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;

theorem
  jauge = r (#) chi(I,I) implies 0 < r
  proof
    assume
A1: jauge = r (#) chi(I,I);
    assume
A2: r <= 0;
    set x = the Element of I;
    x in I;
    then
A3: x in dom chi(I,I) & x in dom jauge by PARTFUN1:def 2;
    then jauge.x = r * (chi(I,I)).x by A1,VALUED_1:def 5
                .= r * 1 by FUNCT_3:def 3
                .= r;
    then jauge.x <= 0 & jauge.x in rng jauge by A2,A3,FUNCT_1:3;
    hence contradiction by PARTFUN3:def 1;
  end;
