# Product of Strictly Positive Real Numbers is Strictly Positive

## Theorem

$x, y \in \R_{>0} \implies x \times y \in \R_{>0}$

## Proof

Let $y > 0$.

$x > z \implies x \times y > z \times y$

Thus setting $z = 0$:

$x > 0 \implies x \times y > 0 \times y$

But from Real Zero is Zero Element:

$0 \times y = 0$

Hence the result:

$x, y > 0 \implies x \times y > 0$

$\blacksquare$