Pure definability is transitive

Verbal statement
A purely definable subgroup of a purely definable subgroup is purely definable.

Statement with symbols
Suppose $$H \le K \le G$$ are groups such that $$K$$ is a purely definable subgroup of $$G$$ and $$H$$ is a purely definable subgroup of $$K$$. Then, $$H$$ is a purely definable subgroup of $$G$$.

Similar facts

 * Pure definability is quotient-transitive
 * Second-order definability is transitive
 * Second-order characteristicity is transitive
 * Monadic second-order definability is transitive
 * Monadic second-order characteristicity is transitive