reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;
reserve f for real-valued Function;

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;
