Pure definability is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., purely definable subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Statement with symbols

Suppose $H \le K \le G$ are groups such that $H$ is a purely definable subgroup of $G$ and $K/H$ is a purely definable subgroup of $G/H$ (note that any purely definable subgroup is characteristic and hence normal, so it makes sense to take the quotient group). Then, $K$ is a purely definable subgroup of $G$.