Strictly Positive Real Numbers are Closed under Multiplication/Proof 1
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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:
- $a \times b \in \R_{> 0}$.
$\blacksquare$