# Theorem

Let H1, H2, ... Hn be subgroups of Group G with operation $\ast$

$H_1 \cap H_2 \cap \cdots \cap H_n$ with $\ast$ is a subgroup of Group G

# $\color{RawSienna}(H_1 \cap H_2) \subseteq G$

 1. $H_1 \subseteq G$ H1 is subgroup of G 2. $H_2 \subseteq G$ H2 is subgroup of G 3. $(H_1 \cap H_2) \subseteq G$ 1. and 2.

# $\color{RawSienna}H_1 \cap H_2$ with $\color{RawSienna}\ast$ is a Group

## Closure

 4. Choose $x,y \in (H_1 \cap H_2)$ 5. $x \ast y \in H_1$ closure of H1 6. $x \ast y \in H_2$ closure of H2 7. $x \ast y \in (H_1 \cap H_2)$ 5. and 6.

## Associativity

 8. $\ast$ is associative on G. Group G's operation is $\ast$ 9. $(H_1 \cap H_2) \subseteq G$ 3. 10. $\ast$ is associative on $(H_1 \cap H_2)$ 8. and 9.

## Identity

 11. $e_{G} \in H_1$ and $e_{G} \in H_2$ Subgroup H1 and H2 inherit identity from G 12. $\forall g \in G: e_{G} \ast g = g \ast e_{G} = g$ eG is identity of G, 13. $\forall \; g \in (H_1 \cap H_2): e_{G} \ast g = g \ast e_{G} = g$ $(H_1 \cap H_2) \subseteq G$ and 9. 14. $(H_1 \cap H_2)$ has identity eG definition of identity

## Inverse

 15. Choose $g \in (H_1 \cap H_2) \subseteq G$ 16. $g \in H_1$, $g \in H_2$, and $g \in G$ 17. gH1-1 in H1, and gH2-1 in H2. G, H1, and H2 are groups 18. $g^{-1}_{H1} \in G$ $H_{1} \subseteq G$ 19. $g^{-1}_{H1} \ast g = g \ast g^{-1}_{H1} = e_{G}$ G and H1 shares identity e 20. gH1-1 is inverse of g in G 19. and definition of inverse 21. Let gG-1 be inverse of g has in G 22. gG-1 = gH1-1 inverse is unique 22. gG-1 = gH2-1 similar to 21. 23. $g^{-1} = g^{-1}_{H1} = g^{-1}_{H2} \in (H_1 \cap H_2)$ 24. g has inverse g-1 in $(H_1 \cap H_2)$