
theorem Th13:
  for X being non empty Subset of ExtREAL holds inf(- X) = - sup X
proof
  let X be non empty Subset of ExtREAL;
  set a = inf(- X);
  set b = sup X;
  a is LowerBound of - X by XXREAL_2:def 4;
  then - a is UpperBound of -(-X) by Th12; then
A1: - (- a) <= - b by XXREAL_3:38,XXREAL_2:def 3;
  b is UpperBound of X by XXREAL_2:def 3;
  then - b is LowerBound of - X by Th11;
  then - b <= a by XXREAL_2:def 4;
  hence thesis by A1,XXREAL_0:1;
end;
