TITLE: The best way to use notation in the proof
QUESTION [9 upvotes]: I have two simple questions regarding the writing styles.

Assume that I make a remark~$1.1$ and I define two sets say $A:=\text{something}$ and $B:=\text{something}$. My question is that I will use the sets $A$ and $B$ in many proofs in the paper. Should I say at the beginning of each proof, take $A$ and $B$ as in the remark~$1.1$? OR just write $A$ and $B$ and the reader should check what $A$ and $ B$ are in the paper?


What is the best way to rewrite the following " Let $A$ and $B$ as in the Theorem~1 and $D$ and $E$ as in the remark~2. Let $\mathcal H$ be a family as in the Theorem~2 with $A.$ Also, let $\mathcal G$ be a family as in the Proposition~3."

Any help will be greatly appreciated.

REPLY [5 votes]: As Ethan Bolker said, there is no general rule for this. My personal recommendation is that, after the definition of $A$ and $B$, you write a little paragraph saying something like:
“After this point and unless specified otherwise, the symbols $A$ and $B$ will stand for the sets defined above.”
That way there is no confusion.