Relation Induced by Strict Positivity Property is Compatible with Multiplication

Theorem
Let $\left({D, +, \times}\right)$ be an ordered integral domain where $P$ is the positivity property.

Let the relation $<$ be defined on $D$ as:


 * $\forall a, b \in D: a < b \iff P \left({-a + b}\right)$

Then $<$ is compatible with $\times$ in the following sense:


 * $\forall x, y, z \in D: x < y, P \left({z}\right) \implies \left({z \times x}\right) < \left({z \times y}\right)$


 * $\forall x, y, z \in D: x < y, P \left({z}\right) \implies \left({x \times z}\right) < \left({y \times z}\right)$

Proof
If $x < y$ then $P \left({- x + y}\right)$.

Hence:

The other result follows from the fact that $\times$ is commutative in an integral domain.