theorem Th19:
  for O being closed Subset of partition_topology(D) holds O is open
  proof
    let O be closed Subset of partition_topology(D);
    [#]partition_topology(D) \ O in UniCl D by PRE_TOPC:def 2,PRE_TOPC:def 3;
    then
A1: (X \ O)` in UniCl D by PROB_1:def 1;
    O = X /\ O by XBOOLE_1:18,XBOOLE_1:19;
    hence thesis by A1,XBOOLE_1:48;
  end;
