reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem Th43:
  for X being non empty Subset of REAL st X is bounded_below holds
  lower_bound X = - upper_bound --X
proof
  let X be non empty Subset of REAL;
  set r = - upper_bound --X;
A1: now
    let t;
    assume
A2: for s st s in X holds s >= t;
    now
      let s;
      assume s in --X;
      then -s in -- --X;
      then -s >= t by A2;
      then - -s <= -t by XREAL_1:24;
      hence s <= -t;
    end;
    then -r <= -t by SEQ_4:45;
    hence r >= t by XREAL_1:24;
  end;
  assume X is bounded_below;
  then
A3: --X is bounded_above by Lm3;
  now
    let s;
    assume s in X;
    then -s in --X;
    then -s <= -r by A3,SEQ_4:def 1;
    hence s >= r by XREAL_1:24;
  end;
  hence thesis by A1,SEQ_4:44;
end;
