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For context, this is used in defining $\phi\vee0$ in a proof that the dual of a vector lattice is a vector lattice.

Given a linear functional $\phi$ on a vector lattice $V$, define $\phi^+$ on $V^+$ as $\phi^+: v \mapsto \sup\{\phi(u) : 0<u<v\}$.

How can I show $\phi^+(v+w) = \phi^+(v) + \phi^+(w)\ \forall v,w \in V^+$ ?

Thanks!

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