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Recently Reny and Wooders () showed that there is some point in the intersection of sets in Shapley's () generalization of the Knaster-Kuratowski-Mazurkiwicz Theorem with the property that the collection of all sets containing that point is partnered as well as balanced. In this paper we provide a further extension by showing that the collection of all such sets can be chosen to be strictly balanced, implying the Reny-Wooders result. Our proof is topological, based on thedoi:10.1287/moor.25.4.539.12117 fatcat:k6ffnrahdzd7xc62k7lwae4fwm