## 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 $x = \eqclass {a, b} {}$, $y = \eqclass {c, d} {}$ and $z = \eqclass {e, f} {}$ for some $x, y, z\in \Z$.

Then:

 $\displaystyle x + z$ $=$ $\displaystyle y + z$ $\displaystyle \leadsto \ \$ $\displaystyle \eqclass {a, b} {} + \eqclass {e, f} {}$ $=$ $\displaystyle \eqclass {c, d} {} + \eqclass {e, f} {}$ $\displaystyle \leadsto \ \$ $\displaystyle \eqclass {a + e, b + f} {}$ $=$ $\displaystyle \eqclass {c + e, d + f} {}$ Definition of Integer Addition $\displaystyle \leadsto \ \$ $\displaystyle a + e$ $=$ $\displaystyle c + e$ Definition of Integer $\, \displaystyle \land \,$ $\displaystyle b + f$ $=$ $\displaystyle d + f$ Definition of Integer $\displaystyle \leadsto \ \$ $\displaystyle a$ $=$ $\displaystyle c$ Natural Number Addition is Cancellable $\, \displaystyle \land \,$ $\displaystyle b$ $=$ $\displaystyle d$ Natural Number Addition is Cancellable $\displaystyle \leadsto \ \$ $\displaystyle \eqclass {a, b} {}$ $=$ $\displaystyle \eqclass {c, d} {}$ Definition of Integer

$\blacksquare$