Axiom:Axiomatization of 1-Based Natural Numbers

Axioms
The following axioms are intended to capture the behaviour of the ($1$-based) natural numbers $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations of addition $+$ and multiplication $\times$ as they pertain to $\N_{>0}$:

Note
The above axiom schema specifies the old-fashioned definition of the natural numbers as:
 * $\text{The set of natural numbers} = \set {1, 2, 3, \ldots}$

as opposed to the more modern approach which defines them as:
 * $\text{The set of natural numbers} = \set {0, 1, 2, 3, \ldots}$

In order to eliminate confusion, on the set $\set {1, 2, 3, \ldots}$ will be denoted as $\N_{> 0}$ or $\N_{\ne 0}$ or $\N_{\ge 1}$.

When $\N$ is used, $\N = \set {0, 1, 2, 3, \ldots}$ is to be understood.