Algebra question

Hi guys,

I am working on an algebra problem and am not quite sure if this is right. It "feels" correct but at the same time I kind of feeling like I might be begging the question.

$\text{\underline{Theorem:} Let G be a group with the following property:}$ $\text{Whenever a, b, c}$ $\in$ $\text{G and a*b = c*a implies b = c, then G is Abelian}$

$\text{\underline{Proof:} Suppose a*b = c for some a, b, c }$ $\in$ $\text{ G.Applying a both sides by a on the right gives a*b*a = c*a.}$
$\text{The associative property of groups gives a*(b*a) = c*a. By hypothesis,}$
$\text{we can cancel a from both sides yielding b*a = c. Hence, b*a = c}$ $\text{and a*b = c implies a*b = b*a. Therefore, G is Abelian.}$ $\blacksquare$