Cancellation Laws/Proof 3

Corollary to Group has Latin Square Property

Let $G$ be a group.

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

Then the following hold:

Right cancellation law
$b a = c a \implies b = c$
Left cancellation law
$a b = a c \implies b = c$

Proof

Suppose $x = b a = c a$.

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

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

$\blacksquare$