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 Th21:
  len tagged_of TD = len division_of TD
  proof
    consider D be Division of I,
             T be Element of set_of_tagged_Division(D) such that
A1: TD = [D,T] by COUSIN:def 3;
    consider s be non empty non-decreasing FinSequence of REAL such that
A2: T = s and
A3: dom s = dom D and
    for i being Nat st i in dom s holds s.i in divset(D,i) by COUSIN:def 2;
    tagged_of(TD) = T & division_of(TD) = D by A1,Th20;
    hence thesis by A2,A3,FINSEQ_3:29;
  end;
