Normality is strongly UL-intersection-closed

Statement with symbols
Suppose $$G$$ is a group, $$I$$ is an indexing set, and for each $$i \in I$$, we have subgroups $$H_i \le K_i \le G$$ such that $$H_i$$ is normal in $$K_i$$. Then, the intersection of the $$H_i$$s is normal in the intersection of the $$K_i$$s.

Applications
Combining this with the fact that UL-intersection-closedness is a [[composition-closed subgroup metaproperty, we can conclude that the property of being $$k$$-subnormal, for any fixed $$k$$, is also strongly intersection-closed.

Weaker facts

 * Normality is strongly intersection-closed
 * Normality satisfies transfer condition
 * Normality satisfies intermediate subgroup condition