  We consider complete positivity of dynamics regarding subsystems of an open
composite quantum system, which is subject of a completely positive dynamics.
By "completely positive dynamics", we assume the dynamical maps called the
completely positive and trace preserving maps, with the constraint that domain
of the map is the whole Banach space of the system's density matrices. We
provide a technically simple and conceptually clear proof for the subsystems'
completely positive dynamics. Actually, we prove that every subsystem of a
composite open system can be subject of a completely positive dynamics if and
only if the initial state of the composite open system is tensor-product of the
initial states of the subsystems. An algorithm for obtaining the Kraus form for
the subsystem's dynamical map is provided. As an illustrative example we
consider a pair of mutually interacting qubits. The presentation is performed
such that a student with the proper basic knowledge in quantum mechanics should
be able to reproduce all the steps of the calculations.
