Product of Strictly Positive Real Numbers is Strictly Positive

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Theorem

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


Proof

Let $y > 0$.

From Real Number Axioms: $\R O2$: compatible with multiplication:

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

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:

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

$\blacksquare$


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