Strictly Positive Real Numbers are Closed under Division

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

Proof
From the definition of division:
 * $a \div b := a \times \paren {\dfrac 1 b}$

where $\dfrac 1 b$ is the inverse for real number multiplication.

From Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group, the algebraic structure $\struct {\R_{>0}, \times}$ forms a group.

Thus it follows that:
 * $\forall a, b \in \R_{>0}: a \times \paren {\dfrac 1 b} \in \R$

Therefore real number division is closed in $\R_{>0}$.

Also see

 * Strictly Positive Real Numbers are Closed under Multiplication