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?
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