reserve n,m for Nat,
  x,X,X1 for set,
  s,g,r,p for Real,
  S,T for RealNormSpace,
  f,f1,f2 for PartFunc of S, T,
  s1,s2,q1 for sequence of S,
  x0,x1, x2 for Point of S,
  Y for Subset of S;

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,
