Strictly Positive Real Numbers are Closed under Multiplication/Proof 2

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 $b > 0$.

From Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication:

$a > c \implies a \times b > c \times b$

Thus setting $c = 0$:

$a > 0 \implies a \times b > 0 \times b$

But from Real Zero is Zero Element:

$0 \times b = 0$

Hence the result:

$a, b > 0 \implies a \times b > 0$

$\blacksquare$