in case that no H shell flash in the HEFM had occurred. A first comparison with observed extremely metal-poor objects is made in § \[cobs\]. Conclusions and a brief discussion close the paper.

The stellar evolution code.
===========================

All calculations reported about in this paper were done with the Garching stellar evolution code [@wsch:2000], which is based on the original program developed by @KWH. The program in its present version is capable of calculating precise solar models [@S00; @SW99] as well as following the evolution of low-mass stars into the latest phases of their evolution [@JPhD; @SW99].

Numerical details. {#numtreat}
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The evolution through the He-flash can be followed in an acceptable amount of computing time thanks to the grid routine implemented by @WaWe, which ensures a sufficiently high accuracy in the linearization of the stellar structure equations. Moreover, the algorithm to determine the under-correction factor, i.e. the actually applied fraction of the correction found in each iteration step, has been improved considerably. By further limiting the time-step with a constraint on the change of the He-burning luminosity ($L_{\rm
He}$), the critical phase of the He-flash ($\log(L_{\rm He}/L_\odot) >7$) can be followed with about 200 models, and with at most 