Subdistributivity relation between intersection operator and composition operator

Finite version
Suppose $$p_1,p_2,q$$ are subgroup properties. Let $$\cap$$ denote the fact about::intersection operator on subgroup properties, and $$*$$ denote the fact about::composition operator on subgroup properties. Then:

$$(p_1 \cap p_2) * q \le (p_1 * q) \cap (p_2 * q)$$.

Corollaries

 * Finite-intersection-closedness is left residual-closed: This is a corollary of the finite version.
 * Intersection-closedness is left residual-closed: This is a corollary of the infinite version.