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

theorem
  union(Int Cl F) c= union(Cl Int Cl F)
proof
  now
    let x be object;
    assume x in union(Int Cl F);
    then consider A being set such that
A1: x in A and
A2: A in Int Cl 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 F by A2,Def1;
    consider D being Subset of T such that
A5: B = Cl D and
A6: D in F by A4,PCOMPS_1:def 2;
    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;
  hence thesis;
end;
