reserve y,w for set;
reserve T for non empty TopSpace;

theorem Th3:
  { A where A is a_partition of the carrier of T : A is closed }
  is Part-Family of the carrier of T
proof
  set S = { A where A is a_partition of the carrier of T : A is closed };
A1: now
    let B be set;
    assume B in { A where A is a_partition of the carrier of T : A is closed };
    then ex A being a_partition of the carrier of T st B = A & A is closed;
    hence B is a_partition of the carrier of T;
  end;
  S c= bool bool the carrier of T
  proof
    let B be object;
    assume B in S;
    then ex A being a_partition of the carrier of T st B = A & A is closed;
    hence thesis;
  end;
  hence thesis by A1,EQREL_1:def 7;
end;
