Complex Addition is Closed/Proof 2

Theorem
The set of complex numbers $\C$ is closed under addition:
 * $\forall z, w \in \C: z + w \in \C$

Proof
From the formal definition of complex numbers, we have:


 * $z = \left({x_1, y_1}\right)$
 * $w = \left({x_2, y_2}\right)$

where $x_1, x_2, y_1, y_2 \in \R$.

Then from the definition of complex addition:
 * $z + w = \left({x_1 + x_2, y_1 + y_2}\right)$

From Real Numbers under Addition form Abelian Group, real addition is closed.

So: $\left({x_1 + x_2}\right) \in \R$ and $\left({y_1 + y_2}\right) \in \R$

Hence the result.