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
  Y c= X & f is_bounded_on X implies f is_bounded_on Y
proof
  assume that
A1: Y c= X and
A2: f is_bounded_on X;
  consider r be Real such that
A3: for x be Element of M st x in X /\ dom f holds ||.f/.x.|| <= r by A2;
  take r;
  let x be Element of M;
  assume x in Y /\ dom f;
  then x in Y & x in dom f by XBOOLE_0:def 4;
  then x in X /\ dom f by A1,XBOOLE_0:def 4;
  hence thesis by A3;
end;
