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 Th24:
  for r being non zero Real holds Fr Ball(x,r) = Sphere(x,r)
proof
  let r be non zero Real;
  set P = Ball(x,r);
  thus Fr P = Cl P \ P by TOPS_1:42
    .= cl_Ball(x,r) \ P by Th23
    .= Sphere(x,r) by Th19;
end;
