Cancellation Laws/Proof 3

Corollary to Latin Square Property
Let $G$ be a group.

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

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

These are respectively called the right and left cancellation laws.

Proof
Suppose $x = b a = c a$.

By the Latin Square Property, there exists exactly one $y \in G$ such that $x = y a$.

That is, $x = b a = c a \implies b = c$.

Similarly, suppose $x = a b = a c$.

Again by the Latin Square Property, there exists exactly one $y \in G$ such that $x = a y$.

That is, $a b = a c \implies b = c$.