Definition:Natural Number Multiplication/1-Based

Definition

$(\text A)$	$:$	$\ds \exists_1 1 \in \N_{> 0}:$	$\ds a \times 1 = a = 1 \times a $
$(\text B)$	$:$	$\ds \forall a, b \in \N_{> 0}:$	$\ds a \times \paren {b + 1} = \paren {a \times b} + a $
$(\text C)$	$:$	$\ds \forall a, b \in \N_{> 0}:$	$\ds a + \paren {b + 1} = \paren {a + b} + 1 $
$(\text D)$	$:$	$\ds \forall a \in \N_{> 0}, a \ne 1:$	$\ds \exists_1 b \in \N_{> 0}: a = b + 1 $
$(\text E)$	$:$	$\ds \forall a, b \in \N_{> 0}:$	$\ds $Exactly one of these three holds:
			$\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} $
$(\text F)$	$:$	$\ds \forall A \subseteq \N_{> 0}:$	$\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} $

The operation $\times$ in this axiomatization is called multiplication.

\((\text A)\)	$:$	\(\ds \exists_1 1 \in \N_{> 0}:\)	\(\ds a \times 1 = a = 1 \times a \)
\((\text B)\)	$:$	\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds a \times \paren {b + 1} = \paren {a \times b} + a \)
\((\text C)\)	$:$	\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds a + \paren {b + 1} = \paren {a + b} + 1 \)
\((\text D)\)	$:$	\(\ds \forall a \in \N_{> 0}, a \ne 1:\)	\(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \)
\((\text E)\)	$:$	\(\ds \forall a, b \in \N_{> 0}:\)	\(\ds \)Exactly one of these three holds:\( \)
			\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)
\((\text F)\)	$:$	\(\ds \forall A \subseteq \N_{> 0}:\)	\(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)