let T be TopSpace; :: thesis: (T ^alpha) /\ (D(c,alpha) T) = the topology of T
thus (T ^alpha) /\ (D(c,alpha) T) c= the topology of T :: according to XBOOLE_0:def 10 :: thesis: the topology of T c= (T ^alpha) /\ (D(c,alpha) T)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (T ^alpha) /\ (D(c,alpha) T) or x in the topology of T )
assume A1: x in (T ^alpha) /\ (D(c,alpha) T) ; :: thesis: x in the topology of T
then x in T ^alpha by XBOOLE_0:def 4;
then consider A being Subset of T such that
A2: x = A and
A3: A is alpha-set of T ;
x in D(c,alpha) T by A1, XBOOLE_0:def 4;
then consider Z being Subset of T such that
A4: x = Z and
A5: Int Z = alphaInt Z ;
A = alphaInt A by A3, Th2;
then Z is open by A2, A4, A5;
hence x in the topology of T by A4, PRE_TOPC:def 2; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the topology of T or x in (T ^alpha) /\ (D(c,alpha) T) )
assume A6: x in the topology of T ; :: thesis: x in (T ^alpha) /\ (D(c,alpha) T)
then reconsider K = x as Subset of T ;
K is open by A6, PRE_TOPC:def 2;
then A7: K = Int K by TOPS_1:23;
then K c= Int (Cl (Int K)) by PRE_TOPC:18, TOPS_1:19;
then A8: K is alpha-set of T by Def1;
then Int K = alphaInt K by A7, Th2;
then A9: K in { B where B is Subset of T : Int B = alphaInt B } ;
K in T ^alpha by A8;
hence x in (T ^alpha) /\ (D(c,alpha) T) by A9, XBOOLE_0:def 4; :: thesis: verum