Gauges and Cages. Part I

For simplicity, we use the following convention: i, i1, i2, j, j1, j2, k, m, n, t denote natural numbers, D denotes a non empty subset of E2 T, E denotes a compact non vertical non horizontal subset of E2 T, C denotes a compact connected non vertical non horizontal subset of E2 T, G denotes a Go-board, p, q, x denote points of E2 T, and r, s denote real numbers. The following propositions are true: (1) For all real numbers s1, s3, s4, l such that s1 ¬ s3 and s1 ¬ s4 and 0 ¬ l and l ¬ 1 holds s1 ¬ (1 − l) · s3 + l · s4. (2) For all real numbers s1, s3, s4, l such that s3 ¬ s1 and s4 ¬ s1 and 0 ¬ l and l ¬ 1 holds (1 − l) · s3 + l · s4 ¬ s1. (3) If n > 0, then m mod m = 0. (4) If j > 0 and i mod j = 0, then i ÷ j = i j . (5) If n > 0, then i ÷ i = i n i .