I have just provided the German translation of the Metamath book as PDF to 
be integrated into the Metamath homepage, see GitHub Pull Request #21 
(https://github.com/metamath/metamath-website-seed/pull/21). After this PR 
is accepted and merged, the German translation of the Metamath book should 
be available for download on the Homepage 
https://us.metamath.org/index.html#book.

Here some remarks:

   - The objective of the translation was to stay as close as possible to 
   the original text. 
   - New appendices "Metamath 100" and "Glossary" (translations of the most 
   important words/phrases in both directions German -> English, English -> 
   German) are added (not contained in the original version)
   - Some updates were made reflecting the changes of Metamath since 2019, 
   mostly in footnotes
   - Errata (see 
   https://github.com/metamath/metamath-book/blob/master/errata.md) are mostly 
   considered/corrected

Everybody speaking German (not only native speakers) and being interested 
in this translation is invited to give feedback for improvements and 
additions. This can be done either here in the Google group, or as issues 
in Github (https://github.com/metamath/metamath-book/issues), which would 
be the preferred way.

I plan to provide the LaTeX-sources of the German translation also on 
Github, maybe in a folder "german" of the repository 
https://github.com/metamath/metamath-book. Then,  the files can be edited 
collaboratively. Furthermore, if the Metamath book is updated, the original 
(which is also located there) and the German translation can be modified 
simultaneously (like set.mm and iset.mm regarding common parts).

As Mario mentioned before, the LaTex-file of the original Metamath-book is 
also available in the repository 
https://github.com/metamath/metamath-website-seed (folder latex). I can 
also place a copy of the German translation there, but the "master" of the 
document should be in https://github.com/metamath/metamath-book.

I hope that all German speaking readers will enjoy this translation, and 
maybe some more German speaking people become interested in Metamath in 
this way.

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