Normality-largeness is transitive

Property-theoretic statement
The subgroup property of being a normality-large subgroup satisfies the subgroup metaproperty of being transitive.

Verbal statement
Any normality-large subgroup of a normality-large subgroup is normality-large.

Statement with symbols
Suppose $$H$$ is a normality-large subgroup of $$K$$ and $$K$$ is a normality-large subgroup of $$G$$. Then, $$H$$ is a normality-large subgroup of $$G$$.

Hands-on proof
Given: A group $$G$$, a normality-large subgroup $$K$$ of $$G$$, and a subgroup $$H$$ of $$K$$ that is normality-large in $$K$$

To prove: $$H$$ is a normality-large subgroup of $$G$$

Proof: We need to show that if $$N$$ is a nontrivial normal subgroup of $$G$$, then $$H \cap N$$ is nontrivial.

First, observe that since $$K$$ is normality-large in $$G$$, $$K \cap N$$ is a nontrivial subgroup of $$K$$. Now, since $$H$$ is normality-large in $$K$$, $$H \cap (K \cap N)$$ is a nontrivial subgroup of $$H$$. Since $$H$$ is contained in $$K$$, $$H \cap K \cap N = H \cap N$$, so $$H \cap N$$ is nontrivial.