reserve T for TopSpace,
  B for Subset of T;

theorem Th10:
  PSO T /\ D(c,ps)(T) = the topology of T
proof
  thus PSO T /\ D(c,ps)(T) c= the topology of T
  proof
    let x be object;
    assume
A1: x in PSO T /\ D(c,ps)(T);
    then x in PSO T by XBOOLE_0:def 4;
    then consider A being Subset of T such that
A2: x = A and
A3: A is pre-semi-open;
    x in D(c,ps)(T) by A1,XBOOLE_0:def 4;
    then consider Z being Subset of T such that
A4: x = Z and
A5: Int Z = psInt Z;
    A = psInt A by A3,Th5;
    hence thesis by A4,PRE_TOPC:def 2,A2,A5;
  end;
  let x be object;
  assume
A6: x in the topology of T;
  then reconsider K = x as Subset of T;
A7: Int Cl K c= Cl Int Cl K by PRE_TOPC:18;
  K is open by A6,PRE_TOPC:def 2;
  then
A8: K = Int K by TOPS_1:23;
  then K c= Int Cl K by PRE_TOPC:18,TOPS_1:19;
  then K c= Cl Int Cl K by A7;
  then
A9: K is pre-semi-open;
  then Int K = psInt K by A8,Th5;
  then
A10: K in {B: Int B = psInt B};
  K in PSO T by A9;
  hence thesis by A10,XBOOLE_0:def 4;
end;
