Element of Group is in its own Coset/Left

Theorem
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x \in G$. Let:
 * $x H$ be the left coset of $x$ modulo $H$.

Then:
 * $x \in x H$

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

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

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

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

The result follows by definition of left coset.

Also see

 * Element of Group is in its own Right Coset