(\[eqn:Cap1\_I\])-(\[eqn:Cap1\_III\]) in ${\mathcal{C}}_1$, which are shown in the converse proof of Theorem \[thm:cap\_set1\]. This completes the proof.

We compare the different terms in *Set1* and *Set2*. Since $X_1$ and $X_2$ are independent, we obtain

[rl]{} I(X\_2;Y\_1|X\_1)=H(X\_2|X\_1)-H(X\_2|X\_1,Y\_1)&H(X\_2)-H(X\_2|X\_1,Y\_1)\
&H(X\_2)-H(X\_2|Y\_1)= I(X\_2;Y\_1)

where (a) follows from the independence of $X_1$ and $X_2$, and (b) is a consequence of the fact that conditioning does not increase the entropy. Hence, condition (\[eqn:set2\_III\]) implies condition (\[eqn:set1\_III\]). Similarly, condition (\[eqn:set2\_II\]) implies condition (\[eqn:set1\_II\]). Moreover, the second terms of conditions (\[eqn:set2\_II\]) and (\[eqn:set2\_IV\]) give the first term in condition (\[eqn:set1\_IV\]). Also, the second term of condition (\[eqn:set2\_III\]) along with the first term of (\[eqn:set2\_IV\]) give the second term in condition (\[eqn:set1\_IV\]). Therefore, *Set2* implies *Set1*, and the conditions of *Set1* are weaker compared to thoes of *Set2*. In fact, we use *Set2* and ${\mathcal{C}}_2$ to derive the capacity results for the Gaussian case in the next section.

Gaussian three-user C-IFC {#sec:Gaussian}
=========================

In this section, we consider the Gaussian three-user C-IFC and characterize capacity results for the Gaussian case. Moreover, we present some numerical examples. The Gaussian three-user C-IFC, as depicted in Fig. \[fig:Gauschannelmodel\], at time $i=1,\ldots,n$ and at each Rx$r$, for $r\in\{1,2,3\}$, can be mathematically modeled as

[rcl]{} 