redundancy of two-sidedness in definition of group

In the definition of group, one usually supposes that there is a two-sided identity element and thatany element has a two-sided inverse (cf. group (http://planetmath.org/Group)).

The group may also be defined without the two-sidednesses:

A group is a pair of a non-empty set $G$ and its associative binary operation$(x,y)\mapsto xy$ such that

1) the operation has a right identity element $e$;

2) any element $x$ of $G$ has a right inverse $x^{-1}$.

We have to show that the right identity $e$ is also a left identityand that any right inverse is also a left inverse.

Let the above assumptions on $G$ be true.  If $a^{-1}$ is the rightinverse of an arbitrary element $a$ of $G$, the calculation

$$a^{-1}a=a^{-1}ae=a^{-1}a{a}^{-1}{(a^{-1})}^{-1}=a^{-1}e{(a^{-1})}^{-1}=a^{-1}{(a^{-1})}^{-1}=e$$

shows that it is also the left inverse of $a$.  Using this result,we then can write

$$ea=(a{a}^{-1})a=a(a^{-1}a)=ae=a,$$

whence $e$ is a left identity element, too.