$$
\begin{array}{l} \frac {1}{2} \operatorname * {P r} [ \mathrm {S} (A) = \mathrm {S} (B) | w _ {1} ^ {A}, w _ {1} ^ {B} ] + \operatorname * {P r} [ \mathrm {S} (A) > \mathrm {S} (B) | w _ {1} ^ {A}, w _ {1} ^ {B} ] \\ = \frac {1}{2} \Pr [ S (A) = + \infty \cap S (B) = + \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] + \frac {1}{2} \Pr [ S (A) = - \infty \cap S (B) = - \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] \\ + \frac {1}{2} \Pr [ S (A) = S (B) \cap S (A) \neq \infty \cap S (B) \neq \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] + \Pr [ S (A) = + \infty \cap S (B) \neq + \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] \\ + \Pr [ S (A) \neq - \infty \cap S (B) = - \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] + \Pr [ S (A) > S (B) \cap S (A) \neq \infty \cap S (B) \neq \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] \\ \leq \frac {1}{2} \Pr [ S (A) = + \infty \cap S (B) = + \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] + \frac {1}{2} \Pr [ S (A) = - \infty \cap S (B) = - \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] \\ + \operatorname * {P r} \bigl [ \mathrm {S} (A) = + \infty \cap \mathrm {S} (B) \neq + \infty | w _ {1} ^ {A}, w _ {1} ^ {B} \bigr ] + \operatorname * {P r} \bigl [ \mathrm {S} (A) \neq - \infty \cap \mathrm {S} (B) = - \infty | w _ {1} ^ {A}, w _ {1} ^ {B} \bigr ] \\ + \Pr \left[ \mathrm {S} (A) \geq \mathrm {S} (B) \cap \mathrm {S} (A) \neq \infty \cap \mathrm {S} (B) \neq \infty | w _ {1} ^ {A}, w _ {1} ^ {B} \right] \\ = \frac {1}{2} \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(\hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \frac {1}{2} \left(1 - \hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(1 - \left(\hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}}\right) \\ \left. + \left(1 - \left(1 - \hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}}\right) \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \Pr [ S (A) \geq S (B) \cap S (A) \neq \infty \cap S (B) \neq \infty | w _ {1} ^ {A}, w _ {1} ^ {B} ] \right] \\ = \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} + \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} - \frac {1}{2} \left[ \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(\hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \left(1 - \hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} \right] \\ + \Pr \left[ \mathrm {S} (A) \geq \mathrm {S} (B) \cap \mathrm {S} (A) \neq \infty \cap \mathrm {S} (B) \neq \infty \mid w _ {1} ^ {A}, w _ {1} ^ {B} \right] \\ \leq \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} + \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} - \frac {1}{2} \left[ \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(\hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \left(1 - \hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} \right] \\ + \mathrm {P r} \left[ \frac {\hat {v} ^ {A} - \hat {q} ^ {A}}{\sqrt {\det (\hat {\mathbf {U}} ^ {A})}} \geq \frac {\hat {v} ^ {B} - \hat {q} ^ {B}}{\sqrt {\det (\hat {\mathbf {U}} ^ {B})}} \Big | w _ {1} ^ {A}, w _ {1} ^ {B} \right] \\ \leq \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} + \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} - \frac {1}{2} \left[ \left(\hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(\hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} + \left(1 - \hat {w} _ {1} ^ {A}\right) ^ {n ^ {A}} \left(1 - \hat {w} _ {1} ^ {B}\right) ^ {n ^ {B}} \right] \\ + \exp \left\{- \frac {2 \left(w _ {1} ^ {B} - w _ {1} ^ {A}\right) ^ {2}}{\left(\frac {1}{n ^ {A} (1 - \lambda^ {A}) ^ {2}} + \frac {1}{n ^ {B} (1 - \lambda^ {B}) ^ {2}}\right)} \right\} \tag {Lemma 1} \\ \end{array}
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