Natural Number Addition is Cancellable

Theorem
Let $\N$ be the natural numbers.

Let $+$ be addition on $\N$.

Then:
 * $\forall a, b, c \in \N: a + c = b + c \implies a = b$
 * $\forall a, b, c \in \N: a + b = a + c \implies b = c$

That is, $+$ is cancellable on $\N$.