reserve a, b, r, s for Real;

theorem Th26:
  for X being Subset of REAL st X is bounded_below & not
  lower_bound X in X holds X c= right_open_halfline(lower_bound X)
proof
  let X be Subset of REAL such that
A1: X is bounded_below and
A2: not lower_bound X in X;
  let x be object;
  assume
A3: x in X;
  then reconsider x as Real;
  lower_bound X <= x by A1,A3,SEQ_4:def 2;
  then lower_bound X < x by A2,A3,XXREAL_0:1;
  hence thesis by XXREAL_1:235;
end;
