reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;
reserve I for non empty closed_interval Subset of REAL,
       TD for tagged_division of I,
        D for Division of I,
        T for Element of set_of_tagged_Division(D),
        f for PartFunc of I,REAL;
reserve f for Function of I,REAL;

theorem Th26:
  for r being Real st I = {r} holds
  (for D being Division of I holds D = <* r *> )
  proof
    let r be Real;
    assume
A1: I = {r};
A2: I = [.r,r.] by A1,XXREAL_1:17;
    let D be Division of I;
    len D = 1
    proof
      assume 1 <> len D;
      then 2 <= len D by NAT_1:23;
      then 1 <= len D & 1 <= 2 <= len D by XXREAL_0:2;
      then
A3:   1 in dom D & 2 in dom D by FINSEQ_3:25;
      then D.1 in I & D.2 in I by INTEGRA1:6;
      then D.1 = r & D.2 = r by A1,TARSKI:def 1;
      hence contradiction by A3,VALUED_0:def 13;
    end;
    hence thesis by A2,COUSIN:64;
  end;
