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