
theorem Th14:
  for X be non empty Subset of ExtREAL holds sup(- X) = - inf X
proof
  let X be non empty Subset of ExtREAL;
  set a = sup(- X);
  set b = inf X;
  a is UpperBound of - X by XXREAL_2:def 3;
  then - a is LowerBound of - (- X) by Th11;
  then
A1: - b <= - (-a) by XXREAL_3:38,XXREAL_2:def 4;
  b is LowerBound of X by XXREAL_2:def 4;
  then - b is UpperBound of - X by Th12;
  then a <= - b by XXREAL_2:def 3;
  hence thesis by A1,XXREAL_0:1;
end;
