reserve T for TopSpace,
  B for Subset of T;

theorem Th15:
  PSO T /\ D(p,ps)(T) = PO T
proof
  thus PSO T /\ D(p,ps)(T) c= PO T
  proof
    let x be object;
    assume x in PSO T /\ D(p,ps)(T); then
A0: x in PSO T & x in D(p,ps)(T) by XBOOLE_0:def 4; then
    consider B being Subset of T such that
A1: x = B & B is pre-semi-open;
A3: B = psInt B by A1,Th5;
    consider B1 being Subset of T such that
A2: x = B1 & pInt B1 = psInt B1 by A0;
    pInt B = B by A2,A3,A1; then
    B is pre-open by Th4; then
    x in {B where B is Subset of T: B is pre-open} by A1;
    hence thesis;
  end;
  let x be object;
  assume x in PO T;
  then consider K being Subset of T such that
A1: x = K and
A2: K is pre-open;
 Int Cl K c= Cl Int Cl K by PRE_TOPC:18;
  then K c= Cl Int Cl K by A2;
  then
A4: K is pre-semi-open;
  then K = psInt K by Th5;
  then pInt K = psInt K by A2,Th4;
  then
A5: K in {B: pInt B = psInt B};
  K in PSO T by A4;
  hence thesis by A1,A5,XBOOLE_0:def 4;
end;
