reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th26:
  for A being bounded Subset of TOP-REAL n, a being Point of TOP-REAL n
  ex r being positive Real st A c= Ball(a,r)
proof
  let A be bounded Subset of TOP-REAL n;
  let a be Point of TOP-REAL n;
  reconsider C = A as bounded Subset of Euclid n by JORDAN2C:11;
  consider r being Real, x being Element of Euclid n such that
A1: 0 < r and
A2: C c= Ball(x,r) by METRIC_6:def 3;
  reconsider r as positive Real by A1;
  reconsider x1 = x as Point of TOP-REAL n by TOPREAL3:8;
  take s = r+|.x1-a.|;
  let p be object;
  assume
A3: p in A;
  then reconsider p1 = p as Point of TOP-REAL n;
  p = p1;
  then reconsider p as Point of Euclid n by TOPREAL3:8;
A4: dist(p,x) < r by A2,A3,METRIC_1:11;
A5: |.p1-x1.| = dist(p,x) by SPPOL_1:39;
A6: |.p1-a.| <= |.p1-x1.| + |.x1-a.| by TOPRNS_1:34;
  |.p1-x1.| + |.x1-a.| < s by A4,A5,XREAL_1:6;
  then |.p1-a.| < s by A6,XXREAL_0:2;
  hence thesis by TOPREAL9:7;
end;
