Theorem. If $H_1,\,H_2,\,\ldots,\,H_n$ are subgroups of $G$ , then $$\left[G:\bigcap_{i=1}^nH_i\right] \leqq \prod_{i=1}^n[G:H_i].$$

Proof. We prove here only the case $n = 2$ ; the general case may be handled by the induction.

Let $H_1\!\cap\!H_2 := K$ . Let $R$ be the set of the right cosets of $K$ and $R_i$ the set of the right cosets of $H_i$ ($i = 1,\,2$ ). Define the relation $\varrho$ from $R$ to $R_1\!\times\!R_2$ as $$\varrho \;:=\; \{\left(Kx,\,(H_1x,\,H_2x)\right)\vdots\;\; x \in G \}.$$ We then have the equivalent conditions $$Kx \;=\; Ky,$$ $$xy^{-1} \in K,$$ $$xy^{-1} \in H_1 \quad\land\quad xy^{-1} \in H_2,$$ $$H_1x \;=\; H_1y \quad\land\quad H_2x \;=\; H_2y,$$ $$(H_1x,\,H_2x) \;=\; (H_1y,\,H_2y),$$ whence $\varrho$ is a mapping and even injective, $\varrho:\, R \to R_1\!\times\!R_2$ . i.e. it is a bijection from $R$ onto the subset $\{\varrho(x)\vdots\;\; x \in R\}$ of $R_1\!\times\!R_2$ . Therefore, $$\card(R) \;\leqq\; \card(R_1\!\times\!R_2) \;=\; \card(R_1)\cdot\card(R_2).$$

As a consequence one obtains the

Theorem (Poincaré). The index of the intersection of finitely many subgroups with finite indices is finite.