The result of equation 6.44 is a specific case of a more general result. It is easily seen that any 
on a set of random variables where there are two of the random variables 
and 
which are conditionally independent given any other random variable(s) is the mutual information between the two random variables. I.e. if 
for 
then 
. This occurs in any chain of random variables, where fixing any random variable in the middle of the chain effectively removes any dependence that the random variable at one end of the chain has on the random variable at the other end of the chain. Other interesting results of this nature are obtainable. For instance, 
for four random variables where three of them decouple given the fourth (imagine a three pointed star tree for the dependency of coupling, with the fourth variable at the center of the tree) reduces to 
on the three that decouple when the fourth is specified, and so on. Another result of this nature is based on similar topological considerations: if a random variable is a cut point of the dependency tree (removing it splits the tree into two nonempty parts), then the information correlation function on all of the random variables reduces to a lower order information correlation function on the subsets of random variables that are generated by removing the cut point random variable.