Direct Product of Normal Subgroups is Normal

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

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

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

Then:
 * $\paren {H \times H'} \lhd \paren {G \times G'}$

where $H \times H'$ denotes the group direct product of $H$ and $H'$

Proof
Let $\tuple {x, x'} \in G \times G'$ and $\tuple {y, y'} \in H \times H'$.

Then:

Hence:
 * $\paren {H \times H'} \lhd {G \times G'}$