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 Th44:
  for X being non empty Subset of REAL st X is bounded_above holds
  upper_bound X = - lower_bound --X
proof
  let X be non empty Subset of REAL;
  set r = - lower_bound --X;
A1: now
    let s;
    assume
A2: for t st t in X holds t <= s;
    now
      let t;
      assume t in --X;
      then -t in -- --X;
      then -t <= s by A2;
      then - -t >= -s by XREAL_1:24;
      hence t >= -s;
    end;
    then -r >= -s by SEQ_4:43;
    hence r <= s by XREAL_1:24;
  end;
  assume X is bounded_above;
  then
A3: --X is bounded_below by Lm2;
  now
    let t;
    assume t in X;
    then -t in --X;
    then -t >= -r by A3,SEQ_4:def 2;
    hence t <= r by XREAL_1:24;
  end;
  hence thesis by A1,SEQ_4:46;
end;
