Real Linear-Metric Space and Isometric Functions

Let V be a non empty metric structure. We say that V is convex if and only if the condition (Def. 1) is satisfied. (Def. 1) Let x, y be elements of the carrier of V and r be a real number. Suppose 0 ¬ r and r ¬ 1. Then there exists an element z of the carrier of V such that ρ(x, z) = r · ρ(x, y) and ρ(z, y) = (1 − r) · ρ(x, y). Let V be a non empty metric structure. We say that V is internal if and only if the condition (Def. 2) is satisfied. (Def. 2) Let x, y be elements of the carrier of V and p, q be real numbers. Suppose p > 0 and q > 0. Then there exists a finite sequence f of elements of the carrier of V such that (i) π1f = x, (ii) πlen ff = y, (iii) for every natural number i such that 1 ¬ i and i ¬ len f − 1 holds ρ(πif, πi+1f) < p, and (iv) for every finite sequence F of elements of R such that lenF = len f − 1 and for every natural number i such that 1 ¬ i and i ¬ lenF holds πiF = ρ(πif, πi+1f) holds |ρ(x, y) − ∑ F | < q. One can prove the following proposition