Equivalence of definitions of local powering-invariant subgroup

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * 1) Whenever $$h \in H$$ and $$n \in \mathbb{N}$$ are such that there is a unique $$x \in G$$ such that $$x^n = h$$, we must have $$x \in H$$.
 * 2) Whenever $$h \in H$$ and $$p$$ is a prime number such that there is a unique $$x \in G$$ such that $$x^p = h$$, we must have $$x \in H$$.

Facts used

 * 1) uses::Unique nth root implies unique mth root for m dividing n

(1) implies (2)
This is obvious.

(2) implies (1)
We use Fact (1) to show uniqueness of $$p^{th}$$ root for some prime $$p$$ dividing $$n$$. Then, use condition (2) to obtain that that $$p^{th}$$ root is also in $$H$$. Now repeat the process with this new element -- we're now finding a $$(n/p)^{th}$$ root of the element. Each step gets rid of one prime divisor.