Theorem 4.15

Statement:

If $G$ is a group with binary operation $*$, then the left and right cancellation laws hold in $G$, that is, $a * b = a * c$ implies $b = c$, and $b * a = c * a$ implies $b = c$ for all $a, b, c \in G$.

Proof:

Suppose $a * b = a * c$. Then by $\mathscr{G}_{3}$, there exists $a'$, and

(1)
$$a' * (a * b) = a' * (a * c).$$

By the associative law, $\mathscr{G}_{1}$

(2)
$$(a' * a) * b = (a' * a) * c.$$

By the definition of $a'$ in $\mathscr{G}_{3}$, $a' * a = e$, so

(3)
$$e * b = e * c.$$

By the definition of $e$ in $\mathscr{G}_{2}$,

(4)
$$b = c.$$

Similarly, from $b * a = c * a$ one can deduce that $b = c$ upon multiplication on the right by $a'$ and use of the axioms for a group.