A generalization of a Theorem of A. Rogers

Generalizing a Theorem due to A. Rogers \cite{ro1}, we are going to prove that if for a pair of convex bodies $K_{1},K_{2}\subset \Rn$, $n\geq 3$, there exists a hyperplane $H$ and a pair of different points $p_1$ and $p_2$ in $\Rn \backslash H$ such that for each $(n-2)$-plane $M\subset H$, there exists a \textit{mirror} which maps the hypersection of $K_1$ defined by $\aff\{ p_1,M\}$ onto the hypersection of $K_2$ defined by $\aff\{ p_2,M\}$, then there exists a \textit{mirror} which maps $K_1$ onto $K_2$.