Sum of Strictly Positive Real Numbers is Strictly Positive

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Theorem

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


Proof

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

$\blacksquare$


Sources