Weakly closed implies relatively normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup-of-subgroup properties. That is, it states that every subgroup-of-subgroup satisfying the first subgroup-of-subgroup property (i.e., weakly closed subgroup) must also satisfy the second subgroup-of-subgroup property (i.e., relatively normal subgroup)
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Statement

Suppose H \le K \le G are such that H is a Weakly closed subgroup (?) of K relative to G. Then, H is a Normal subgroup (?) of K. In other words, H is a Relatively normal subgroup (?) in K with respect to G.