Relation Induced by Strict Positivity Property is Compatible with Multiplication

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

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


 * $\forall a, b \in D: a < b \iff \map P {-a + b}$

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


 * $\forall x, y, z \in D: x < y, \map P z \implies \paren {z \times x} < \paren {z \times y}$


 * $\forall x, y, z \in D: x < y, \map P z \implies \paren {x \times z} < \paren {y \times z}$

Proof
If $x < y$ then $\map P {-x + y}$.

Hence:

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