1$, each user can achieve a symmetric rate given by $$R_{sym} = \frac{1}{2}\log(P).$$ b) If $a^2 \geq 2$, and there exists integer $N_1 > 0$ such that $$P_a^{N_1} < P < P_a^{N_1+1},$$ then each user can achieve a symmetric rate given by $$R_{sym} = \frac{N_1}{2}\log(a^2-1) + \frac{1}{2}\log\left(1 + \frac{(2a^2+1)(P - P_a^{N_1})}{1 + (2a^2+1)P_a^{N_1}}\right).$$ c) If $a^2 \geq 2$, and there exists integer $N_1 > 0$ such that $P_a^{N_1} = P$, then each user can achieve a symmetric rate given by $$R_{sym} = \frac{N_1}{2}\log(a^2-1).$$ d) If $a^2 \leq \frac{1}{3}$ and $P \leq \frac{1-a^2}{2a^4}$, each user can achieve a symmetric rate of $$R_{sym} = \mathcal{R}_{HK}(P, a, 1).$$ e) If $a^2 \leq \frac{1}{3}$ and there exists integer $N_2 > 0$ such that $$P_a^{N_2} < P < P_a^{N_2+1}$$ then each user can achieve a symmetric rate of $$\begin{aligned}
\begin{array}{l}
R_{sym} = \max_{i = N_2 - 1 : N_2} \frac{i}{2}\log\left(\frac{1-a^2}{2a^2}\right) + \\
 \qquad\qquad\qquad\quad\mathcal{R}_{hk} (P - P_a^i, (2a^2+1)P_a^i, a)\end{array}
$$ f) If $a^2 \leq \frac{1}{3}$ and there exists integer $N_2 > 0$ such that $P = P_a^{N_2}$, then each user can achieve a symmetric rate given by $$R_{sym} = \frac{N_2}{2}\log\left(\frac{1-a^2}{2a^2}\right).$$

The proof of the above Lemma is very similar to the proof of Theorem 1 and 