 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th34:
  for A being non empty closed_interval Subset of REAL,
  D being Division of A holds D in set_of_tagged_Division(D)
  proof
    let A be non empty closed_interval Subset of REAL,
    D be Division of A;
    for i be Nat st i in dom D holds D.i in divset(D,i)
    proof
      let i be Nat;
      assume
A1:   i in dom D;
      consider a,b be Real such that
A2:   divset(D,i) = [.a,b.] by MEASURE5:def 3;
      a <= b by A2,XXREAL_1:29; then
A3:   upper_bound divset(D,i) = b & b in [.a,b.] by A2,JORDAN5A:19,XXREAL_1:1;
      1 <= i by A1,FINSEQ_3:25; then
      per cases by XXREAL_0:1;
      suppose i = 1;
        hence thesis by A3,A2,A1,INTEGRA1:def 4;
      end;
      suppose 1 < i;
        hence thesis by A3,A2,A1,INTEGRA1:def 4;
      end;
    end;
    hence thesis by Def2;
  end;
