reserve a, b, r, s for Real;

theorem Th28:
  for X being interval Subset of REAL st X is not bounded_above &
  X is not bounded_below holds X = REAL
proof
  let X be interval Subset of REAL;
  assume that
A1: X is not bounded_above and
A2: X is not bounded_below;
  thus X c= REAL;
  let x be object;
  assume x in REAL;
  then reconsider x as Real;
  x is not UpperBound of X by A1;
  then consider r being ExtReal such that
A3: r in X & r > x by XXREAL_2:def 1;
  reconsider r as Real by A3;
  x is not LowerBound of X by A2;
  then consider s being ExtReal such that
A4: s in X & s < x by XXREAL_2:def 2;
  reconsider s as Real by A4;
  [.s,r.] c= X & x in [.s,r.] by A3,A4,XXREAL_1:1,XXREAL_2:def 12;
  hence thesis;
end;
