reserve T for TopSpace,
  B for Subset of T;

theorem Th16:
  PSO T /\ D(alpha,p)(T) = SO T
proof
  thus PSO T /\ D(alpha,p)(T) c= SO T
  proof
    let x be object;
    assume x in PSO T /\ D(alpha,p)(T); then
A0: x in PSO T & x in D(alpha,p)(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 & alphaInt B1 = pInt B1 by A0;
    sInt B = psInt B by A2,A1,Th1; then
    sInt B = B by A3; then
    B is semi-open by Th3; then
    x in SO T by A1;
    hence thesis;
  end;
  let x be object;
  assume x in SO T;
  then consider K being Subset of T such that
A1: x = K and
A2: K is semi-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
 Cl Int K c= Cl Int Cl K by TOPS_1:26;
  then K c= Cl Int Cl K by A2;
  then
A4: K is pre-semi-open;
  then K = psInt K by Th5;
  then sInt K = psInt K by A2,Th3;
  then alphaInt K = pInt K by Th1;
  then
A5: K in {B: alphaInt B = pInt B};
  K in PSO T by A4;
  hence thesis by A1,A5,XBOOLE_0:def 4;
end;
