Transitive normality is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Suppose $H\leq K\leq G$ are groups, such that $K$ is a transitively normal subgroup of $G$ and $H$ is a transitively normal subgroup of $K$, then $H$ is a transitively normal subgroup of $G$.