Proof of Additive and Multiplicative Inverses Are Unique

Negative. Let $x\in r$. By Commutative Rings §ring there is $w\in r$ with $x+w=0$. Let $w,w'\in r$ with $x+w=0$ and $x+w'=0$. Then, by the axioms of Commutative Rings §ring,

$$w=w+0=w+(x+w')=(w+x)+w'=(x+w)+w'=0+w'=w'+0=w'.$$

Hence there is exactly one $w\in r$ with $x+w=0$.

Reciprocal. Let $r$ be a field and let $x\in r$ with $x\neq0$. By Fields §field there is $y\in r$ with $x\cdot y=1$, and $r$ is a commutative ring. Let $y,y'\in r$ with $x\cdot y=1$ and $x\cdot y'=1$. Then, by the axioms of Commutative Rings §ring,

$$y=y\cdot1=y\cdot(x\cdot y')=(y\cdot x)\cdot y'=(x\cdot y)\cdot y'=1\cdot y'=y'\cdot1=y'.$$

Hence there is exactly one $y\in r$ with $x\cdot y=1$.