Let $P$ be a transition matrix on $E$, $\left(X_n\right)_{n \in \mathbb{N}}$ a Markov chain of transition matrix $P$, and $\mu$ a probability measure on $E$.The initial law of $X$ is $\mu$.
Show that for all $n, m \in \mathbb{N}, x, y \in E$ such that $\mathbb{P}_\mu\left(X_m=x\right)>0$
$\mathbb{P}_\mu\left(X_{n+m}=y \mid X_m=x\right)=\mathbb{P}_x\left(X_n=y\right)=P^n(x, y)$
I know how to explain the equation in words but I don't know exactly how to prove it using the properties. The equation tells us that the portability of getting to point $y$ knowing you were at point $x$ at time $m$ is the same as the portability of starting at point $x$ to be at point $y$ at time $n$. It's like a "shift" in the Markov chain.
Do we have to apply the strong markov property?