Cancellation Laws/Proof 1

Theorem
Let $G$ be a group.

Let $a, b, c \in G$.

Then:
 * $b a = c a \implies b = c$
 * $a b = a c \implies b = c$

Proof
Let $a, b, c \in G$ and let $a^{-1}$ be the inverse of $a$.

Suppose $b a = c a$.

Then:

Thus, the right cancellation law holds.

The proof of the left cancellation law is analogous.