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
  for f being Point of Euclid 2, x being Point of TOP-REAL 2 st x in
  Ball(f,a) holds not |[x`1-2*a,x`2]| in Ball(f,a)
proof
  let f be Point of Euclid 2, x be Point of TOP-REAL 2 such that
A1: x in Ball(f,a);
A2: a > 0 by A1,TBSP_1:12;
  set p = |[x`1-2*a,x`2]|;
  reconsider z = p, X = x as Point of Euclid 2 by TOPREAL3:8;
A3: dist(X,z) = (Pitag_dist 2).(X,z) by METRIC_1:def 1
    .= sqrt ((x`1 - p`1)^2 + (x`2 - p`2)^2) by TOPREAL3:7
    .= sqrt ((x`1 - (x`1-2*a))^2 + (x`2 - p`2)^2)
    .= sqrt (0^2 + 2*(x`1 - x`1)*(2*a) + (2*a)^2 + (x`2 - x`2)^2)
    .= 2*a by A2,SQUARE_1:22;
  assume |[x`1-2*a,x`2]| in Ball(f,a);
  then
A4: dist(f,z) < a by METRIC_1:11;
  dist(f,X) < a by A1,METRIC_1:11;
  then dist(f,X) + dist(f,z) < a + a by A4,XREAL_1:8;
  hence contradiction by A3,METRIC_1:4;
end;
