\begin{definition}[Definition:Proper Divisor/Integer]
Let $\struct {\Z, +, \times}$ be the ring of integers.
Let $x, y \in \Z$.
Then '''$x$ divides $y$''' is defined as:
:$x \divides y \iff \exists t \in \Z: y = t \times x$
Then $x$ is a '''proper divisor''' of $y$ {{iff}}:
:$(1): \quad x \divides y$
:$(2): \quad \size x \ne \size y$
:$(3): \quad x \ne \pm 1$
That is:
:$(1): \quad x$ is a divisor of $y$
:$(2): \quad x$ and $y$ are not equal in absolute value
:$(3): \quad x$ is not equal to either $1$ or $-1$.
\end{definition}