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