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;

theorem Th83:
  (f1|X is bounded_above & f2|Y is bounded_above implies (f1+f2)|
  (X /\ Y) is bounded_above) & (f1|X is bounded_below & f2|Y is bounded_below
implies (f1+f2)|(X /\ Y) is bounded_below) & (f1|X is bounded & f2|Y is bounded
  implies (f1+f2)|(X /\ Y) is bounded)
proof
  (f1+f2)|(X /\ Y) = f1|(X /\ Y)+f2|(X /\ Y) by Th44
    .= f1|(X /\ Y)+f2|Y|X by RELAT_1:71
    .= f1|X|Y+f2|Y|X by RELAT_1:71;
  hence thesis;
end;
