The best way to use notation in the proof

Question

I have two simple questions regarding the writing styles.

1. 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?

2. 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.

Answer 1

There is no rule, and no uniform "best" way to do this.

If you refer to an object often, you need not point the reader back to the definition each time. If you define it, use it once right away and then again much further on then the back reference might help.

In any case don't put a definition in a remark. Make it a definition.

Answer 2

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.