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

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (SPO T) /\ (D(p,sp) T) or x in PO T )
assume x in (SPO T) /\ (D(p,sp) T) ; :: thesis: x in PO T
then A0: ( x in SPO T & x in D(p,sp) T ) by XBOOLE_0:def 4;
then consider B being Subset of T such that
A1: ( x = B & B is semi-pre-open ) ;
A3: B = spInt B by A1, Th6;
consider B1 being Subset of T such that
A2: ( x = B1 & pInt B1 = spInt B1 ) by A0;
pInt B = B by A2, A3, A1;
then B is pre-open by Th4;
hence x in PO T by A1; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in PO T or x in (SPO T) /\ (D(p,sp) T) )
assume x in PO T ; :: thesis: x in (SPO T) /\ (D(p,sp) T)
then consider K being Subset of T such that
A1: x = K and
A2: K is pre-open ;
A3: Int (Cl K) c= (Cl (Int K)) \/ (Int (Cl K)) by XBOOLE_1:7;
K c= Int (Cl K) by A2;
then K c= (Cl (Int K)) \/ (Int (Cl K)) by A3;
then A4: K is semi-pre-open ;
then K = spInt K by Th6;
then pInt K = spInt K by A2, Th4;
then A5: K in { B where B is Subset of T : pInt B = spInt B } ;
K in SPO T by A4;
hence x in (SPO T) /\ (D(p,sp) T) by A1, A5, XBOOLE_0:def 4; :: thesis: verum