  A recent result of the authors shows a so-called I-MMSE-like relationship
that, for the two-user Gaussian interference channel, an I-MMSE relationship
holds in the limit, as n $\to \infty$, between the interference and the
interfered-with receiver, assuming that the interfered-with transmission is an
optimal point-to-point sequence (achieves the point-to-point capacity). This
result was further used to provide a proof of the "missing corner points" of
the two-user Gaussian interference channel. This paper provides an information
theoretic proof of the above-mentioned I-MMSE-like relationship which follows
the incremental channel approach, an approach which was used by Guo, Shamai and
Verd\'u to provide an insightful proof of the original I-MMSE relationship for
point-to-point channels. Finally, some additional applications of this result
are shown for other multi-user settings: the Gaussian multiple-access channel
with interference and specific K-user Gaussian Z-interference channel settings.
