Relation Induced by Strict Positivity Property is Transitive

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

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


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

Then $<$ is a transitive relation.

Proof
Let $a < b$ and $b < c$.

Thus:

And so $<$ is seen to be transitive.