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 Th54:
  f|Y is constant implies f is_bounded_on Y
proof
  assume f|Y is constant;
  then consider r being VECTOR of V such that
A1: for c be Element of M st c in Y /\ dom f holds f/.c = r by PARTFUN2:35;
  now
     reconsider p=||.r.|| as Real;
    take p;
    let c be Element of M;
    assume c in Y /\ dom f;
    hence ||.f/.c.|| <= p by A1;
  end;
  hence thesis;
end;
