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

\(\displaystyle x\) \(>\) \(\displaystyle 0\)
\(\, \displaystyle \land \, \) \(\displaystyle y\) \(>\) \(\displaystyle 0\) Real Number Ordering is Compatible with Addition
\(\displaystyle \implies \ \ \) \(\displaystyle x + y\) \(>\) \(\displaystyle 0 + 0\) Real Number Inequalities can be Added
\(\displaystyle \) \(=\) \(\displaystyle 0\) Real Number Axioms: $\R A3$: Identity

$\blacksquare$


Sources