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:


 * $z = \tuple {x_1, y_1}$
 * $w = \tuple {x_2, y_2}$

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

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

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

So:
 * $\paren {x_1 + x_2} \in \R$ and $\paren {y_1 + y_2} \in \R$

and hence the result.