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

theorem
  for I being non empty closed_interval Subset of REAL,
      D being Division of I holds
    divset(D,1) = [.lower_bound I,D.1.] &
    for j be Nat st j in dom D & j <> 1 holds divset(D,j) = [.D.(j-1),D.j.]
  proof
    let I be non empty closed_interval Subset of REAL,
        D be Division of I;
    rng D <> {};
    then 1 in dom D by FINSEQ_3:32;
    then lower_bound divset(D,1) = lower_bound I &
      upper_bound divset(D,1) = D.1 by INTEGRA1:def 4;
    hence divset(D,1) = [.lower_bound I,D.1.] by INTEGRA1:4;
    thus for j be Nat st j in dom D & j <> 1 holds divset(D,j)=[.D.(j-1),D.j.]
    proof
      let j be Nat;
      assume that
A1:   j in dom D and
A2:   j <> 1;
      lower_bound divset(D,j) = D.(j-1) &
        upper_bound divset(D,j) = D.j by A1,A2,INTEGRA1:def 4;
      hence thesis by INTEGRA1:4;
    end;
  end;
