  We point out the curious phenomenon of order by projection in a class of
lattice Fermi systems near half filling. Enhanced pairing correlations of
extended s-wave Cooper pairs result from the process of projecting out s-wave
Cooper pairs, with negligible effect on the ground state energy. The Hubbard
model is a particularly nice example of the above phenomenon, which is revealed
with the use of rigorous inequalities including the Uncertainty Principle
Inequality. In addition, we present numerical evidence that at half filling, a
related but simplified model shows ODLRO of extended s-wave Cooper pairs.
