Complex Multiplication is Closed/Proof 2

Proof
From the formal definition of complex numbers, we define the following:


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

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

From Real Numbers form Field:
 * $x_1 x_2 - y_1 y_2 \in \R$

and:
 * $x_1 y_2 + x_2 y_1 \in \R$

Hence the result.