\begin{align}\label{eq:cond-freq-exp}
f^{(1\to L)}_{i_{1\to L}}(\alpha|\mu_1) &= \frac{v^{-1}m^L\left(1+\frac{U^{(1\to L)}_{i_{1\to L}}(\mu_1;\alpha)-m^L/v}{m^L/v}\right)}{n_c v^{-1} m^L \left(1+\frac{D^{(1\to L)}_{i_{1\to L}}(\mu_1)-n_c m^L / v}{m^L}\right)} \\
&= \frac{1}{n_c} + \frac{1}{n_c}\frac{U^{(1\to L)}_{i_{1\to L}}(\mu_1;\alpha)-m^L/v}{m^L/v} - \frac{1}{n_c}\frac{D^{(1\to L)}_{i_{1\to L}}(\mu_1)-n_c m^L / v}{m^L / v} \nonumber \\
&= \frac{1}{n_c} + \frac{v}{n_c m^L}\left(U^{(1\to L)}_{i_{1\to L}}(\mu_1;\alpha)-\frac{1}{n_c}D^{(1\to L)}_{i_{1\to L}}(\mu_1)\right).
\end{align}
