Hi there

I was reading through this webpage:

Arithmetic Rules and I got to the author's attempt to prove the statement

$\displaystyle b + (-a) = b - a$ where $\displaystyle a, b \in \mathbb{R}$.

I am, however, not convinced by his argument.

Up to this point he had estabished the following to be true:

- $\displaystyle a + 0 = a$
- $\displaystyle a + b = b + a$
- $\displaystyle (a + b) + c = a + (b + c)$
- $\displaystyle a = b \implies a + c = b + c$
- $\displaystyle a + b = c \iff c - b = a$
- $\displaystyle -a = 0 - a$
- $\displaystyle (-a) + a = 0$
- $\displaystyle 0 = -0$

He gives as a proof:

$\displaystyle -a = 0 - a$

$\displaystyle b + (-a) = b + 0 - a$

$\displaystyle b + (-a) = b - a$

Am I correct in thinking that $\displaystyle 0-a$ should be in parentheses when $\displaystyle b$ is added to it, and since the author is yet to establish that

$\displaystyle a + (b - c) = (a + b) - c$ the proof is invalid?

Would anyone be able to tell me whether or not I am correct in thinking this, and if not then explain what I am not understanding?