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 object ; :: according to TARSKI:def 3 :: thesis: ( not x in (PSO T) /\ (D(p,ps) T) or x in PO T )
assume x in (PSO T) /\ (D(p,ps) T) ; :: thesis: x in PO 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 x in PO T ; :: thesis: verum

end;

let x be object ; :: 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
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 where B is Subset of T : pInt B = psInt B } ;
K in PSO T by A4;
hence x in (PSO T) /\ (D(p,ps) T) by A1, A5, XBOOLE_0:def 4; :: thesis: verum