reserve n for Nat,
  i,j for Nat,
  r,s,r1,s1,r2,s2,r9,s9 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board,
  x,y for set,
  v for Point of Euclid 2;

theorem Th2:
  for p being Point of Euclid n, q being Point of TOP-REAL n, r
  being Real st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q
proof
  let p be Point of Euclid n, q be Point of TOP-REAL n, r be Real;
  reconsider A = Ball (p, r) as Subset of TOP-REAL n by TOPREAL3:8;
A1: 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 TOPMETR:14;
  then
A2: A is open by A1,PRE_TOPC:30;
  assume p = q & r > 0;
  hence thesis by A2,Th1,CONNSP_2:3;
end;
