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

theorem Th33:
  for f be PartFunc of the carrier of S,REAL st dom f<>{} & (dom f
) is compact & f is_continuous_on (dom f) ex x1,x2 st x1 in dom f & x2 in dom f
  & f/.x1 = upper_bound (rng f) & f/.x2 = lower_bound (rng f)
proof
  let f be PartFunc of the carrier of S,REAL;
  assume dom f <> {} & dom f is compact & f is_continuous_on (dom f);
  then
A1: rng f <> {} & rng f is compact by Th31,RELAT_1:42;
  then consider x being Element of S such that
A2: x in dom f & upper_bound (rng f) = f.x by PARTFUN1:3,RCOMP_1:14;
  take x;
  consider y being Element of S such that
A3: y in dom f & lower_bound (rng f ) = f.y by A1,PARTFUN1:3,RCOMP_1:14;
  take y;
  thus thesis by A2,A3,PARTFUN1:def 6;
end;
