reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem
  D = Ball(e,r) implies D is non horizontal
proof
  reconsider p = e as Point of TOP-REAL 2 by TOPREAL3:8;
  assume
A1: D = Ball(e,r);
  then
A2: r > 0 by TBSP_1:12;
  take p, q = |[p`1,p`2-r/2]|;
  thus p in D by A1,TBSP_1:11,12;
  reconsider f = q as Point of Euclid 2 by TOPREAL3:8;
  dist(e,f) = (Pitag_dist 2).(e,f) by METRIC_1:def 1
    .= sqrt ((p`1 - q`1)^2 + (p`2 - q`2)^2) by TOPREAL3:7
    .= sqrt ((p`1 - p`1)^2 + (p`2 - q`2)^2)
    .= sqrt (0 + (p`2 - (p`2-r/2))^2)
    .= r/2 by A2,SQUARE_1:22;
  then dist(e,f) < r by A1,TBSP_1:12,XREAL_1:216;
  hence q in D by A1,METRIC_1:11;
  r/2 > 0 by A2,XREAL_1:139;
  then p`2-r/2 < p`2-0 by XREAL_1:15;
  hence thesis;
end;
