$$
\begin{array}{l} \left| \left(\mathcal {T} ^ {*}\right) ^ {h} q ^ {\pi} (s, a) - \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h} q ^ {\pi} (s, a) \right| = \\ \left| r (s, a) + \gamma \sum_ {s ^ {\prime}} p \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \hat {r} (s, a) - \gamma \sum_ {s ^ {\prime}} \hat {p} \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, \right. \right. (35) \\ \leq \epsilon_ {r} + \gamma \left| \sum_ {s ^ {\prime}} p \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \sum_ {s ^ {\prime}} \hat {p} \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \right| (36) \\ = \epsilon_ {r} + \gamma \Bigg | \sum_ {s ^ {\prime}} p (s ^ {\prime} | s, a) \max  _ {a ^ {\prime} \in \mathcal {A}} (\mathcal {T} ^ {*}) ^ {h - 1} q ^ {\pi} (s ^ {\prime}, a ^ {\prime}) + \sum_ {s ^ {\prime}} \hat {p} (s ^ {\prime} | s, a) \max  _ {a ^ {\prime} \in \mathcal {A}} (\mathcal {T} ^ {*}) ^ {h - 1} q ^ {\pi} (s ^ {\prime}, a ^ {\prime}) \\ - \sum_ {s ^ {\prime}} \hat {p} \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \sum_ {s ^ {\prime}} \hat {p} \left(s ^ {\prime} \mid s, a\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \Bigg | (37) \\ = \epsilon_ {r} + \gamma \left| \sum_ {s ^ {\prime}} \hat {p} \left(s ^ {\prime} \mid s, a\right) \left(\max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right)\right) \right. \\ + \sum_ {s ^ {\prime}} \left(p \left(s ^ {\prime} \mid s, a\right) - \hat {p} \left(s ^ {\prime} \mid s, a\right)\right) \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \Bigg | (38) \\ \leq \epsilon_ {r} + \gamma \max  _ {s ^ {\prime}, a ^ {\prime}} \left| \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \left(\mathcal {T} _ {\hat {m}} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \right| \\ + \gamma \sum_ {s ^ {\prime}} \left| p \left(s ^ {\prime} \mid s, a\right) - \hat {p} \left(s ^ {\prime} \mid s, a\right) \right| \left| \max  _ {a ^ {\prime} \in \mathcal {A}} \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \right| (39) \\ \leq \epsilon_ {r} + \gamma \max  _ {s ^ {\prime}, a ^ {\prime}} \left| \left(\mathcal {T} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) - \left(\mathcal {T} _ {m} ^ {*}\right) ^ {h - 1} q ^ {\pi} \left(s ^ {\prime}, a ^ {\prime}\right) \right| + \gamma \epsilon_ {p} v _ {\max } ^ {*}. (40) \\ \end{array}
$$