The Correspondence Between n-dimensional Euclidean Space and the Product of n Real Lines

The terminology and notation used in this paper have been introduced in the following papers: [2], [6], [10], [4], [7], [18], [8], [13], [1], [3], [5], [15], [16], [17], [21], [22], [9], [19], [20], [11], [14], and [12]. For simplicity, we use the following convention: x, y are sets, i, n are natural numbers, r, s are real numbers, and f1, f2 are n-long real-valued finite sequences. Let s be a real number and let r be a non positive real number. One can check the following observations: ∗ ]s− r, s+ r[ is empty, ∗ [s− r, s+ r[ is empty, and ∗ ]s− r, s+ r] is empty. Let s be a real number and let r be a negative real number. Observe that [s− r, s+ r] is empty. Let f be an empty yielding function and let us consider x. Observe that f(x) is empty. Let us consider i. Observe that i 7→ 0 is empty yielding. Let f be an n-long complex-valued finite sequence. One can check the following observations: