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So in writing a math paper, if I am writing a proof of a theorem and I need to refer a lemma that comes later in the same paper. What is the proper way to say that? For example if I am proving a theorem in page 10 and I need a lemma in page 15, how should I say that in the proof?

  • How do other authors do that? Have you had a look at some papers to find out? – Snijderfrey Nov 12 '19 at 6:33
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    Are you structuring the paper correctly? – Solar Mike Nov 12 '19 at 6:36
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The proper way here is to rearrange your paper so that this does not happen. It might be possible to put the theorem into the paper twice. Once in the introduction, possibly without some technical details but with more context and intuition around it and once after the required lemmata a proven with all the technical details.

The main reason is that the proofs of your various lemmata and theorems depend on one another but you want the dependence to form a tree and not have any loops. If you have a theorem that depends on a lemma proved later on, it could look like the proof of the lemma also depends on the theorem stated earlier on.

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    Linear structure for a paper is not the only alternative, and is often not the best. A separate section with the proofs of techical lemmas can be a good idea in many papers, for example. (There should be no logical loops, but this does not imply that the proofs have to be presented in order.) – Tommi Nov 12 '19 at 10:36
  • The logical order is not always the best pedagogical order. A lemma may be better-motivated if the proof of the theorem comes first, and motivating the reader should be a top priority. It is not uncommon. It just requires careful communication of what the logical order is. – Jair Taylor Nov 17 '19 at 5:03
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if I am writing a proof of a theorem and I need to refer a lemma that comes later in the same later. What is the proper way to say that?

Many approaches are possible, it depends on the precise setting. E.g., when the lemma is relevant to that particular theorem, you could write:

\begin{theorem}\label{mytheorem}
...
\end{theorem}

\begin{proof}

..., by the following lemma:

\begin{lemma}\label{mylemma}
...
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{mylemma}]
\end{proof}

Continuing the proof of Theorem~\ref{mytheorem}, ...

\end{proof}
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  • But what if I am using only one part of the lemma and also the proof of the lemma itself is given later? – A. Rahman Nov 12 '19 at 8:49
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    If you are using only one part of the lemma, then the lemma surely isn't only relevant to the particular theorem, since the other part is relevant elsewhere. (So my example mightn't be relevant.) To get better answers (and to allow me to improve mine), I think you need to provide more context in your question – user2768 Nov 12 '19 at 9:53

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