Element in Coset iff Product with Inverse in Subgroup

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

Let $$x, y \in G$$.

Let:
 * $$y H$$ denote the left coset of $$H$$ by $$y$$;
 * $$H y$$ denote the right coset of $$H$$ by $$y$$.

Then:
 * $$x \in y H \iff x^{-1} y \in H$$
 * $$x \in H y \iff x y^{-1} \in H$$

Proof

 * $$x \in y H \iff x^{-1} y \in H$$:

$$ $$ $$ $$ $$


 * $$x \in H y \iff x y^{-1} \in H$$:

$$ $$ $$ $$