
theorem
  for T being non empty TopStruct, A being Subset of T, p being Point of
  T holds p in Cl A iff for K being Basis of p, Q being Subset of T st Q in K
  holds A meets Q
proof
  let T be non empty TopStruct, A be Subset of T, p be Point of T;
  thus p in Cl A implies for K being Basis of p, Q being Subset of T st Q in K
  holds A meets Q by Lm1;
  assume for K being Basis of p, Q being Subset of T st Q in K holds A meets Q;
  then
  ex K being Basis of p st for Q being Subset of T st Q in K holds A meets
  Q by Lm2;
  hence thesis by Lm3;
end;
