  For the Hubbard model on the two-dimensional copper-oxide lattice, equal-time
four-point correlation functions at positive temperature are proved to decay
exponentially in the thermodynamic limit if the magnitude of the on-site
interactions is smaller than some power of temperature. This result especially
implies that the equal-time correlation functions for singlet Cooper pairs of
various symmetries decay exponentially in the distance between the Cooper pairs
in high temperatures or in low-temperature weak-coupling regimes. The proof is
based on a multi-scale integration over the Matsubara frequency.
