  Previously one of us presented a conjecture [APF-4 Proceedings] to model
antiferromagnetism and high temperature superconductivity and their
'unification' by quantum group symmetry rather than the corresponding classical
symmetry in view of the critique by Baskaran and Anderson of Zhang's classical
SO(5) model. This conjecture was further sharpened, experimental evidence and
the important role of 1-d systems [stripes] was emphasized and moreover the
relationship between quantum groups and strings via WZWN models were given in
[Phys. Lett A272, (2000)]. In this brief note we give and discuss mathematical
proof of this conjecture, which completes an important part of this idea, since
previously an explicit simple mathematical proof was lacking. Moreover an
independent calculation [IC/99/2] which constructs the generators forming SO(5)
algebra not only supports our previous conjecture but provides a check on our
calculations. It is important to note that in terms of physics that the
arbitariness [freedom] of the d-wave factor g$^{2}$(k) is tied to quantum group
symmetry whereas in order to recover classical SO(5) one must set it to unity
in an adhoc manner. We intuitively expect that this freedom may be related to
psuedogap behaviour in cuprates.
