let T be non empty TopSpace; :: thesis: for F being Subset-Family of T holds Int (Cl (Int F)) = { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) }
let F be Subset-Family of T; :: thesis: Int (Cl (Int F)) = { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) }
set P = { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } ;

now :: thesis: for C being object st C in { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } holds
C in bool the carrier of T

let C be object ; :: thesis: ( C in { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } implies C in bool the carrier of T )
assume C in { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } ; :: thesis: C in bool the carrier of T
then ex A being Subset of T st
( C = A & ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) ) ;
hence C in bool the carrier of T ; :: thesis: verum

end;

then reconsider P = { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } as Subset-Family of T by TARSKI:def 3;
reconsider P = P as Subset-Family of T ;
for X being object holds
( X in Int (Cl (Int F)) iff X in P )

proof

let X be object ; :: thesis: ( X in Int (Cl (Int F)) iff X in P )

A1: now :: thesis: ( X in P implies X in Int (Cl (Int F)) )

assume A2: X in P ; :: thesis: X in Int (Cl (Int F))
then reconsider C = X as Subset of T ;
ex D being Subset of T st
( D = C & ex B being Subset of T st
( D = Int (Cl (Int B)) & B in F ) ) by A2;
then consider B being Subset of T such that
A3: C = Int (Cl (Int B)) and
A4: B in F ;
Int B in Int F by A4, Def1;
then Cl (Int B) in Cl (Int F) by PCOMPS_1:def 2;
hence X in Int (Cl (Int F)) by A3, Def1; :: thesis: verum

end;

now :: thesis: ( X in Int (Cl (Int F)) implies X in P )

assume A5: X in Int (Cl (Int F)) ; :: thesis: X in P
then reconsider C = X as Subset of T ;
consider B being Subset of T such that
A6: C = Int B and
A7: B in Cl (Int F) by A5, Def1;
consider D being Subset of T such that
A8: B = Cl D and
A9: D in Int F by A7, PCOMPS_1:def 2;
ex E being Subset of T st
( D = Int E & E in F ) by A9, Def1;
hence X in P by A6, A8; :: thesis: verum

end;

hence ( X in Int (Cl (Int F)) iff X in P ) by A1; :: thesis: verum

end;

hence Int (Cl (Int F)) = { A where A is Subset of T : ex B being Subset of T st
( A = Int (Cl (Int B)) & B in F ) } by TARSKI:2; :: thesis: verum