
theorem Th22:
  for X being non empty set, Y being non empty Subset of ExtREAL,
  F being Function of X,Y holds F is bounded_below iff - F is bounded_above
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_below;
    then - inf F <- (-infty) by XXREAL_3:38;
    hence - F is bounded_above by Th19,XXREAL_3:5;
  end;
  assume - F is bounded_above;
  then - inf F <+infty by Th19; then
  - (+infty) <- (- inf F) by XXREAL_3:38;
  hence thesis by XXREAL_3:def 3;
end;
