Proof. First establish the lattice property. Choose two payoff vectors $p'=(w',\pi')$ and $p''=(w'',\pi'')$. We need to show that $p'\vee p''$ satisfies the dual constraints and complementary slackness. \begin{equation*} p'\vee p''=(\max\{w'_l,w''_l\},\min\{\pi'_f, \pi''_f\})_{l,f\in\ll\cup\ff}. \end{equation*} Then for all $lf$ pairs, \begin{equation*} \begin{aligned} w'_l&\geq v_{lf}-\pi'_f\\ w''_l&\geq v_{lf}-\pi''_f. \end{aligned}\tag{$*$} \end{equation*} and so \begin{equation*} \begin{aligned} \max\{w'_l,w''_l\}&\geq\max\{v_{lf}-\pi'_f,v_{lf}-\pi''_f\}\\ &=v_{lf}-\min\{\pi'_f,\pi''_f\} \end{aligned}\tag{$**$} \end{equation*} So $p'\vee p''\geq v_{lf}$. A similar argument holds for $p'\wedge p''$. So the meet and join of $p'$ and $p''$ satisfy the dual constraints.