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

theorem Th45:
  for a,b being Real st a < b holds <*a,b*> is non empty increasing
    FinSequence of REAL
  proof
    let a,b be Real;
    assume
A1: a < b;
    set s = <*a,b*>;
A2: rng s c= REAL;
    s is increasing
    proof
      now
        let e1,e2 be ExtReal;
        assume that
A3:     e1 in dom s and
A4:     e2 in dom s and
A5:     e1 < e2;
        dom s = Seg len s by FINSEQ_1:def 3
             .= Seg 2 by FINSEQ_1:44;
        then (e1 = 1 or e1 = 2) & (e2 = 1 or e2 = 2)
          by TARSKI:def 2,A3,A4,FINSEQ_1:2;
        then s.e1 = a & s.e2 = b by A5;
        hence s.e1 < s.e2 by A1;
      end;
      hence thesis;
    end;
    hence thesis by A2,FINSEQ_1:def 4;
  end;
