Complex Multiplication is Closed/Proof 1

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


 * $z = x_1 + i y_1$
 * $w = x_2 + i y_2$

where $i = \sqrt {-1}$ and $x_1, x_2, y_1, y_2 \in \R$.

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

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.