is due to the memoryless property of the channel and the fact that conditioning does not increase the entropy (like parts (d)-(f) in (\[eqn:cap\_set1\_fanoI\])).

By applying a similar technique based on (\[eqn:set1\_I\^n2\]) and (\[eqn:set1\_II\^n\]), we obtain:

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

Using a standard time-sharing argument [@ElgKim11 P. 85] for (\[eqn:cap\_set1\_fanoI\])-(\[eqn:cap\_set1\_fanoV\]) completes the proof.

Next, we derive the second set of strong interference conditions (*Set2*), under which the region of Theorem \[thm:ach2\] is the capacity region. For every $p(.)\in{\mathcal{P}}$, *Set2* includes (\[eqn:set1\_I\]) and the following conditions:

[rcl]{} I(X\_1,X\_3;Y\_1|X\_2)&& {I(X\_1;Y\_2),I(X\_1,X\_3;Y\_3|X\_2)}\[eqn:set2\_II\]\
I(X\_2,X\_3;Y\_2|X\_1)&& {I(X\_2;Y\_1),I(X\_2,X\_3;Y\_3|X\_1)}\[eqn:set2\_III\]\
{I(X\_1;Y\_2)I(X\_1;Y\_3)}&&{I(X\_2;Y\_1)I(X\_2;Y\_3)} \[eqn:set2\_IV\]

\[thm:cap\_set2\] The capacity region of the three-user C-IFC, satisfying (\[eqn:set1\_I\]) and (\[eqn:set2\_II\])-(\[eqn:set2\_IV\]), is given by

[rl]{} \_2 =\_[p(.)]{} {(R\_1,R\_2,R\_3): &R\_1 0, R\_2 0, R\_3 0\
&R\_[3]{}I(X\_3;Y\_3|X\_1,X\_2) \[eqn:Cap2\_I\]\
&R\_[1]{}+R\_[3]{}I(X\_1,X\_3;Y\_1|X\_2) \[eqn:Cap2\_II\]\
&R\_[2]{}+R\_[3]{}I(X\_2,X\_3;Y\_2|X\_1) \[eqn:Cap2\_III\]}}.

Consider the region of Theorem \[thm:ach2\]. Using the second terms of conditions (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]), the bounds in (\[eqn:ach1\_I\])-(\[eqn:ach1\_III\]) reduce to ${\mathcal{C}}_2$. Based on the first term of condition (\[eqn:set2\_II\]), the bound in (\[eqn:ach2\_I\]) is redundant due to (\[eqn:Cap1\_II\]). Similarly, (\[eqn:Cap1\_III\]) and (\[eqn:set2\_III\]) make the bound in (\[eqn:ach2\_II\]) redundant. Moreover, considering (\[eqn:ach2\_I\]) (or (\[eqn:ach2\_II\])), (\[eqn:set2\_IV\]), and the second bound in (\[eqn:ach1\_III\]) (or (\[eqn:ach1\_II\])), the bound in (\[eqn:ach2\_III\]) becomes redundant.

The bounds in ${\mathcal{C}}_2$ are same as the bounds 