Quotient Group of Direct Products

Theorem
Let $G$ and $G'$ be groups.

Let:
 * $H \triangleleft G$
 * $H' \triangleleft G'$

where $\triangleleft$ denotes the relation of being a normal subgroup.

Then:
 * $(1): \quad \left({H \times H'}\right) \triangleleft \left({G \times G'}\right)$
 * $(2): \quad \left({H \times H'}\right) / \left({G \times G'}\right)$ is isomorphic to $\left({G / H}\right) \times \left({G' / H'}\right)$

where:
 * $H \times H'$ denotes the group direct product of $H$ and $H'$
 * $G / H$ denotes the quotient group of $G$ by $H$.