the set of all possible codeword sequences, it follows that $$\begin{aligned}
\frac{1}{n}H(\rvu^n)&= \frac{1}{n}\log_2 |\cC|  \notag\\
&= I(\rvu;\rvyr) - 2\eps_n \label{eq:EquivTerm1}\end{aligned}$$

Next, as verified below, the channel to the eavesdropper $(\rvu^n,\rvs^n) \rightarrow \rvye^n$, is memoryless: $$\begin{aligned}
&p_{\rvye^n|\rvu^n,\rvs^n}(y_e^n|u^n,s^n) \\&= \sum_{x^n \in \cX^n}
p_{\rvye^n|\rvu^n,\rvs^n,\rvx^n}(y_e^n|u^n,s^n,x^n)p_{\rvx^n|\rvu^n,\rvs^n}(x^n|u^n,s^n)\\
&= \sum_{x^n \in \cX^n} \prod_{i=1}^n
p_{\rvy_{e}|\rvu,\rvs,\rvx}(y_{e,i}|u_i,s_i,x_i)p_{\rvx|\rvu,\rvs}(x_i|u_i,s_i)\\
&= \prod_{i=1}^n
\sum_{x_i \in \cX}p_{\rvy_{e}|\rvu,\rvs,\rvx}(y_{e,i}|u_i,s_i,x_i)p_{\rvx|\rvu,\rvs}(x_i|u_i,s_i)\\
&= \prod_{i=1}^n p_{\rvy_{e}|\rvu,\rvs}(y_{e,i}|u_i,s_i)\\\end{aligned}$$ The second step above follows from the fact that the channel is memoryless and the symbol $\rvx_i$ at time $i$ is generated as a function of $(\rvu_i,\rvs_i)$. Hence we have that $$\begin{aligned}
&\frac{1}{n}H(\rvye^n|\rvs^n,\rvu^n) = \sum_{i=1}^n H(\rvy_{e,i}|\rvs_i,\rvu_i).\label{eq:EquivTerm2}\end{aligned}$$

Furthermore note that $$\frac{1}{n}H(\rvye^n) \le \sum_{i=1}^n H(\rvy_{ei}).
\label{eq:YeBound}$$

Finally, in order to lower bound the term $I(\rvs^n;\rvye^n|\rvu^n)$ we let $J$ to be a random variable which equals 1 if $(\rvs^n,\rvu^n)$ are jointly typical. Note that $\Pr(J=1) = 1 -
o_n(1)$. $$\begin{aligned}
&\frac{1}{n}I(\rvs^n;\rvye^n|\rvu^n) = \frac{1}{n}H(\rvs^n|\rvu^n)
- \frac{1}{n}H(\rvs^n|\rvu^n,\rvye^n) \notag\\
&\ge \frac{1}{n}H(\rvs^n|\rvu^n,J=1)\Pr(J=1)
- \frac{1}{n}H(\rvs^n|\rvu^n,\rvye^n)\notag\\
&\ge \frac{1}{n}H(\rvs^n|\rvu^n,J=1) -
\frac{1}{n}H(\rvs^n|\rvu^n,\rvye^n) - o_n(1) \notag\\
&\ge H(\rvs|\rvu) - \frac{1}{n}H(\rvs^n|\rvu^n,\rvye^n) - o_n(1)
\label{eq:NoMem}\\
&\ge H(\rvs|\rvu) -
\frac{1}{n}\sum_{i=1}^nH(\rvs_i|\rvu_i,\rvy_{e,i}) -
o_n(1)\label{eq:mutInfBound}\end{aligned}$$ where  follows from the fact that $\rvs^n$ is an i.i.d. sequence and hence conditioned on the fact that $(\rvs^n,\rvu^n)$ is a pair of typical sequence there are $2^{n H(\rvs|\rvu)- no_n(1)}$ possible sequences $\rvs^n$.

Substituting , ,  and  in 