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

theorem
  union(Int Cl Int F) c= union(Cl Int F)
proof
    let x be object;
    assume x in union(Int Cl Int F);
    then consider A being set such that
A1: x in A and
A2: A in Int 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 = Int B and
A4: B in Cl Int F by A2,Def1;
    consider D being Subset of T such that
A5: B = Cl D and
A6: D in Int F by A4,PCOMPS_1:def 2;
    consider E being Subset of T such that
A7: D = Int E and
A8: E in F by A6,Def1;
    ex P being set st x in P & P in Cl Int F
    proof
      take Cl Int E;
A9:   Int E in Int F by A8,Def1;
      A c= Cl Int E by A3,A5,A7,Th1;
      hence thesis by A1,A9,PCOMPS_1:def 2;
    end;
    hence x in union(Cl Int F) by TARSKI:def 4;
end;
