  The validity of substituting a c-number $z$ for the $k=0$ mode operator $a_0$
is established rigorously in full generality, thereby verifying one aspect of
Bogoliubov's 1947 theory. This substitution not only yields the correct value
of thermodynamic quantities like the pressure or ground state energy, but also
the value of $|z|^2$ that maximizes the partition function equals the true
amount of condensation in the presence of a gauge-symmetry breaking term -- a
point that had previously been elusive.
