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 Th25:
  for A being closed Subset of TOP-REAL n, p being Point of TOP-REAL n st
  not p in A ex r being positive Real st Ball(p,r) misses A
proof
  let A be closed Subset of TOP-REAL n, p be Point of TOP-REAL n;
  assume not p in A;
  then
A1: p in A` by SUBSET_1:29;
  reconsider e = p as Point of Euclid n by TOPREAL3:8;
A2: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider AA = A` as Subset of TopSpaceMetr Euclid n;
  AA is open by A2,PRE_TOPC:30;
  then consider r being Real such that
A3: r > 0 and
A4: Ball(e,r) c= A` by A1,TOPMETR:15;
  reconsider r as positive Real by A3;
  take r;
  Ball(p,r) = Ball(e,r) by TOPREAL9:13;
  hence thesis by A4,SUBSET_1:23;
end;
