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 Th27:
  for f being Function of I,REAL st I is trivial holds
  f is HK-integrable & HK-integral(f) = 0
  proof
    let f be Function of I,REAL;
    assume
A1: I is trivial;
    reconsider J = 0 as Real;
A2: now
      let epsilon be Real;
      assume
A3:   epsilon > 0;
      reconsider jauge = chi(I,I) as positive-yielding Function of I,REAL
        by Th17,Th11;
      take jauge;
      thus for TD be tagged_division of I st TD is jauge-fine holds
        |.tagged_sum(f,TD) - J.| <= epsilon
      proof
        let TD be tagged_division of I;
        assume TD is jauge-fine;
        consider x be object such that
A4:     I = {x} by A1,ZFMISC_1:131;
        x in I by A4,TARSKI:def 1;
        then reconsider x as Real;
A4Bis:  division_of TD = <* x *> by A4,Th26;
A5:     len tagged_volume(f,TD) = len TD &
          for i be Nat st i in dom TD holds
             tagged_volume(f,TD).i = f.((tagged_of TD).i)
               * vol(divset(division_of TD,i)) by Def4;
A6:     tagged_volume(f,TD).1
          = f.((tagged_of TD).1) * vol(divset(division_of TD,1))
            by A5,FINSEQ_5:6;
        divset(division_of TD,1) = [. lower_bound I, (division_of TD).1 .]
          by COUSIN:50
                                .= [. x,(division_of TD).1 .] by A4,SEQ_4:9
                                .= [. x,x .] by A4Bis
                                .= {x} by XXREAL_1:17;
        then
A7:     vol divset(division_of TD,1) = 0 by COUSIN:41;
        len tagged_volume(f,TD) = len TD by Def4
                               .= 1 by A4Bis,FINSEQ_1:40;
        then tagged_volume(f,TD) = <* tagged_volume(f,TD).1 *>
          by FINSEQ_1:40;
        then Sum tagged_volume(f,TD) = 0 by A6,A7,RVSUM_1:73;
        hence thesis by A3;
      end;
    end;
    then f is HK-integrable;
    hence thesis by A2,DEFCC;
  end;
