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

proof

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

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in SO T or x in (PSO T) /\ (D(alpha,p) T) )
assume x in SO T ; :: thesis: x in (PSO T) /\ (D(alpha,p) 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 where B is Subset of T : alphaInt B = pInt B } ;
K in PSO T by A4;
hence x in (PSO T) /\ (D(alpha,p) T) by A1, A5, XBOOLE_0:def 4; :: thesis: verum