Sorry if this was already posted. I couldn't find any answer anywhere.

Also, sorry if this is the wrong forum.

In just the 2nd page of Chapter 1 of Spivak's Calculus 3ed edition,

Spivak describes to us 3 properties of numbers and then he uses these properties to prove that: a+x=a.

Right on the start he adds (-a) to both sides. Hence (-a)+(a+x)=(-a)+a.

My question is, how do you prove that it is a valid thing to do, to add the same number to equal sides of the equation?

Thanks!

And I'm sorry if this is a silly question!

BTW, the properties P(x) are these:

P(1) : If a, b and c are any numbers, then a+(b+c)=(a+b)+c.

P(2) : If a is any number, then a+0=0+a=0

P(3) : For any number a, there is a number -a such that a+(-a)=(-a)+a=0