# Relation Induced by Strict Positivity Property is Transitive

## 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 a transitive relation.

## Proof

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

Thus:

 $\displaystyle$  $\displaystyle a < b, b < c$ $\displaystyle$ $\leadsto$ $\displaystyle \map P {-a + b}, \map P {-b + c}$ Definition of $<$ $\displaystyle$ $\leadsto$ $\displaystyle \map P {\paren {-a + b} + \paren {-b + c} }$ Definition of Strict Positivity Property $\displaystyle$ $\leadsto$ $\displaystyle \map P {-a + c}$ Properties of $+$ in $D$ $\displaystyle$ $\leadsto$ $\displaystyle a < c$ Definition of $<$

And so $<$ is seen to be transitive.

$\blacksquare$