theorem Th112:
  for X being Subset of REAL, r st X <> {} & for r9 st r9 in X
  holds r <= r9 holds lower_bound X >= r
proof
  let X be Subset of REAL, r such that
A1: X <> {} and
A2: for r9 st r9 in X holds r <= r9;
  for r9 be ExtReal st r9 in X holds r <= r9 by A2;
  then r is LowerBound of X by XXREAL_2:def 2;
  then
A3: X is bounded_below;
  now
    let r9 be Real;
    assume r9 > 0;
    then consider r1 be Real such that
A4: r1 in X and
A5: r1 < lower_bound X + r9 by A1,A3,Def2;
    r <= r1 by A2,A4;
    hence lower_bound X + r9 >= r by A5,XXREAL_0:2;
  end;
  hence thesis by XREAL_1:41;
end;
