 Finite intersection of finite-index subgroups

For finite-index subgroups $H,K$ of a group $G$, the map of sets of left cosetsis well defined and injective: equality of both components is equivalent to the difference $g^{\prime -1}g$ belonging to $H\cap K$. Hence the displayed subgroup index bound holds, without a normal subgroup hypothesis. By mathematical induction, the index of a subgroup for a finite intersection is at most the product of the individual subgroup indices.