 $$\begin{aligned} \label{double double local extremes asymptotics 0} &\mathbb{P}(A\cap C)+\mathbb{P}( B\cap C)-\mathbb{P}((A\cup B)\cap C)\nonumber\\ &=R(u)H_{\alpha_0}\left(\frac{\mathbb{T}_0}{q_{\alpha_0,c_0}}\right)H_{\alpha_0}\left(\frac{\mathbb{T}_0}{q_{\alpha_0,c_0}},