Mapping between Q d and Q d+2

Beckman, F.S and Quarles D.A, have proven in [1], the following theorem: Each function from the d-Euclidean space, (d≥2) into itself and which maintains distance 1 is an isometry. In addition,in [7, 8, [I had proven the parallel theoremfor rational spaces to the previous theorem, namely: every mapping from d-rational into itself, and which maintains distance 1 is an isometry provided d ≥ 5. Furthermore, in[6], I had proven: If d ≥ 5, and if, ω (d) = ω (d+1),then every unit- distance preserving mappingQdinto Qd+1 is an isometry, add to that,gave a form for the dimensionfor which Here in this article, I will expand one last result to show that in some special dimensiond, every unit distance-preserving mapping is an isometry. In addition, which are the dimension for which?