  In this note, we show how difficult the brute-force Fourier-Motzkin
elimination is, even in a simple case with three eliminating variables.
Specifically, we first give a theorem, which plays quite an important role in
the study of information-theoretic security for a multiple access wiretap
(MAC-WT) channel, and then prove it for the case with three users by directly
using the Fourier-Motzkin procedure. It is shown that large amounts of
inequalities are generated in the elimination procedure while most of them are
redundant. Actually, the number of generated inequalities grows doubly
exponentially with the number of users or eliminating variables. It thus
becomes unmanageable to directly prove the theorem in this brute-force way.
Besides the great complexity, another disadvantage of the direct strategy is
that it works only if the number of users is given. Obviously, this makes the
strategy inappropriate for the proof of the theorem since it is a general
result for any number of users. It is thus urgent and challenging to generally
prove the theorem.
