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I discovered that a student of mine uses constructs like

\begin{theorem} 
  ...a statement...
  \begin{proof}
    ...proof of that statement...
  \end{proof}
\end{theorem} 

My first reaction was negative but then I realized that it seems to render fine and I do not have any substantial argument against it. In fact there might be a certain rationale behind this usage: you begin a statement and only end it after finishing to deal with it, proof included.

Is such usage considered incorrect by any reason?

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    For the usual definitions of therorem that do little more than setting the font it should be ok. If the theorem uses some hanging indentation, for instance, then the proof may or may not look like intended. This way of nesting is unconventional, but not wrong, in my opinion.
    – gernot
    Feb 27 at 11:09
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The manual of https://ctan.org/pkg/ntheorem states that this might rise problems with endmarks, see pages 30-31. There is also way to solve this problem but one might easy forget to apply the hack which gives inconsistent results. It might be fine as long as you do not use endmarks but in case one decides to do otherwise later it might be problematic. There might be other issues.

The document testmath.tex of https://ctan.org/pkg/amsmath does not nest the proof environments inside theorems and alike, too.

Both might indicate that nesting proofs is not a standard use case and yield more problems or is not expected by programmers of packages. The doument https://ctan.org/pkg/lshort or the manual of https://ctan.org/pkg/amsthm apparently do not address the question.

One issue I think about is the following: The editor might want to make the document fancier by adding colored boxes around theorems and images of רְאֵם to fundamental theorem, 槌の子 to theorems, and lemmings to lemmata in the margin to "enhance" the readability and embig the appeal to a broader audience. The inclusion would possibly frame the proof as well which the editor might not want.

Therefore, I do not recommend nesting proof environments.

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