Additive Cancellation and Elementary Additive Identities in a Field

Let $K$ be a field, with additive identity $0$ and with the additive inverse $-x$ of an element $x$ as in that definition, and write $x-y$ for $x+(-y)$. Let $x,y,z\in K$. Then the following hold.

1. (Uniqueness of additive inverses) If $x+y=0$, then $y=-x$ and $x=-y$.

2. (Cancellation) If $x+z=y+z$, then $x=y$.

3. (Vanishing differences) $x-x=0$, and $x-y=0$ if and only if $x=y$.

4. (Differences involving the additive identity) $-0=0$, $x-0=x$, and $0-x=-x$.

5. (Double inverse) $-(-x)=x$.

6. (Additive inverse of a sum and of a difference) $-(x+y)=(-x)+(-y)$ and $-(x-y)=y-x$.