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An edit to normed ring is being discussed in another thread, here.
added the original reference:
For what it’s worth, the extra axiom in not stated there, either.
In Bosch, Güntzer & Remmert 1984 the axiom is also not stated, but they prove that it follows from their axioms (Prop. 3).
added pointer to:
After the definition (here), I have replaced the pointer to Berkovich 2009, def. 1.2.1 and the claim that Berkovich is missing an axiom (instead the issue seems to be that Berkovich discusses the non-Archimedean case) by pointer to Ozaki, Kashiwagi & Tsuboi 1953.
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