Natural Number Addition is Cancellable

Theorem
Let $\N_{> 0}$ be the 1-based natural numbers:
 * $\N_{> 0} = \left\{{1, 2, 3, \ldots}\right\}$

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

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