
theorem Th24:
  for M being non empty MetrSpace holds for X being Subset of
  TopSpaceMetr(M) st X <> {} & X is compact holds dist_max(X) is continuous
proof
  let M be non empty MetrSpace;
  let X be Subset of TopSpaceMetr(M);
  assume
A1: X <> {} & X is compact;
  for P being Subset of R^1 st P is open holds (dist_max(X))"P is open
  proof
    let P be Subset of R^1;
    assume
A2: P is open;
    for p being Point of M st p in (dist_max(X))"P ex r being Real
    st r>0 & Ball(p,r) c= (dist_max(X))"P
    proof
      let p be Point of M;
      set y = upper_bound((dist(p)).:(X));
      y in REAL by XREAL_0:def 1;
      then reconsider y as Point of RealSpace by METRIC_1:def 13;
      assume p in (dist_max(X))"P;
      then
A3:   (dist_max(X)).p in P by FUNCT_1:def 7;
      reconsider P as Subset of TopSpaceMetr(RealSpace) by TOPMETR:def 6;
      y in P by A3,Def5;
      then consider r being Real such that
A4:   r>0 and
A5:   Ball(y,r) c= P by A2,TOPMETR:15,def 6;
      reconsider r as Real;
      take r;
      Ball(p,r) c= (dist_max(X))"P
      proof
        let z be object;
        assume
A6:     z in Ball(p,r);
        then reconsider z as Point of M;
        set q = upper_bound((dist(z)).:(X));
        q in REAL by XREAL_0:def 1;
        then reconsider q as Point of RealSpace by METRIC_1:def 13;
        dist(p,z) < r by A6,METRIC_1:11;
        then |.upper_bound((dist(p)).:(X)) - upper_bound((dist(z)).:(X)).|+
        dist(p,z) < r+dist(p,z) by A1,Th20,XREAL_1:8;
        then |.upper_bound((dist(p)).:(X)) - upper_bound((dist(z)).:(X)).| <
        r by XREAL_1:6;
        then dist(y,q) < r by TOPMETR:11;
        then
A7:     q in Ball(y,r) by METRIC_1:11;
        dom (dist_max(X)) = the carrier of TopSpaceMetr(M) by FUNCT_2:def 1;
        then
A8:    dom (dist_max(X)) = the carrier of M by TOPMETR:12;
        q = (dist_max(X)).z by Def5;
        hence thesis by A5,A7,A8,FUNCT_1:def 7;
      end;
      hence thesis by A4;
    end;
    hence thesis by TOPMETR:15;
  end;
  hence thesis by Lm2,TOPS_2:43;
end;
