reserve a, b, r, s for Real;

theorem Th27:
  for X being non empty interval Subset of REAL st X is
  bounded_below & X is not bounded_above & not lower_bound X in X holds X =
  right_open_halfline(lower_bound X)
proof
  let X be non empty interval Subset of REAL such that
A1: X is bounded_below and
A2: X is not bounded_above and
A3: not lower_bound X in X;
  thus X c= right_open_halfline(lower_bound X) by A1,A3,Th26;
  let x be object;
  assume
A4: x in right_open_halfline(lower_bound X);
  then reconsider x as Real;
  x is not UpperBound of X by A2;
  then consider r being ExtReal such that
A5: r in X & x < r by XXREAL_2:def 1;
  lower_bound X < x by A4,XXREAL_1:235;
  then lower_bound X - lower_bound X < x - lower_bound X by XREAL_1:14;
  then consider s such that
A6: s in X & s < lower_bound X + (x - lower_bound X) by A1,SEQ_4:def 2;
  reconsider r as Real by A5;
  [.s,r.] c= X & x in [.s,r.] by A5,A6,XXREAL_1:1,XXREAL_2:def 12;
  hence thesis;
end;
