In most of our preceding articles (I-IV), we split the 2nd law into two parts: a) the law of evolution of an adiabatically isolated system Σ (= Σ_o), and b) the law of equilibrium of an isolated Σ (= Σ_oo). This article is mainly devoted to the 2nd law part b, where we demonstrate that the maximum of the entropy functional S[. . .] may be found by the use of Lagrange multipliers expressing the conservation of the energy functional H[. . .] and, in general, of several other conserved extensive quantities.