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* Since the operation of H is the same as the operation of G, the operation is associative since G is a group.
* Since H is not empty there exists an element x in H.
* Let x be an element of H. Since the identity e is in H it follows that ex<sup>-1</sup> = x<sup>-1</sup> in H, so the inverse of an element in H is in H.
* Finally, let x and y be elements in H, then since y is in H it follows that y<sup>-1</sup> is in H. Hence x(y<sup>-1</sup>)<sup>-1</sup> = xy is in H and so H is closed under the operation.
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