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

theorem Th35:
  for a,b being Real, Iab being non empty closed_interval Subset of REAL
  st Iab = [.a,b.] holds <*b*> is Division of Iab
  proof
    let a,b be Real, Iab be non empty closed_interval Subset of REAL;
    assume
A1: Iab = [.a,b.];
A2: a <= b by A1,XXREAL_1:29;
    set D = <*b*>;
A3: rng D c= Iab
    proof
      let x be object;
      assume x in rng D;
      then x in {b} by FINSEQ_1:39;
      then x = b by TARSKI:def 1;
      hence thesis by A1,A2,XXREAL_1:1;
    end;
    D.(len D) = upper_bound Iab
    proof
      dom D = Seg 1 by FINSEQ_1:def 8;
      then D.(len D) = <*b*>.1 by FINSEQ_1:def 3
                    .= b;
      hence thesis by A1,A2,JORDAN5A:19;
    end;
    hence thesis by A3,INTEGRA1:def 2;
  end;
