Correct usage of the phrase "In the sequel"? History? Alternatives? While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does this phrase mean?
(Someone recently argued to me that "sequel" was actually supposed to refer to a forthcoming second part of a paper, which I found highly unlikely, but I'd just like to make sure. )
My main questions: 
At what points in the text, and for what kinds of X, is it appropriate to use the phrase "In the sequel, X" in a paper? In a book? Is it ever acceptable to introduce definitions via 
"In the sequel, we introduce the concept of a "blah", which is a thing satisfying ..." 
at the start of a paper or book without a formal
"Definition. A "blah" is a thing, satsifying ..."
in the main text of the paper or book? Finally, out of curiosity, I'm wondering how long this phrase has been around, if it's considered out of date or if it's still a popular phrase, and what some good alternatives are.

Mathematics is generally seen as a precise and specific language.  Surely if there is ambiguity in the phrase "in the sequel" it should be avoided.  There are far more accurate ways of expressing both the meanings that phrase could carry so why create needless confusion?
