let T be non empty TopSpace; :: thesis: for F being Subset-Family of T holds union (Cl (Int F)) c= union (Cl (Int (Cl F)))
let F be Subset-Family of T; :: thesis: union (Cl (Int F)) c= union (Cl (Int (Cl F)))
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in union (Cl (Int F)) or x in union (Cl (Int (Cl F))) )
assume x in union (Cl (Int F)) ; :: thesis: x in union (Cl (Int (Cl F)))
then consider A being set such that
A1: x in A and
A2: A in Cl (Int F) by TARSKI:def 4;
reconsider A = A as Subset of T by A2;
consider B being Subset of T such that
A3: A = Cl B and
A4: B in Int F by A2, PCOMPS_1:def 2;
consider D being Subset of T such that
A5: B = Int D and
A6: D in F by A4, Def1;
ex P being set st
( x in P & P in Cl (Int (Cl F)) )

proof

take Cl (Int (Cl D)) ; :: thesis: ( x in Cl (Int (Cl D)) & Cl (Int (Cl D)) in Cl (Int (Cl F)) )
Cl D in Cl F by A6, PCOMPS_1:def 2;
then A7: Int (Cl D) in Int (Cl F) by Def1;
A c= Cl (Int (Cl D)) by A3, A5, Th2;
hence ( x in Cl (Int (Cl D)) & Cl (Int (Cl D)) in Cl (Int (Cl F)) ) by A1, A7, PCOMPS_1:def 2; :: thesis: verum

end;

hence x in union (Cl (Int (Cl F))) by TARSKI:def 4; :: thesis: verum