Cancellation law on a commutative and associative binary operation on a set $S$

Question

I need to Show: Let$*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$.(This z may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and ac=bc, then $a=b$.

Answer 1

Suppose ac=bc. Find $x,y$ so that $acx=a$ and $ay=b$. Then $$a=acx=bcx=aycx=acxy=ay=b.$$