The stipulations you have posited do not require equilibrium for each reaction, only steady-state.  This means that each individual reaction may be out of balance, but the entire system is balanced: A1 becomes A2 at precisely the same rate that A3 becomes A1, and so on, and the macroscopic rate of change of the concentration of each of the components of the system is zero.  The condition you express there at the end is that for which each of the individual reactions is also in equilibrium.

As the word says, detailed balance principle is a principle and cannot be demonstrated. The statement should be that the detailed balance principle is a sufficient condition to have equilibrium. In gas phase it is a consequence of micro reversibility, i.e. the trajectory of a collision has the same probability if time and space are reversed. This is true for simple geometries, but for complex large molecules this is not. In general the existence of equilibrium is used as a justification of the validity of the detailed balance principle. I think there is no demonstration that DB is a necessary condition for equilibrium.

Thank you Marc. Am I correct?  "The Gibbs energy of the chemical cycle, which ends with the same substances at the same conditions, is zero." So, the product of dissociation constants (k13/k31)(k32/k23)(k21/k12) = 1 as sum of Gibbs energies of elementary reactions in cycle is zero. Using equation Gij=-R T ln(kij/kji), where Gij is Gibbs energy of reaction AiAj and kij/kji is the dissociation constant.  

I don't think that's correct.  State functions are only path independent when systems are in equilibrium, and claiming that each reaction in this system is in equilibrium is begging the question.  Given the equations Marek posted, there is no requirement that k12 A1 - k21 A2 = 0.  If, instead, k12 A1 - k21 A2 = r, there is still a steady-state solution for the system, for which all time derivatives are zero.

Yes, you're right -- the cycle is the key.

As Gianpiero Colonna wrote it is a principle that cannot be demonstrated, such as the principles of thermodynamics. A principle is accepted simply because it is consistent with experimental observations.

This principle has implications in chemical kinetics, since it requires relationships between kinetic constants. An example is given in the first part of the electronic supplement to our paper "Kinetics of iodous acid disproportionation", Int.J.Chem.KInet. 2013, 45,525.

Analysis of your problem and answer to your question is contained in the attached article.