reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;

theorem
  for X being non empty set,D being a_partition of X holds
  OpenClosedSet(partition_topology(D)) = the topology of partition_topology(D)
  proof
    let X be non empty set,D be a_partition of X;
    thus OpenClosedSet(partition_topology(D)) c=
      the topology of partition_topology(D)
    proof
      let x be object;
      assume x in OpenClosedSet(partition_topology(D));
      then ex y be Subset of partition_topology(D) st x = y & y is open closed;
      hence thesis;
    end;
    let x be object;
    assume
A2: x in the topology of partition_topology(D);
    then reconsider y = x as Subset of partition_topology(D);
    y is open by A2;
    then y is open closed by Th18;
    hence thesis;
  end;
