Complex Multiplication is Closed/Proof 2

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


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

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

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.