
theorem Th12:
  for X being non empty Subset of ExtREAL holds for a being R_eal
  holds a is LowerBound of X iff - a is UpperBound of - X
proof
  let X be non empty Subset of ExtREAL;
  let a be R_eal;
  hereby
    assume
A1: a is LowerBound of X;
    for x being ExtReal st x in - X holds x <= - a
    proof
      let x be ExtReal;
      assume
A2:   x in - X;
      reconsider x as Element of ExtREAL by XXREAL_0:def 1;
      - x in - (- X) by A2;
      then - (- x) <= - a by XXREAL_3:38,A1,XXREAL_2:def 2;
      hence thesis;
    end;
    hence - a is UpperBound of - X by XXREAL_2:def 1;
  end;
  assume
A3: - a is UpperBound of - X;
  for x being ExtReal st x in X holds a <= x
  proof
    let x be ExtReal;
    assume
A4: x in X;
    reconsider x as Element of ExtREAL by XXREAL_0:def 1;
    - x in - X by A4;
    hence thesis by XXREAL_3:38,A3,XXREAL_2:def 1;
  end;
  hence thesis by XXREAL_2:def 2;
end;
