Strictly Positive Real Numbers are Closed under Multiplication

Theorem
The set $\R_{>0}$ of strictly positive real numbers is closed under multiplication:
 * $\forall a, b \in \R_{> 0}: a \times b \in \R_{> 0}$

Proof
Let $a, b \in \R_{> 0}$

We have that the Real Numbers form Ordered Integral Domain.

It follows from Positive Elements of Ordered Ring that:
 * $a \times b \in \R_{> 0}$.

Also see

 * Strictly Positive Real Numbers are Closed under Division