Modularity satisfies intermediate subgroup condition

Verbal statement
A modular subgroup of a group is also a modular subgroup inside every intermediate subgroup.

Modular subgroup
A subgroup $$A$$ of a group $$G$$ is termed modular in $$G$$ if for any subgroups $$B, C \le G$$ such that $$A \le C$$, we have:

$$\langle A, B \cap C \rangle = \langle A,B \rangle \cap C$$.

Intermediate subgroup condition
A subgroup property $$p$$ is said to satisfy the intermediate subgroup condition if whenever $$A \le D \le G$$ are groups such that $$A$$ satisfies property $$p$$ in $$G$$, $$A$$ also satisfies property $$p$$ in $$D$$.

Related subgroup properties satisfying intermediate subgroup condition

 * Permutability satisfies intermediate subgroup condition
 * Normality satisfies intermediate subgroup condition
 * Ellipticity satisfies intermediate subgroup condition

Proof
Given: A group $$G$$, subgroups $$A \le D \le G$$. $$A$$ is modular in $$G$$.

To prove: $$A$$ is modular in $$D$$: whenever $$B, C \le D$$ are such that $$A \le C$$, we have:

$$\langle A, B \cap C \rangle = \langle A, B \rangle \cap C$$.

Proof: Since $$B,C$$ are subgroups of $$D$$, they are also subgroups of $$G$$, and the condition $$A \le C$$ holds by assumption. Thus, by modularity of $$A$$ in $$G$$, we conclude that:

$$\langle A, B \cap C \rangle = \langle A, B \rangle \cap C$$.