Natural Number Multiplication is Cancellable

Theorem
Let $\N$ be the natural numbers.

Let $\times$ be multiplication on $\N$.

Then:
 * $\forall a, b, c \in \N_{>0}: a \times c = b \times c \implies a = b$
 * $\forall a, b, c \in \N_{>0}: a \times b = a \times c \implies b = c$

That is, $\times$ is cancellable on $\N_{>0}$.