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.