reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem
  i <= j implies len Gauge(D,i) <= len Gauge(D,j)
proof
  assume i <= j;
  then
A1: 2|^i <= 2|^j by PREPOWER:93;
  len Gauge(D,i) = 2|^i + 3 & len Gauge(D,j) =2|^j + 3 by JORDAN8:def 1;
  hence thesis by A1,XREAL_1:6;
end;
