Non-Zero Integers are Cancellable for Multiplication/Proof 2

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Theorem

Every non-zero integer is cancellable for multiplication.


That is:

$\forall x, y, z \in \Z, x \ne 0: x y = x z \iff y = z$


Proof

Let $y, z \in \Z: y \ne z$.

\(\displaystyle y\) \(\ne\) \(\displaystyle z\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle y - z\) \(\ne\) \(\displaystyle 0\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x \paren {y - z}\) \(\ne\) \(\displaystyle 0\) Ring of Integers has no Zero Divisors‎
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x y - x z\) \(\ne\) \(\displaystyle 0\) Integer Multiplication Distributes over Subtraction

The result follows by transposition.

$\blacksquare$


Sources