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

proof

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

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

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

end;

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

assume A5: X in 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 = Cl B and
A7: B in Int F by A5, PCOMPS_1:def 2;
ex D being Subset of T st
( B = Int D & D in F ) by A7, Def1;
hence X in P by A6; :: thesis: verum

end;

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

end;

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