I am currently studying Introduction to Hilbert Spaces with Applications, by Debnath and Mikusinski. Chapter 1.2 Vector Spaces says the following:
(a) $x + y = y + x$;
(b) $(x + y) + z = x + (y + z)$;
(c) For every $x, y \in E$ there exists a $z \in E$ such that $x + z = y$;
From (c) it follows that for every $x \in E$ there exists a $z_x \in E$ such that $x + z_x = x$. We will show that there exists exactly one element $z \in E$ such that $x + z = x$ for all $x \in E$. That element is denoted by $0$ and called the zero vector.
Let $x, y \in E$. By (c), there exists a $w \in E$ such that $x + w = y$. If $x + z_x = x$ for some $z_x \in E$, then by (a) and (b),
$$y + z_x = (x + w) + z_x = (x + z_x) + w = x + w = y.$$
This shows that, if $x + z = x$ for some $x \in E$, then $y + z = y$ for any other element $y \in E$. We still need to show that such an element is unique. Indeed, if $z_1$ and $z_2$ are two such elements, then $z_1 + z_2 = z_1$ and $z_1 + z_2 = z_2$. Thus, $z_1 = z_2$.
I find the uniqueness "proof" suspicious. The authors just claim that $z_1 + z_2 = z_1$ and $z_1 + z_2 = z_2$, and that therefore $z_1 = z_2$, without really "proving" anything. Is this actually a valid proof? I would greatly appreciate it if people would please take the time to clarify this.
Note that they have established that $x + z_1 = x$ and that therefore for all $y \in E$ we have $y + z_1 = y$. Thus they can choose their $y$ to be $z_2$ and thus get $z_2 + z_1 = z_2$. The same can be repeated for $z_2$.