Currently I am reading Notes on Markov Chains.
Context: We consider an amount of $S$ dollars which is to be shared between two players A and B. We let $X_n$ represent the wealth of Player A at time $n \in \mathbb{N}.$ We let $R_A$ to be Player A loses all his capital at some time, that is, $$R_A = \bigcup_{n\in\mathbb{N}} \{X_n=0\}.$$
At page $54,$ Lemma $2.2$ is stated as follows:
For all $k=1,2,...,S-1,$ we have $$\mathbb{P}(R_A|X_1 =k\pm 1, X_0=k) = \mathbb{P}(R_A|X_1 =k\pm 1) = \mathbb{P}(R_A| X_0=k\pm 1).$$ In other words, the ruin probability depends on the data of the starting point and not on the starting time.
My question is
Question: How to show the second equality directly, that is, $$\mathbb{P}(R_A|X_1 =k\pm 1) = \mathbb{P}(R_A| X_0=k\pm 1)?$$
Any hint is appreciated.