reserve T for TopSpace,
  B for Subset of T;

theorem Th13:
  PSO T /\ D(alpha,ps)(T) = T^alpha
proof
  thus PSO T /\ D(alpha,ps)(T) c= T^alpha
  proof
    let x be object;
    assume
A1: x in PSO T /\ D(alpha,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(alpha,ps)(T) by A1,XBOOLE_0:def 4;
    then consider Z being Subset of T such that
A4: x = Z and
A5: alphaInt Z = psInt Z;
    A = psInt A by A3,Th5;
    then Z is alpha-set of T by A2,A4,A5,Th2;
    hence thesis by A4;
  end;
  let x be object;
  assume x in T^alpha;
  then consider K being Subset of T such that
A6: x = K and
A7: K is alpha-set of T;
  Cl Int K c= Cl K by PRE_TOPC:19,TOPS_1:16;
  then
A8: Int Cl Int K c= Int Cl K by TOPS_1:19;
  Int Cl K c= Cl Int Cl K by PRE_TOPC:18;
  then
A9: Int Cl Int K c= Cl Int Cl K by A8;
  K c= Int Cl Int K by A7,Def1;
  then K c= Cl Int Cl K by A9;
  then
A10: K is pre-semi-open;
  then K = psInt K by Th5;
  then alphaInt K = psInt K by A7,Th2;
  then
A11: K in {B: alphaInt B = psInt B};
  K in PSO T by A10;
  hence thesis by A6,A11,XBOOLE_0:def 4;
end;
