Intersections of Intervals and Balls in E

For simplicity, we follow the rules: n denotes a natural number, a, b, r denote real numbers, x, y, z denote points of En T , and e denotes a point of En. The following propositions are true: (1) x − y − z = x − z − y. (2) If x + y = x + z, then y = z. (3) If n is non empty, then x 6= x + 1.REALn. (4) For every set x such that x = (1 − r) · y + r · z holds x = y iff r = 0 or y = z and x = z iff r = 1 or y = z. (5) For every finite sequence f of elements of R holds |f | = ∑ f. (6) For every non empty metric space M and for all points z1, z2, z3 of M such that z1 6= z2 and z1 ∈ Ball(z3, r) and z2 ∈ Ball(z3, r) holds r > 0.