Relation Induced by Strict Positivity Property is Trichotomy

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 $\forall a, b \in D:$ exactly one of the following conditions applies:
 * $a < b$
 * $a = b$
 * $a > b$

That is, $<$ is a trichotomy.

Proof
Take any $a, b \in D$ and consider $-a + b$.

From the trichotomy law of ordered integral domains, exactly one of the following applies:


 * $(1): \quad \map P {-a + b}$
 * $(2): \quad \map P {-\paren {-a + b} }$
 * $(3): \quad -a + b = 0$

Taking each of these in turn and taking into account that $\struct {D, +}$ is the additive group of $\struct {D, +, \times}$:

The result has been proved.