Suppose that I am writing a proof to a theorem that has the following structure:
Let $x \in X$, then "some statement involving $x$".
I would usually start the proof by repeating the conditions of the theorem, that is, I would usually start:
Let $x \in X$. Then...
Let's assume that the proof follows directly after the theorem, there is no intermittent text.
One of the maxims of good writing is to avoid unnecessary repetitions. This leads me to my question:
Is it necessary to repeat the conditions of a theorem in its proof, in the sense that it improves legibility?
Some of my thoughts so far.
Unconditional reasons to repeat the conditions of a theorem in its proof:
- It is logically necessary. A proof should be readable as a self-sufficient statement independently from the statement of the theorem.
- It helps the reader distinguish the conditions of the theorem from the statement of the theorem.
Conditional reasons to repeat the conditions of a proof:
- If the theorem statement is long, the the conditions were presumably written down some time ago, hence one eliminates the necessity of skipping back and forth in the text.
- If the theorem proves several related but separable claims then it helps the reader identify which point you are about to prove.
- In the case that the theorem reads something along the lines of:
If $x \in X$, then .... If moreover $x \in Y \subseteq X$, then ....
It establishes if you are talking about $x \in X$, or about the more special case of $x \in Y$.
- This is simply false, a proof need not be independently readable.
- If the theorem statement is well written, then this should be already clear.
- Reading mathematics is already non-linear, the attempt to force it to be linear is at best futile and at worst counter-productive.
- In this case the reader would be better served by an itemized list of the separate claims and an itemized list of proofs.
- I have no refutation for this point.
In conclusion, I think that 5. is the most convincing reason to repeat the conditions of a theorem in its proof. This tempts me to do this always for a consistency in style, but I am not sure if this is misguided.