\begin{equation}\label{eq proof TC 2 reduced euqation}
    2\mu^\ddagger_{\Tilde{a}}-1=\theta\left[\frac{\tau}{\tau+(1-\tau)\left(1-\alpha\frac{\mu_0(1-\mu^\ddagger_{\Tilde{a}})}{(1-\mu_0)\mu^\ddagger_{\Tilde{a}}}+(1-\alpha)\frac{\mu_0}{1-\mu_0}\right)}-\frac{\tau}{\tau+(1-\tau)\alpha\frac{1}{\mu^\ddagger_{\Tilde{a}}}}\right].
\end{equation}
