Power of Product in Abelian Group/Additive Notation

Theorem
Let $\struct {G, +}$ be an abelian group.

Then:
 * $\forall x, y \in G: \forall k \in \Z: k \cdot \paren {x + y} = \paren {k \cdot x} + \paren {k \cdot y}$

Proof
By definition of abelian group, $x$ and $y$ commute.

That is:
 * $x + y = y + x$

The result follows from Power of Product of Commutative Elements in Group.