Unity of Ordered Integral Domain is Strictly Positive

Theorem
Let $\struct {D, +, \times \le}$ be an ordered integral domain whose unity is $1_D$.

Then:
 * $\map P {1_D}$

where $P$ is the (strict) positivity property.

Proof
We have by definition of the unity that:
 * $\forall a \in D: 1_D \times a = a = a \times 1_D$

This particularly applies to $1_D$ itself:
 * $1_D = 1_D \times 1_D$

But then by Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive:
 * $\map P {1_D \times 1_D} \implies \map P {1_D}$