, .., X_{n-1}|X_n] = H[X_n , ..., X_2|X_1]$.

I know that

$H[X_1, .., X_{n-1}|X_n]=H[X_1, ..,X_n]-H[X_n]$ and $H[X_n , ..., X_2|X_1]=H[X_1, ..,X_n]-H[X_1]$

But why are these two terms equal?