reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;

theorem
  union(Cl Int F) c= union(Cl Int Cl F)
proof
    let x be object;
    assume x in union(Cl Int 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 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;
      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 thesis by A1,A7,PCOMPS_1:def 2;
    end;
    hence x in union(Cl Int Cl F) by TARSKI:def 4;
end;
