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;

theorem Th42:
  jauge = r (#) chi(I,I) & D is jauge-fine
  implies delta(division_of D) <= r
  proof
    assume that
A1: jauge = r (#) chi(I,I) and
A2: D is jauge-fine;
A3: now
      let i be Nat;
      assume
A4:   i in dom division_of D;
      consider D9 be Division of I,
               T9 be Element of set_of_tagged_Division(D9) such that
A5:   D = [D9,T9] & for i being Nat st i in dom D9 holds
      vol divset(D9,i) <= jauge.(T9.i) by A2,COUSIN:def 4;
A6:   T9 = tagged_of D & D9 = division_of D by A5,Th20; then
A7:   vol divset(division_of D,i) <= jauge.((tagged_of D).i) by A5,A4;
A8:   dom (r (#) chi(I,I)) = dom chi(I,I) by VALUED_1:def 5
                          .= I by FUNCT_3:def 3;
      i in Seg len division_of D by A4,FINSEQ_1:def 3;
      then i in Seg len tagged_of D by Th21; then
A9:   i in dom T9 by A6,FINSEQ_1:def 3;
      rng T9 c= I by Th22; then
A10:  (tagged_of D).i in I by  A9,A6,FUNCT_1:3;
      now
        let x be object;
        assume
A11:    x in dom (r (#) chi(I,I));
        then (r (#) chi(I,I)).x = r * (chi(I,I)).x by VALUED_1:def 5
                               .= r * 1 by A11,FUNCT_3:def 3
                               .= r;
        hence jauge.x = r by A1;
      end;
      hence vol divset(division_of D,i) <= r by A7,A8,A10;
    end;
    reconsider g = chi(I,I) as Function of I,REAL by Th11;
A12: for i be Nat st i in dom division_of D holds
      upper_volume(g,division_of D).i <= r
    proof
      let i be Nat;
      assume
A13:  i in dom division_of D;
      then vol divset(division_of D,i) <= r by A3;
      hence thesis by A13,INTEGRA1:20;
    end;
    delta(division_of D) <= r
    proof
      assume r < delta(division_of D);
      then
A14:  r < max rng upper_volume(g,division_of D) by INTEGRA3:def 1;
      sup rng upper_volume(g,division_of D)
        in rng upper_volume(g,division_of D) by XXREAL_2:def 6;
      then consider x be object such that
A15:  x in dom upper_volume(g,division_of D) and
A16:  (upper_volume(g,division_of D)).x
        = sup rng upper_volume(g,division_of D) by FUNCT_1:def 3;
A17:  dom upper_volume(g,division_of D)
        = Seg len upper_volume(g,division_of D) by FINSEQ_1:def 3
       .= Seg len division_of D by INTEGRA1:def 6
       .= dom division_of D by FINSEQ_1:def 3;
      reconsider i = x as Nat by A15;
      thus contradiction by A14,A15,A17,A16,A12;
    end;
    hence thesis;
  end;
