## Theorem

The operation of addition on the set of integers $\Z$ is cancellable:

$\forall x, y, z \in \Z: x + z = y + z \implies x = y$

## Proof

Let $x = \left[\!\left[{a, b}\right]\!\right]$, $y = \left[\!\left[{c, d}\right]\!\right]$ and $z = \left[\!\left[{e, f}\right]\!\right]$ for some $x, y, z\in \Z$.

Then:

 $\displaystyle x + z$ $=$ $\displaystyle y + z$ $\displaystyle \implies \ \$ $\displaystyle \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{e, f}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{c, d}\right]\!\right] + \left[\!\left[{e, f}\right]\!\right]$ $\displaystyle \implies \ \$ $\displaystyle \left[\!\left[{a + e, b + f}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{c + e, d + f}\right]\!\right]$ Definition of Integer Addition $\displaystyle \implies \ \$ $\displaystyle a + e$ $=$ $\displaystyle c + e$ Definition of Integer $\, \displaystyle \land \,$ $\displaystyle b + f$ $=$ $\displaystyle d + f$ Definition of Integer $\displaystyle \implies \ \$ $\displaystyle a$ $=$ $\displaystyle c$ Natural Number Addition is Cancellable $\, \displaystyle \land \,$ $\displaystyle b$ $=$ $\displaystyle d$ Natural Number Addition is Cancellable $\displaystyle \implies \ \$ $\displaystyle \left[\!\left[{a, b}\right]\!\right]$ $=$ $\displaystyle \left[\!\left[{c, d}\right]\!\right]$ Definition of Integer

$\blacksquare$