
theorem Th11:
  for X being non empty Subset of ExtREAL, a being R_eal holds
  a is UpperBound of X iff - a is LowerBound of - X
proof
  let X be non empty Subset of ExtREAL;
  let a be R_eal;
  hereby
    assume
A1: a is UpperBound of X;
    for x being ExtReal st x in - X holds -a <= x
    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 - a <= - -x by XXREAL_3:38,A1,XXREAL_2:def 1;
      hence thesis;
    end;
    hence - a is LowerBound of - X by XXREAL_2:def 2;
  end;
  assume
A3: - a is LowerBound of - X;
  for x being ExtReal st x in X holds x <=a
  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 2;
  end;
  hence thesis by XXREAL_2:def 1;
end;
