and the independence of the messages, we can bound $R_1+R_3$ as

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

where (a) follows from the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], (b) follows from the fact that $M_3\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow Y_3^n$ forms a Markov chain, (c) is obtained from (\[eqn:set1\_I\^n1\]), (d) follows from the chain rule, and (e) follows from the memoryless property of the channel and the fact that conditioning does not increase the entropy (like parts (d)-(f) in (\[eqn:cap\_set1\_fanoI\])). Applying similar steps using (\[eqn:Fano\_delta\]) for $u\in\{2,3\}$ and (\[eqn:set1\_I\^n2\]), we can show that,

[l]{} n(R\_2+R\_3)-n(\_[2n]{}+\_[3n]{})\_[i=1]{}\^[n]{}I(X\_[2,i]{},X\_[3,i]{};Y\_[2,i]{}|X\_[1,i]{}).\[eqn:cap\_set1\_fanoIII\]

Finally, using (\[eqn:Fano\_delta\]) for $u\in\{1,2,3\}$ and the independence of the messages, the sum-rate bounds can be obtained as

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

where (a) follows from the encoding functions $f_1$, $f_2$ and $f_3$, defined in Definition \[def:code\], and the fact that conditioning does not increase the entropy, (b) is due to the fact that conditioning does not increase the entropy, (c) follows since $(M_2,M_3)\rightarrow (X_1^n,X_2^n,X_3^n)\rightarrow (Y_2^n,Y_3^n)$ forms a Markov chain, (d) is obtained from (\[eqn:set1\_I\^n1\]) and (\[eqn:set1\_III\^n\]), (e) follows from the chain rule, and (f) 