$$
\begin{array}{l} \mathbb {P} \left[ p _ {1} \neq \tilde {v} _ {t} ^ {u}, p _ {1} \neq v _ {m} \right] \\ \leq \mathbb {P} \left[ p _ {1} \neq v _ {m} \right] \\ = \mathbb {P} \left[ p _ {1} \neq v _ {m}, \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \frac {\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {p _ {1}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]}{\sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} <   1 \right] \\ \leq \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {p _ {1}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] <   \sum_ {h = 1} ^ {H} \left[ w _ {h} f _ {v _ {m}} ^ {h} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] \right] \\ \leq \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {h = 1} ^ {H} w _ {h} \left[ \tilde {\eta} _ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \epsilon \right] <   \sum_ {h = 1} ^ {H} w _ {h} \left[ \tilde {\eta} _ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \epsilon \right] \right] \\ = \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \tilde {\eta} _ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \epsilon <   \tilde {\eta} _ {m} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \epsilon \right] \\ = \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \sum_ {v _ {j} \in \mathcal {V}} \tau_ {v _ {j} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \sum_ {v _ {j} \in \mathcal {V}} \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + 2 \epsilon \right] \\ = \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \tau_ {p _ {1} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \sum_ {v _ {j} \neq p _ {1}} \tau_ {v _ {j} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \right. \tag {13} \\ \tau_ {p _ {1} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \sum_ {v _ {j} \neq p _ {1}} \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + 2 \epsilon \Bigg ] \\ = \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \left[ \tau_ {p _ {1} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \tau_ {p _ {1} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] <   \right. \\ \sum_ {v _ {j} \neq p _ {1}} \left[ \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \tau_ {v _ {j} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + 2 \epsilon \Bigg ] \\ = \mathbb {P} \left[ \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \geq \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right), \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \frac {\sum_ {v _ {j} \neq p _ {1}} \left[ \tau_ {v _ {j} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \tau_ {v _ {j} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right] \eta_ {v _ {j}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + 2 \epsilon}{\left[ \tau_ {p _ {1} p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) - \tau_ {p _ {1} v _ {m}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \right]} \right] \\ = \mathbb {P} \left[ \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) \leq \eta_ {p _ {1}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) <   \eta_ {p _ {2}} \left(\tilde {\mathbf {x}} _ {t} ^ {u}\right) + \right. \\ \frac {\sum_ {v _ {j} \neq p _ {1}} \left[ \tau_ {v _ {j} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {v _ {j} p _ {1}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \eta_ {v _ {j}} (\tilde {\mathbf {x}} _ {t} ^ {u}) + 2 \epsilon - \left[ \tau_ {p _ {1} p _ {1}} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right] \eta_ {p _ {2}} (\tilde {\mathbf {x}} _ {t} ^ {u})}{\left[ \tau_ {p _ {1} p _ {1}} (\tilde {\mathbf {x}} _ {t} ^ {u}) - \tau_ {p _ {1} v _ {m}} (\tilde {\mathbf {x}} _ {t} ^ {u}) \right]} \\ \end{array}
$$