of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]) make the (\[eqn:ach2\_I\]) and (\[eqn:ach2\_II\]) redundant and (\[eqn:set2\_IV\]) is used to make the (\[eqn:ach2\_III\]) redundant. However, (\[eqn:set1\_I\]) is used to prove the converse part for the rates in (\[eqn:Cap2\_II\]) and (\[eqn:Cap2\_III\]).

These results are summarized in Table \[tbl:str\_cond2\].

                                                           Achievability                                                                                                   Converse
  -------- ------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------
   *Set1*                   (\[eqn:set1\_IV\]) + second terms of (\[eqn:set1\_II\]),(\[eqn:set1\_III\])                                     (\[eqn:set1\_I\]) $\rightarrow$ (\[eqn:Cap1\_II\]),(\[eqn:Cap1\_III\])
              $\rightarrow$ second terms of (\[eqn:ach1\_II\]),(\[eqn:ach1\_III\]) + third term of (\[eqn:ach1\_IV\])     (\[eqn:set1\_I\]) + first terms of (\[eqn:set1\_II\]),(\[eqn:set1\_III\]) $\rightarrow$ (\[eqn:Cap1\_IV\])
   *Set2*   second terms of (\[eqn:set2\_II\]),(\[eqn:set2\_III\]) $\rightarrow$ (\[eqn:ach1\_II\]),(\[eqn:ach1\_III\])                     (\[eqn:set1\_I\]) $\rightarrow$ (\[eqn:Cap2\_II\]),(\[eqn:Cap2\_III\])
             first terms of (\[eqn:set2\_II\]),(\[eqn:set2\_III\]) $\rightarrow$ (\[eqn:ach2\_I\]),(\[eqn:ach2\_II\])    
                                       (\[eqn:set2\_IV\]) $\rightarrow$ (\[eqn:ach2\_III\])                              

Assume that the following set of strong interference conditions (*Set1*) holds for every $p(.)\in{\mathcal{P}}$:

[rcl]{} I(X\_3;Y\_3|X\_1,X\_2)&& {I(X\_3;Y\_1|X\_1,X\_2),I(X\_3;Y\_2|X\_1,X\_2)}\[eqn:set1\_I\]\
I(X\_1,X\_3;Y\_1|X\_2)&& {I(X\_1;Y\_2|X\_2),I(X\_1,X\_3;Y\_3|X\_2)}\[eqn:set1\_II\]\
I(X\_2,X\_3;Y\_2|X\_1)&& {I(X\_2;Y\_1|X\_1),I(X\_2,X\_3;Y\_3|X\_1)}\[eqn:set1\_III\]\
{I(X\_1,X\_2,X\_3;Y\_1),I(X\_1,X\_2,X\_3;Y\_2)}&&I(X\_1,X\_2,X\_3;Y\_3) \[eqn:set1\_IV\]

In fact, under these conditions interfering signals at the receivers are strong enough that all messages can be jointly decoded by all the receivers.

\[thm:cap\_set1\] The capacity region of the three-user C-IFC, satisfying (\[eqn:set1\_I\])-(\[eqn:set1\_IV\]), is given by

[rl]{} \_1 =\_[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) \[eqn:Cap1\_I\]\
&R\_[1]{}+R\_[3]{}I(X\_1,X\_3;Y\_1|X\_2) \[eqn:Cap1\_II\]\
&R\_[2]{}+R\_[3]{}I(X\_2,X\_3;Y\_2|X\_1) \[eqn:Cap1\_III\]\
&R\_[1]{}+R\_[2]{}+R\_[3]{} {I(X\_1,X\_2,X\_3;Y\_1),I(X\_1,X\_2,X\_3;Y\_2)}}. \[eqn:Cap1\_IV\]

\[remark:cap\_set1\_I\] The message of the cognitive user ($M_3$) can be decoded at Rx1 and Rx2, under condition (\[eqn:set1\_I\]). Rx1 can decode $M_2$ considering the condition of the 