is mixed during the HEFM into the helium flash driven convective zone (HECZ), it is crucial to treat the mixing as a time-dependent process and to consider carefully the simultaneous mixing and burning processes. In the present investigation, we therefore improved the treatment of mixing and burning in our program. The equations for microscopic diffusion, convection (see below) and nuclear burning are solved in one common scheme [@PhD]; this also implies that the hydrogen and helium networks are solved simultaneously. This is necessary for the correct treatment of the nuclear evolution during those phases, when H and He burning are operating at the same time, as is the case in the core of evolved low-mass main sequence-stars [“CNO-flash”, @WCSS00; @FIH90] of Pop. III and during the HEFM.

In our approach convection is treated as a fast diffusive process where the diffusion constant is proportional to the convective velocity obtained from a convection theory [@Lan85]. When the HECZ touches the H-rich layers at the lower boundary of the H-burning shell, the ingested hydrogen interacts with $^{12}$C to produce $^{14}$N (CNOI), releasing a huge amount of energy ($\log (L_{\rm H}/L_\odot)
\ga 6$). The movement of the outer boundary of the convective zone 