Original sentence: A rigorous treatment of any of these topics begins with a specification of these axioms.Basic theories, such as Arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth).complex vaiables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms
Edited sentence: Any rigorous exposition to any such subject always starts by specifying this list, such as in textbooks, research papers or lectures.Basic theories, such as Arithmetic, real analysis (sometimes pompously referred to as the theory of functions of one real variable in basic levels), linear algebra, and complex analysis (a.What we mean by example in this section, is a mathematical theory that is developed entirely from the ground up from only a list of non-logical axioms (axioms, henceforth).complex vaiables), are often introduced vaguely in mostly technical studies by prescinding of its formal development, but any well form course in these subjects always starts by presenting its list of axioms