Integer Addition is Cancellable

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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

$\blacksquare$


Sources