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 Th34:
  for f 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 such that
A1: dom f <> {} and
A2: dom f is compact and
A3: f is_continuous_on (dom f);
A4: dom f = dom ||.f.|| by NORMSP_0:def 3;
  dom ||.f.|| is compact by A2,NORMSP_0:def 3;
  then
A5: rng ||.f.|| is compact by A3,A4,Th28,Th31;
A6: rng ||.f.|| <> {} by A1,A4,RELAT_1:42;
  then consider x being Element of S such that
A7: x in dom ||.f.|| & upper_bound (rng ||.f.||) = ||.f.||.x by A5,PARTFUN1:3
,RCOMP_1:14;
  consider y being Element of S such that
A8: y in dom ||.f.|| & lower_bound (rng ||.f.||) = ||.f.||.y by A6,A5,
PARTFUN1:3,RCOMP_1:14;
  take x;
  take y;
  thus thesis by A7,A8,NORMSP_0:def 3,PARTFUN1:def 6;
end;
