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 Th44:
  for f being Function of I,REAL
  for epsilon being Real st
  fc = chi(I,I) &
  r1 = min rng upper_volume(fc,D1) &
  r2 = epsilon / (2 * (len D1) *
    |. upper_bound (rng f) - lower_bound (rng f) .| ) &
  0 < r1 & 0 < r2 & s = min(r1,r2) / 2 &
  jauge = s (#) fc & TD is jauge-fine holds
  delta(division_of TD) < min rng upper_volume(fc,D1) &
  delta(division_of TD) < epsilon / (2 * (len D1) *
    |. upper_bound (rng f) - lower_bound (rng f) .| )
  proof
    let f be Function of I,REAL;
    let epsilon be Real;
    assume that
A1: fc = chi(I,I) and
A2: r1 = min rng upper_volume(fc,D1) and
A3: r2 = epsilon / (2 * (len D1) *
      |. upper_bound (rng f) - lower_bound (rng f) .| ) and
A4: 0 < r1 and
A5: 0 < r2 and
A6: s = min(r1,r2) / 2 and
A7: jauge = s (#) fc and
A8: TD is jauge-fine;
A9: delta(division_of TD) <= min(r1,r2) / 2 by A1,A6,A7,A8,Th42;
    min(r1,r2) / 2 < min(r1,r2) / 1 by A4,A5,XREAL_1:76;
    then delta(division_of TD) < min(r1,r2) by A9,XXREAL_0:2;
    hence thesis by A2,A3,XXREAL_0:23;
  end;
