Internal Direct Product Theorem

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $H_1, H_2 \le G$.

Then $G$ is the internal group direct product of $H_1$ and $H_2$ :


 * $(1): \quad G = H_1 \circ H_2$
 * $(2): \quad H_1 \cap H_2 = \set e$
 * $(3): \quad H_1, H_2 \lhd G$

where $H_1 \lhd G$ denotes that $H_1$ is a normal subgroup of $G$.