let T be TopSpace; :: thesis: (PSO T) /\ (D(p,ps) T) = PO T
thus (PSO T) /\ (D(p,ps) T) c= PO T :: according to XBOOLE_0:def 10 :: thesis: PO T c= (PSO T) /\ (D(p,ps) T)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (PSO T) /\ (D(p,ps) T) or x in PO T )
assume A1: x in (PSO T) /\ (D(p,ps) T) ; :: thesis: x in PO 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(p,ps) T by A1, XBOOLE_0:def 4;
then consider Z being Subset of T such that
A4: x = Z and
A5: pInt Z = psInt Z ;
A = psInt A by A3, Th5;
then Z is pre-open by A2, A4, A5, Th4;
hence x in PO T by A4; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in PO T or x in (PSO T) /\ (D(p,ps) T) )
assume x in PO T ; :: thesis: x in (PSO T) /\ (D(p,ps) T)
then consider K being Subset of T such that
A6: x = K and
A7: K is pre-open ;
A8: Int (Cl K) c= Cl (Int (Cl K)) by PRE_TOPC:48;
K c= Int (Cl K) by A7, Def3;
then K c= Cl (Int (Cl K)) by A8, XBOOLE_1:1;
then A9: K is pre-semi-open by Def4;
then K = psInt K by Th5;
then pInt K = psInt K by A7, Th4;
then A10: K in { B where B is Subset of T : pInt B = psInt B } ;
K in PSO T by A9;
hence x in (PSO T) /\ (D(p,ps) T) by A6, A10, XBOOLE_0:def 4; :: thesis: verum