reserve a, b, r, s for Real;

theorem Th25:
  for X being interval Subset of REAL st X is bounded_below & X
  is not bounded_above & lower_bound X in X holds X = right_closed_halfline(
  lower_bound X)
proof
  let X be interval Subset of REAL such that
A1: X is bounded_below and
A2: X is not bounded_above and
A3: lower_bound X in X;
  thus X c= right_closed_halfline(lower_bound X) by A1,Th24;
  let x be object;
  assume
A4: x in right_closed_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 and
A6: x < r by XXREAL_2:def 1;
  reconsider r as Real by A5;
  lower_bound X <= x by A4,XXREAL_1:236;
  then
A7: x in [.lower_bound X,r.] by A6,XXREAL_1:1;
  [.lower_bound X,r.] c= X by A3,A5,XXREAL_2:def 12;
  hence thesis by A7;
end;
