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 Finally, out of curiosity, I'm wondering how long this phrase has been around, if it's considered out of date or if it's still a popular phrase, and what some good alternatives are.

I've taught math for more than a decade at the college level, and I have been a professional copy editor for almost a decade. I find this phrase distracting and potentially confusing or misleading, so my recommendation is to avoid it. I found it in a paper about "communication for power distribution grid," and I have substituted it with a standard, non-ambiguous phrase. Finally, none of Merriam-Webster's definitions of the word "sequel" justify this phrase with the intended meaning.
