Continuity of Barycentric Coordinates in Euclidean Topological Spaces

For simplicity, we adopt the following rules: x denotes a set, n, m, k denote natural numbers, r denotes a real number, V denotes a real linear space, v, w denote vectors of V , A1 denotes a finite subset of V , and A2 denotes a finite affinely independent subset of V . One can prove the following propositions: (1) For all real-valued finite sequences f1, f2 and for every real number r holds (Intervals(f1, r)) a Intervals(f2, r) = Intervals(f1 a f2, r). (2) Let f1, f2 be finite sequences. Then x ∈ ∏ (f1 a f2) if and only if there exist finite sequences p1, p2 such that x = p1 a p2 and p1 ∈ ∏ f1 and p2 ∈ ∏ f2.