reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem Th86:
  for f be PartFunc of the carrier of CNS,REAL st dom f<>{} & (
  dom f) is compact & f is_continuous_on (dom f) holds ex x1,x2 be Point of CNS
  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 CNS,REAL;
  assume dom f <> {} & dom f is compact & f is_continuous_on (dom f);
  then
A1: rng f <> {} & rng f is compact by Th81,RELAT_1:42;
  then consider x being Element of CNS 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 CNS 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;
