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 Th30:
  I c= A & f = chi(A,I) implies f is HK-integrable & HK-integral(f) = vol(I)
  proof
    assume that
A1: I c= A and
A2: f = chi(A,I);
    reconsider J = vol(I) as Real;
A3: now
      let epsilon be Real;
      assume
A4:   epsilon > 0;
      reconsider jauge = chi(I,I) as Function of I,REAL by Th11;
      now
        let r be Real;
        assume r in rng jauge;
        then r in {1} by INTEGRA1:17;
        hence 0 < r;
      end;
      then reconsider jauge as positive-yielding Function of I,REAL
        by PARTFUN3:def 1;
      now
        take jauge;
        hereby
          let TD be tagged_division of I;
          assume TD is jauge-fine;
          f = chi(I,I) by A1,A2,Th24;
          then tagged_sum(f,TD) = vol I by Lm04;
          hence |.tagged_sum(f,TD) - J.| <= epsilon by A4;
        end;
      end;
      hence ex jauge being positive-yielding Function of I,REAL st
      for TD being tagged_division of I st TD is jauge-fine
      holds |.tagged_sum(f,TD) - J.| <= epsilon;
    end;
    then f is HK-integrable;
    hence thesis by A3,DEFCC;
  end;
