I'm currently at Chapter 4, vol. 1 and 1st ed. I have to ask this question because the most important thing about this book is in its minute details. Thanks.
Take *3.3 for example. Acording to this site, I think Proof would be more appropriate.
In standard mathematical terminology, "proof" and "demonstration" mean the same thing. Therefore Russell and Whitehead's use of language was appropriate.
Quod erat demonstrandum.