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
    capOpCl partition_topology(D) = UniCl D
  proof
    let X be non empty set,D be a_partition of X;
    set Y = {A /\ B where A, B is Subset of partition_topology(D):
      A is open & B is closed};
A1: Y c= UniCl D
    proof
      let x be object;
      assume x in Y;
      then consider A,B be Subset of partition_topology(D) such that
A2:   x = A /\ B and
A3:   A is open and
A4:   B is closed;
      B is open by A4,Th19;
      hence thesis by A2,A3,FINSUB_1:def 2;
    end;
    UniCl D c= Y
    proof
      let x be object;
      assume
A5:   x in UniCl D;
      then reconsider y = x as Subset of partition_topology(D);
      X in UniCl D by Th15;
      then reconsider XX = X as Subset of partition_topology(D);
A6:   y = y /\ X by XBOOLE_1:18,XBOOLE_1:19;
      y is open & XX is closed by A5;
      hence thesis by A6;
    end;
    hence thesis by A1;
  end;
