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;

theorem Th23:
  for I being non empty closed_interval Subset of REAL
  for jauge1,jauge2 being positive-yielding Function of I,REAL
  for TD being jauge1-fine tagged_division of I st jauge1 <= jauge2 holds
  TD is jauge2-fine tagged_division of I
  proof
    let I be non empty closed_interval Subset of REAL;
    let jauge1,jauge2 be positive-yielding Function of I,REAL;
    let TD be jauge1-fine tagged_division of I;
    assume
A1: jauge1 <= jauge2;
    consider D be Division of I,
             T be Element of set_of_tagged_Division(D) such that
A2: TD = [D,T] and
A3: for i be Nat st i in dom D holds vol divset(D,i) <= jauge1.(T.i)
      by COUSIN:def 4;
    now
      let i be Nat;
      assume
A4:   i in dom D;
      then
A5:   vol divset(D,i) <= jauge1.(T.i) by A3;
      dom T = Seg len T by FINSEQ_1:def 3
           .= Seg len tagged_of TD by A2,Th20
           .= Seg len division_of TD by Th21
           .= Seg len D by A2,Th20
           .= dom D by FINSEQ_1:def 3; then
A6:   T.i in rng T by A4,FUNCT_1:3;
      rng T c= I by Th22;
      then jauge1.(T.i) <= jauge2.(T.i) by A6,A1;
      hence vol divset(D,i) <= jauge2.(T.i) by A5,XXREAL_0:2;
    end;
    hence thesis by A2,COUSIN:def 4;
  end;
