 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 closed_interval Subset of REAL
    st a < b & Iab = [.a,b.] holds <*a,b*> is Division of Iab
  proof
    let a,b be Real, Iab be non empty closed_interval Subset of REAL;
    assume that
A1: a < b and
A2: Iab = [.a,b.];
      set D = <*a,b*>;
A3:   D is non empty increasing FinSequence of REAL by A1,Th45;
A4:   rng D c= Iab
      proof
        let x be object;
        assume x in rng D;
        then x in {a,b} by FINSEQ_2:127;
        then x = a or x = b by TARSKI:def 2;
        hence thesis by A1,A2,XXREAL_1:1;
      end;
      D.(len D) = upper_bound Iab
      proof
A5:     len D = 2 by FINSEQ_1:44;
        upper_bound Iab = b by A1,A2,JORDAN5A:19;
        hence thesis by A5;
      end;
      hence thesis by A3,A4,INTEGRA1:def 2;
  end;
