Homeomorphism between [:e

The notation and terminology used in this paper have been introduced in the following articles: [20], [9], [25], [7], [8], [4], [16], [24], [21], [19], [13], [18], [23], [1], [2], [10], [5], [17], [11], [3], [22], [14], [12], [6], [26], and [15]. We use the following convention: i, j, n denote natural numbers, f , g, h, k denote finite sequences of elements of R, and M , N denote non empty metric spaces. We now state a number of propositions: (1) For all real numbers a, b such that max(a, b) ¬ a holds max(a, b) = a. (2) For all real numbers a, b, c, d holds max(a + c, b + d) ¬ max(a, b) + max(c, d). (3) For all real numbers a, b, c, d, e, f such that a ¬ b + c and d ¬ e + f holds max(a, d) ¬ max(b, e) + max(c, f). (4) For all finite sequences f , g holds dom g ⊆ dom(f a g). (5) For all finite sequences f , g such that len f < i and i ¬ len f + len g holds i − len f ∈ dom g. (6) For all finite sequences f , g, h, k such that f ag = hak and len f = lenh and len g = len k holds f = h and g = k. (7) If len f = len g or dom f = dom g, then len(f + g) = len f and dom(f + g) = dom f.