I do not have Spivak's book, but if he were trying to prove that a+x=a, he would not start the proof by assuming what he is trying to prove. He would not start with a+x=a then add (-a) to both sides and get (-a)+(a+x)=(-a)+a.

The properties you listed are taken to be true without proof. So is the property about adding the same number both sides of an equation.