proof follows from Theorem \[thm:ach1\].

Consider a $(2^{nR_1},2^{nR_2},2^{nR_3},n)$ code with average error probability $P_e^n\rightarrow 0$, which implies that $P_{e,u}^{(n)}\rightarrow 0$ for $u\in\{1,2,3\}$. Applying Fano’s inequality [@CovTho06], [@ElgKim11 P. 19] results in

[rcl]{} H(M\_u|Y\_u\^n)P\_[e,u]{}\^[(n)]{} log(2\^[nR\_u]{} - 1) + h(P\_[e,u]{}\^[(n)]{})n\_[un]{}\[eqn:Fano\_delta\]

for $u\in\{1,2,3\}$, where $\delta_{un}\rightarrow 0$ as $P_{e,u}^{(n)}\rightarrow 0$. Note that, due to the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\] and the independence of the messages, we have $p(.)\in{\mathcal{P}}$. Now, we derive the bounds in Theorem \[thm:cap\_set1\]. For the first bound, we obtain

[ll]{} nR\_3=H(M\_3)&H(M\_3|M\_1,M\_2)\
&=I(M\_3;Y\_3\^n|M\_1,M\_2)+H(M\_3|Y\_3\^n,M\_1,M\_2)\
&I(M\_3;Y\_3\^n|M\_1,M\_2)+H(M\_3|Y\_3\^n)\[eqn:cap\_set1\_fanoI\_before\]\
&I(M\_3;Y\_3\^n|M\_1,M\_2)+n\_[3n]{}

where (a) follows since $M_1$, $M_2$ and $M_3$ are independent, (b) is due to the fact that conditioning does not increase the entropy and (c) follows from (\[eqn:Fano\_delta\]) for $u=3$. Hence,

[ll]{} nR\_3- n\_[3n]{}&I(M\_3;Y\_3\^n|M\_1,M\_2)\
& I(M\_3,X\_3\^n;Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)\
&=H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
& H(Y\_3\^n|X\_1\^n,X\_2\^n)-H(Y\_3\^n|M\_1,M\_2,X\_1\^n,X\_2\^n,M\_3,X\_3\^n)\
& H(Y\_3\^n|X\_1\^n,X\_2\^n)-H(Y\_3\^n|X\_1\^n,X\_2\^n,X\_3\^n)=I(X\_3\^n;Y\_3\^n|X\_1\^n,X\_2\^n)\
&\_[i=1]{}\^[n]{}I(X\_3\^n;Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{})\[eqn:cap\_set1\_fanoI\]\
&=\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{})-I(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{},X\_3\^n)\
&\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})-I(Y\_[3,i]{}|X\_1\^n,X\_2\^n,Y\_3\^[i-1]{},X\_3\^n)\
&\_[i=1]{}\^[n]{}H(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})-I(Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{},X\_[3,i]{})=\_[i=1]{}\^[n]{}I(X\_[3,i]{};Y\_[3,i]{}|X\_[1,i]{},X\_[2,i]{})

where (a) is due to the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], (b) and (e) are due to the fact that conditioning does not increase the entropy, (c) follows from the fact that $(M_1,M_2,M_3)\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow Y_3^n$ forms a Markov chain, (d) is obtained from the chain rule, and (f) follows from the memoryless property of the channel.

Now, applying (\[eqn:Fano\_delta\]) for $u\in\{1,3\}$ 