# Sum of Strictly Positive Real Numbers is Strictly Positive

## Theorem

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

## Proof

 $\displaystyle x$ $>$ $\displaystyle 0$ $\, \displaystyle \land \,$ $\displaystyle y$ $>$ $\displaystyle 0$ Real Number Ordering is Compatible with Addition $\displaystyle \leadsto \ \$ $\displaystyle x + y$ $>$ $\displaystyle 0 + 0$ Real Number Inequalities can be Added $\displaystyle$ $=$ $\displaystyle 0$ Real Number Axioms: $\R \text A 3$: Identity

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