
theorem Th21:
  for X being non empty set, Y being non empty Subset of ExtREAL,
  F being Function of X,Y holds F is bounded_above iff - F is bounded_below
proof
  let X be non empty set, Y be non empty Subset of ExtREAL, F be Function of X
  ,Y;
  hereby
    assume F is bounded_above;
    then
A1: - (+infty) <- sup F by XXREAL_3:38;
    - (+infty) = -infty by XXREAL_3:def 3;
    hence - F is bounded_below by A1,Th19;
  end;
  assume - F is bounded_below;
  then - inf(- F) <- (-infty) by XXREAL_3:38;
  then - (- sup F) <- (-infty) by Th19;
  hence thesis by XXREAL_3:5;
end;
