A generalization of a Theorem of A. Rogers

Generalizing a Theorem due to A. Rogers ro1, we are going to prove that if for a pair of convex bodies K₁, K₂, n 3, there exists a hyperplane H and a pair of different points p₁ and p₂ in H such that for each (n-2) -plane M H, there exists a mirror which maps the hypersection of K₁ defined by \ p₁, M\ onto the hypersection of K₂ defined by \ p₂, M\, then there exists a mirror which maps K₁ onto K₂.