reserve T for TopSpace,
  B for Subset of T;

theorem Th17:
  PSO T /\ D(sp,ps)(T) = SPO T
proof
  thus PSO T /\ D(sp,ps)(T) c= SPO T
  proof
    let x be object;
    assume x in PSO T /\ D(sp,ps)(T); then
A0: x in PSO T & x in D(sp,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 & spInt B1 = psInt B1 by A0;
    B is semi-pre-open by A1,A2,A3,Th6;
    hence thesis by A1;
  end;
  let x be object;
  assume x in SPO T;
  then consider K being Subset of T such that
A1: x = K and
A2: K is semi-pre-open;
  Cl Int K c= Cl K by PRE_TOPC:19,TOPS_1:16;
  then Int Cl Int K c= Int Cl K by TOPS_1:19;
  then Cl Int Cl Int K c= Cl Int Cl K by PRE_TOPC:19;
  then
A3: Cl Int K c= Cl Int Cl K by TOPS_1:26;
  Int Cl K c= Cl Int Cl K by PRE_TOPC:18;
  then
 Cl Int K \/ Int Cl K c= Cl Int Cl K by A3,XBOOLE_1:8;
  then K c= Cl Int Cl K by A2;
  then
A5: K is pre-semi-open;
  then K = psInt K by Th5;
  then spInt K = psInt K by A2,Th6;
  then
A6: K in {B: spInt B = psInt B};
  K in PSO T by A5;
  hence thesis by A1,A6,XBOOLE_0:def 4;
end;
