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 Th22:
  for I being non empty closed_interval Subset of REAL,
  D being Division of I, T being Element of set_of_tagged_Division(D) holds
  rng T c= I
  proof
    let I be non empty closed_interval Subset of REAL,
    D be Division of I, T be Element of set_of_tagged_Division(D);
    consider s be non empty non-decreasing FinSequence of REAL such that
A1: T = s & dom s = dom D &
    for i be Nat st i in dom s holds s.i in divset(D,i) by COUSIN:def 2;
    now
      let x be object;
      assume x in rng T;
      then consider y be object such that
A2:   y in dom T and
A3:   x = T.y by FUNCT_1:def 3;
      reconsider y as Nat by A2;
      divset(D,y) c= I by A1,A2,INTEGRA1:8;
      hence x in I by A3,A1,A2;
    end;
    hence thesis;
  end;
