The Borsuk-Ulam Theorem

For simplicity, we adopt the following rules: a, b, x, y, z, X, Y , Z denote sets, n denotes a natural number, i denotes an integer, r, r1, r2, r3, s denote real numbers, c, c1, c2 denote complex numbers, and p denotes a point of En T. Let us observe that every element of IQ is irrational. Next we state a number of propositions: (1) If 0 ≤ r and 0 ≤ s and r2 = s2, then r = s. (2) If frac r ≥ frac s, then frac(r − s) = frac r − frac s. (3) If frac r < frac s, then frac(r − s) = (frac r − frac s) + 1.