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 Th5:
  for u being Point of Euclid n, P being Subset of TOP-REAL n holds
  u in Int P iff ex r being Real st r > 0 & Ball(u,r) c= P
proof
  let u be Point of Euclid n, P be Subset of TOP-REAL n;
A1: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider PP = P as Subset of TopSpaceMetr Euclid n;
  u in Int PP iff ex r being Real st r > 0 & Ball(u,r) c= PP by Th4;
  hence thesis by A1,Lm5;
end;
