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

theorem
  for a,b being Real,
      Iab being non empty compact Subset of REAL,
      Dab being Division of Iab st 2 <= len Dab holds
  (Dab/^1) is Division of Iab
  proof
    let a,b be Real, Iab be non empty compact Subset of REAL,
        Dab be Division of Iab;
    assume
A1: 2 <= len Dab;
    set D = (Dab/^1);
A2: 1 <= len Dab by A1,XXREAL_0:2; then
A3: len D = len Dab - 1 by RFINSEQ:def 1;
A5: D is non empty
    proof
      2 - 1 <= len Dab - 1 by A1,XREAL_1:13;
      hence thesis by A3;
    end;
A4: D is non empty increasing
    proof
      D is increasing
      proof
        now
          let e1,e2 be ExtReal;
          assume that
A6:       e1 in dom D and
A7:       e2 in dom D and
A8:       e1 < e2;
          reconsider ne1 = e1, ne2 = e2 as Nat by A6,A7;
A9:       D.e1 = Dab.(ne1 + 1) by A2,A6,RFINSEQ:def 1;
          ne1 in Seg len D by A6,FINSEQ_1:def 3;
          then ne1 + 1 in Seg (len D + 1) by FINSEQ_1:60; then
A10:      ne1 + 1 in dom Dab by A3,FINSEQ_1:def 3;
          ne2 in Seg len D & ne2 in Seg len D by A7,FINSEQ_1:def 3;
          then ne2 + 1 in Seg (len D + 1) by FINSEQ_1:60; then
A11:      ne2 + 1 in dom Dab by A3,FINSEQ_1:def 3;
A12:      D.e2 = Dab.(ne2 + 1) by A2,A7,RFINSEQ:def 1;
          ne1 + 1 < ne2 + 1 by A8,XREAL_1:8;
          hence D.e1 < D.e2 by A9,A12,A10,A11,VALUED_0:def 13;
        end;
        hence thesis;
      end;
      hence thesis by A5;
    end;
A13: rng D c= Iab
    proof
A14:  rng D c= rng Dab by FINSEQ_5:33;
      rng Dab c= Iab by INTEGRA1:def 2;
      hence thesis by A14;
    end;
    D.(len D) = upper_bound Iab
    proof
      2 - 1 <= len Dab - 1 by A1,XREAL_1:13;
      then Seg len D = dom D & len D in Seg len D
        by A3,FINSEQ_1:def 3,FINSEQ_1:3;
      then D.(len D) = Dab.(len D + 1) by A2,RFINSEQ:def 1
                    .= Dab.len Dab by A3;
      hence thesis by INTEGRA1:def 2;
    end;
    hence thesis by A4,A13,INTEGRA1:def 2;
  end;
