Arithmetic of Multiplication on Omega: Recursion Rules, Distributivity, Associativity, Commutativity, No Zero Divisors, Cancellation and Compatibility with the Order

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ and $0$ be as in The Class Omega of Natural Numbers with Zero §omega and The Class Omega of Natural Numbers with Zero §zero, let $S(x)$ denote the successor of a set $x$, let $+$ and $\cdot$ be the addition and the multiplication on $\omega$, products being formed before sums as in Multiplication on Omega §precedence, and let $<$ be the strict order on $\omega$. Let $k,m,n\in\omega$.

¶ $m\cdot0=0$ and $0\cdot m=0$.

¶ $m\cdot S(n)=m\cdot n+m$ and $S(m)\cdot n=m\cdot n+n$.

¶ $m\cdot S(0)=m$ and $S(0)\cdot m=m$.

¶ $k\cdot(m+n)=k\cdot m+k\cdot n$ and $(m+n)\cdot k=m\cdot k+n\cdot k$.

¶ $(k\cdot m)\cdot n=k\cdot(m\cdot n)$.

¶ $m\cdot n=n\cdot m$.

¶ If $m\cdot n=0$, then $m=0$ or $n=0$.

¶ If $k\neq0$, then $m<n$ if and only if $k\cdot m<k\cdot n$.

¶ If $k\neq0$ and $k\cdot m=k\cdot n$, then $m=n$.