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;
reserve f,g for HK-integrable Function of I,REAL,
          r for Real;
reserve f for bounded integrable Function of I,REAL;
reserve jauge for positive-yielding Function of I,REAL;
reserve D for tagged_division of I;
reserve r1,r2,s for Real,
           D,D1 for Division of I,
             fc for Function of I,REAL;

theorem Th47:
  divset(D,1) = [.D.1,D.1.] implies D.1 = lower_bound I
  proof
    assume divset(D,1) = [.D.1,D.1.]; then
A1: lower_bound divset(D,1) = D.1 & upper_bound divset(D,1) = D.1
      by JORDAN5A:19;
A2: divset(D,1) = [.lower_bound I,D.1.] by COUSIN:50;
    rng D <> {};
    then 1 in dom D by FINSEQ_3:32;
    then D.1 in I by INTEGRA1:6;
    then D.1 in [. lower_bound I, upper_bound I.] by INTEGRA1:4;
    then lower_bound I <= D.1 <= upper_bound I by XXREAL_1:1;
    hence thesis by A1,A2,JORDAN5A:19;
  end;
