Školjka

%V0 Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $$\vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert,$$ where $\vert A \vert$ denotes the number of elements in the finite set $A$. Note Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.