  In this work we study the mathematical aspects of the development in the
continuum thermodynamics known as the "Entropy Principle". It started with the
pioneering works of B.Coleman, W.Noll and I. Muller in 60th of XX cent. and got
its further development mostly in the works of G. Boillat, I-Shis Liu and
T.Ruggeri. "Entropy Principle" combines in itself the structural requirement on
the form of balance laws of the thermodynamical system (denote such system
$(\mathcal{C})$) and on the entropy balance law with the convexity condition of
the entropy density. First of these requirements has pure mathematical form
defining so called "supplementary balance laws" (shortly SBL) associated with
the original balance system. Vector space of SBL can be considered as a kind of
natural "closure" of the original balance system. This space includes the
original balance laws, the entropy balance, the balance laws corresponding to
the symmetries of the balance system and some other balance equations. We
consider the case of Rational Extended Thermodynamics where densities, fluxes
and sources of the balance equations do not depend on the derivatives of
physical fields $y^i$. We present the basic structures of RET: Lagrange-Liu
equations,"main fields", and dual formulation of the balance system. We obtain
and start studying the defining system of equations for the density $h^0$ of a
supplementary balance law. This overdetermined linear system of PDE of second
order determines all the densities $h^0$ and with them, due to the formalism of
RET, the fluxes and sources of SBL. Solvability conditions of defining system
delivers the constitutive restrictions on the balance equations of the original
balance system. We illustrate our results by some simple examples of balance
system and by describing all the supplementary balance laws and the
constitutive restrictions for the Cattaneo heat propagation system.
