Simple Proof question

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I'm studying Spivak's calculus and I have a really simple question :

I'm only in the first chapter on "The basic properties of numbers"

So far, we have the following propostion

P1 : (a+b)+c=a+(b+c)

P2 : a+0=0+a=a

P3 : a+(-a)=(-a)+a=0

Now, he tries to prove P2 (He doesn't do it for P3, so it's granted)
He also says :

"The proof of this assertion involves nothing more than subtracting a from both sides of the equation, in other word, adding -a to both sides." Now, that I understand

"as the following detailled proof shows, all three properties P1-P3 must be used to justify this operation." That I don't understand. First, how can you use a proof of something you haven't proven ? Second, when he says all three properties to justify this operation, he means to substract "a" from both sides, right ? If so, I don't understand how they (properties) can be used ...

He starts with this :

If a+x=a

then (-a)+(a+x)=(-a)+a=0

hence ((-a)+a)+x=0

hence 0+x=0

hence x=0

My comments : For the first line, he starts with the assertion that an equation a+x=a exists. Now, he substract "a" from borth sides and with property 3 the right hand sides equals 0. With property 1 we regroup and cancel with property 3.Now we have 0+x=0 and we subtract zero from both sides to have x=0. Where is property 2 used ? How is subtracting "a" from both sides proven with all three properties ?

Thank you

Answer

Spivak wants to show that zero is the unique additive identity on $\mathbb{R}$. That is, he want to prove that if we have $a+x=a$ then $x$ must identical to zero. He assumes P1, P2 and P3 to prove this. In particular, he uses P2 in the last step. If $0+x=0$ then using P2 we can conclude that $x=0$ without P2 we can not conclude this.