  Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real
number. We prove that if x,y \in X and ||x-y||/d is a rational number then
there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following
property: for each strictly convex Y of dimension 2 each map from S(x,y) to Y
preserving the distance d preserves the distance between x and y. It implies
that each map from X to Y that preserves the distance d is an isometry.
