Transitive and transfer condition implies finite-intersection-closed

Statement
A fact about::transitive subgroup property that satisfies the fact about::transfer condition is finite-intersection-closed.

Transitive subgroup property
A subgroup property $$p$$ is termed transitive if whenever $$H \le K \le G$$ are groups such that $$H$$ satisfies property $$p$$ in $$K$$ and $$K$$ satisfies property $$p$$ in $$G$$, $$H$$ also satisfies property $$p$$ in $$G$$.

Transfer condition
A subgroup property $$p$$ is said to satisfy the transfer condition if whenever $$H, K \le G$$ such that $$H$$ satisfies property $$p$$ in $$G$$, $$H \cap K$$ satisfies property $$p$$ in $$K$$.

Finite-intersection-closed subgroup property
A subgroup property $$p$$ is termed finite-intersection-closed if whenever $$H, K$$ are subgroups satisfying property $$p$$ in $$G$$, then $$H \cap K$$ also satisfies property $$p$$ in $$G$$.

Facts used

 * 1) uses::Transitive and transfer condition implies finite-relative-intersection-closed
 * 2) uses::Finite-relative-intersection-closed implies finite-intersection-closed

Proof using given facts
The proof follows directly by piecing together facts (1) and (2).