Prove that if in a certain set with associative operation $*$ defined, $ax = b$ and $ya = b$ have solutions, then that structure is a group.
My attempt:
$$ax = b \Rightarrow (\exists e )(axe = x \land be =b)$$
$$ya = b \Rightarrow(\exists n)(nya = y \land nb = b)$$
Now, I'm stuck - I don't know how to prove that $e$ and $n$ are the same thing and that if $be = nb = b$ then $e=n$.
