Element of Group is in its own Coset

Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.

Let $x \in G$.

Let:


 * $x H$ be the left coset of $x$ modulo $H$;
 * $H x$ be the right coset of $x$ modulo $H$.

Then:
 * $x \in x H$;
 * $x \in H x$.

Proof
Let $e$ be the identity of $G$.


 * $x \in x H$:

From Identity of Subgroup, we have $e \in H$.

From the behaviour of the identity, we have:
 * $x e = x = e x$

That is:
 * $\exists h \in H: x h = x$;
 * $\exists h \in H: x = h x$.

The result follows by definition of coset.