The question is from this paper Applications of Random Sampling in Computational Geometry , II

Let SS be a set of nn subsets of EdE^d, whose elements are called objects\textit{objects}. Let F\mathcal{F} be a set of subsets of EdE^d, called range space. Let bb be a positive integer and S(b)S^{(b)} be the set of subset of SS with bb or fewer objects. Let δ\delta be a relation between F\mathcal{F} and S(b)S^{(b)}. We say a range F∈FF\in \mathcal{F} is defined by some X∈S(b)X\in S^{(b)} if FδXF\delta X. Set of all ranges defined by objects of SS is denoted by FS\mathcal{F}_S. Define FR\mathcal{F}_R similarly for R⊂SR\subset S, this is the collection of ranges FF such that FδXF\delta X for some X∈R(b)X\in R^{(b)}.

Ranges in FS\mathcal{F}_S are defined by bb or fewer elements of SS. For a given F∈FSF\in \mathcal{F}_S , let iFi_F denote the size of the smallest set defining FF, let this set be XFX_F. If there are only one such set for every range defined, we call δ\delta functional\textit{functional}.

Prove that for R⊂SR\subset S, we have F∈FRF\in \mathcal{F}_R iff XF⊂RX_F\subset R.

The only if part is trivial , I am having trouble with the if part. I was trying to come up with a contradiction but in vain.

All these things are defined in the aforementioned paper in page 7-8.

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