reserve X for non empty TopSpace;
reserve x for Point of X;
reserve U1 for Subset of X;

theorem
  for T being non empty TopSpace, A being Subset of T for p being Point
  of T holds p in Cl A iff for G being a_neighborhood of p holds G meets A
proof
  let T be non empty TopSpace, A be Subset of T;
  let p be Point of T;
  hereby
    assume
A1: p in Cl A;
    let G be a_neighborhood of p;
    p in Int G & Int G is open by Def1;
    then A meets Int G by A1,PRE_TOPC:def 7;
    hence G meets A by TOPS_1:16,XBOOLE_1:63;
  end;
  assume
A2: for G being a_neighborhood of p holds G meets A;
  now
    let G be Subset of T;
    assume G is open & p in G;
    then G is a_neighborhood of p by Th3;
    hence A meets G by A2;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
