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;
reserve f1,f2 for real-valued Function;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  (f1|X is bounded_above & f2|Y is constant implies (f1+f2)|(X /\ Y) is
  bounded_above) & (f1|X is bounded_below & f2|Y is constant implies (f1+f2)|(X
/\ Y) is bounded_below) & (f1|X is bounded & f2|Y is constant implies (f1+f2)|(
  X /\ Y) is bounded) by Th83;
