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
  0c(#)f is_bounded_on Y
proof
  now
    take p=0;
    let c be Element of M;
    |.0c.|*||.f/.c.|| <= 0 by COMPLEX1:44;
    then
A1: ||.0c*(f/.c).|| <= 0 by CLVECT_1:def 13;
    assume c in Y /\ dom (0c(#)f);
    then c in dom (0c(#)f) by XBOOLE_0:def 4;
    hence ||.(0c(#)f)/.c.|| <= p by A1,Def2;
  end;
  hence thesis;
end;
