In the formerly proposed theory for sub-linear (linear in a changed addition rule) systems, we developed a sub-linear mathematical theory and demonstrated its capability in several sub-linear systems. In this paper, we assert that we can further construct hybrid systems useful in engineering by combining sub-linear systems with linear systems, and as an example, we show the construction of a divergence-free electrodynamics. Since the energy of photon fields in engineering is tending upward, it would be desirable for us to get rid of the divergence difficulties encountered in the conventional high energy electrodynamics. The most important result is the recognition that Nature herself has a hybrid system composed of sub-linear photon fields and linear electron fields. Mathematically, our electrodynamics is formulated by only one point correction (the insertion of tanh into the electromagnetic energy density) in the conventional electrodynamics of photons and electrons (including positrons).