Coset Equals Subgroup iff Element in Subgroup

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

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

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

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

Proof

 * $$x H = H \iff x \in H$$:

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 * $$H x = H \iff x \in H$$ is proved similarly:

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