Artin presents the symmetric group $S_3$ as the group generated by $x = (123)$ and $y = (12)$, governed by the relations $x^3 = 1$, $y^2 = 1$, and $yx = x^2 y$, which yield $$S_3 = \{1,x,x^2, y, xy, x^2 y\}.$$ It seems to me that there are others way to write this group, such as subsituting $yx$ for $x^2 y$, per the defining relation, but by brute force, I can see that these elements are distinct.
He comments, however, that we can see that these elements are distinct from the cancellation law. I'm not able to see how this is the case. Could someone point me in the right direction? I apologize if this is obvious.