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 Th3:
  for B being Subset of TOP-REAL n, u being Point of Euclid n st B
  = Ball(u,r) holds B is open
proof
  let B be Subset of TOP-REAL n, u be Point of Euclid n;
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider BB = B as Subset of TopSpaceMetr Euclid n;
  assume B = Ball(u,r);
  then BB is open by Lm4;
  hence thesis by A1,PRE_TOPC:30;
end;
