Bivic's calculus is easy to read, yes. It is written for students who are just used to strict proof-based mathematics. I have the third edition here. The first chapter is called "Basic Properties of Numbers". The following is an excerpt from the first paragraph of Chapter 1:
This short chapter simply explains what "the basic properties of numbers" are … we are already familiar with them … its purpose is not to make an extended review of old materials, but to condense these knowledge into some simple and obvious digital properties. Some even seem too obvious to mention, but many different important facts are the result of the facts we want to emphasize.
Explanation: This chapter is a familiar and obvious list of attributes. This chapter is not an extended review, but a condensed list. These properties will be used to prove other things.
Saying that spivak "must prove that a+b is exchangeable" constitutes a fundamental misunderstanding of the first chapter. The attribute (P 1)-(P 12) is an axiom. Bivic didn't prove it. He listed them, commented on them, and then used them to prove some other properties, such as if and then.
The commutative property of addition is (P4). He has no evidence; He made a list. It is worth mentioning that in other texts,+,* and.
I will also mention spivak's delay statement (13 page), the axiom of minimum upper bound, until the end of the second chapter. This is a thoughtful move, and in my opinion, it is also a smart move. It emphasizes that (P 13) is all that is needed to describe the real number system necessary to complete calculus. The more familiar algebraic properties (P 1)-(P 12) already exist in the rational number system. In fact, there is a technical term called ordered domain for digital systems that satisfy (P 1)-(P 12). If (P 13) is taken as the definition of "integrity", Spivak's article emphasizes that the axioms needed by calculus are axioms of completely ordered fields.