= Closedness of projection from projective space
Projection $X\times\mathbb P^n\to X$ sends closed subsets to closed subsets, with geometric points or schemes understood. On an affine base, choose finitely many homogeneous equations for a closed subset. Its fiber is nonempty exactly when every <graded multiplication map for a projective fiber> fails to be surjective, by the <Projective Nullstellensatz>. The image is thus an intersection of closed rank-defect loci. An affine open cover proves the general case. This is the universally closed part of the <proper morphism> property of projective space.
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