theorem
  for f be PartFunc of the carrier of S, REAL for Y st Y <> {} & Y c=
dom f & Y is compact & f is_uniformly_continuous_on Y ex x1,x2 st x1 in Y & x2
in Y & f/.x1 = upper_bound (f.:Y) & f/.x2 = lower_bound (f.:Y) by Th8,
