It should be said that this routine was developed after long trial and error due to lacking understanding of the theory behind. The routine is needed in order to evaluate the thermodynamic state in a space compartment evaluated from the value in the adjacent compartment, comp. section 3.3.6.
Acc. to the proof in appendix 4 the appropriate way to do this by solving the thermodynamic state by the equation system

In doing so the integration (with finite small space intervals in the pressure equation) can be avoided and the accuracy of the calculation should increase compared with the present subroutine. In future this subroutine VX_Mu will replace the presented one for the said purpose.

Nevermind, the present subroutione VTX_PMu works. The temperature value calculated comes back to the preset temperature value with an accuracy below mK !
If we installed the subroutine VX_PF   instead of  VTX_PMu , the algorithm did not converge because the cut-off error of calculation and integration and perhaps other errors led to slightly mismatched starting values which led to the break down of the subroutine.