Modulo Multiplication on Reduced Residue System is Cancellable

Theorem
Let $m \in \Z_{> 0}$ be a (strictly) positive integer.

Let $\Z'_m$ be the reduced residue system modulo $m$:


 * $\Z'_m = \left\{{\left[\!\left[{k}\right]\!\right]_m \in \Z_m: k \perp m}\right\}$

Let $S = \left({\Z'_m, \times_m}\right)$ be the algebraic structure consisting of $\Z'_m$ under modulo multiplication.

Then $\times_m$ is cancellable, in the sense that:
 * $\forall a, b, c \in \Z'_m: a \times_m c = b \times_m c \implies a = b$

and:
 * $\forall a, b, c \in \Z'_m: c \times_m a = c \times_m b \implies a = b$