reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem
  X misses dom f implies f is_bounded_on X
proof
  assume X /\ dom f = {};
  then for x be Element of M holds x in X /\ dom f implies ||.f/.x.|| <= 0;
  hence thesis;
end;
