
theorem
  for X being non empty ext-real-membered set holds sup X = inf
  SetMajorant(X) & inf X = sup SetMinorant(X)
proof
  let X be non empty ext-real-membered set;
  for y being ExtReal st y in SetMajorant X holds sup X <= y
  proof
    let y be ExtReal;
    assume y in SetMajorant X;
    then y is UpperBound of X by Def1;
    hence thesis by XXREAL_2:def 3;
  end;
  then
A1: sup X is LowerBound of SetMajorant X by XXREAL_2:def 2;
  inf X is LowerBound of X by XXREAL_2:def 4;
  then
A2: inf X in SetMinorant X by Def2;
  for y being ExtReal st y in SetMinorant X holds y <= inf X
  proof
    let y be ExtReal;
    assume y in SetMinorant X;
    then y is LowerBound of X by Def2;
    hence thesis by XXREAL_2:def 4;
  end;
  then
A3: inf X is UpperBound of SetMinorant X by XXREAL_2:def 1;
  sup X is UpperBound of X by XXREAL_2:def 3;
  then sup X in SetMajorant X by Def1;
  hence thesis by A1,A2,A3,XXREAL_2:55,56;
end;
