theorem Th78:
  (f|Y is bounded_above & 0<=r implies (r(#)f)|Y is bounded_above)
  & (f|Y is bounded_above & r<=0 implies (r(#)f)|Y is bounded_below)
proof
  (r(#)f)|Y = r(#)(f|Y) by Th49;
  hence thesis;
end;
