Abstract: By considering the connection of approximate isometry with isometry we study the following problem "For what subclass of (F)-normed spaces the metric structure completely determines the linear constructure and further, what form can be taken when we represent an approximate isometry as a perturbation of isometry. We introduce a subclass of so-called midpoint p-constracted (F)-normed spaces. For this subclass some new results are obtained and especially the known results of Mazur-Ulam and Gervitz-Gruber are extended.