Relation Induced by Strict Positivity Property is Compatible with Addition

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 compatible with $+$, that is:


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

Proof
Let $a < b$:

And so $<$ is seen to be compatible with $+$.