I was checking the proof for the following lemma:
Lemma. Suppose that $\pi$ is a stationary policy. Then $V^π$ and $Q^π$ satisfy the following Bellman consistency equations: for all $s \in S$ and $a \in A$, \begin{align} V^{\pi}(s) &= Q^{\pi}(s, \pi(s)); \ Q^{\pi}(s, a) = r(s, a) + \gamma \sum_{s' \in S} P(s' | s, a) V^{\pi}(s'). \end{align}**
To prove that, they state that $\pi(s) = \pi(s')$ for all $s$ and $s'$, which is a simplification to consider the average return following the policy $\pi$. I didn't get this, since a policy maps state to actions, how could this assumption of different states be equal following $\pi$?