-------- --------------------------------- ---------------------------------- ---------------------------------- ----------------------------------- ---------------------------------------------------------------------------------
   *Set1*   first term of (\[eqn:set1\_I\])   second term of (\[eqn:set1\_I\])   first term of (\[eqn:set1\_II\])   first term of (\[eqn:set1\_III\])   (\[eqn:set1\_IV\]) + second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\])
   *Set2*   first term of (\[eqn:set1\_I\])   second term of (\[eqn:set1\_I\])   first term of (\[eqn:set2\_II\])   first term of (\[eqn:set2\_III\])   (\[eqn:set2\_IV\]) + second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\])

Theorem \[thm:ach2\] includes (\[eqn:ach2\_I\])-(\[eqn:ach2\_III\]) instead of (\[eqn:ach1\_IV\]) in Theorem \[thm:ach1\]. In fact, in the *Gaussian case*, the converse proof can not be established for the two first terms in (\[eqn:ach1\_IV\]). Therefore, we propose Theorem \[thm:ach2\] and find the stronger conditions than *Set1*, i.e., *Set2*, which makes the bounds in (\[eqn:ach2\_I\])-(\[eqn:ach2\_III\]) redundant. Hence, we intend to use *Set2* to derive the capacity results for the Gaussian case in Section \[sec:Gaussian\].

In *Set1*, (\[eqn:set1\_IV\]) and the second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) are used to make the second terms of (\[eqn:ach1\_II\]) and (\[eqn:ach1\_III\]), and the third term of (\[eqn:ach1\_IV\]) redundant. However, (\[eqn:set1\_I\]) and the first terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) are used to prove the converse part for the rates in (\[eqn:Cap1\_II\])-(\[eqn:Cap1\_IV\]).

In *Set2*, the second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]) are used to make the second terms of (\[eqn:ach1\_II\]) and (\[eqn:ach1\_III\]) redundant. The first terms 