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

theorem
  union(Int F) c= union(Int Cl Int F)
proof
    let x be object;
    assume x in union(Int F);
    then consider A being set such that
A1: x in A and
A2: A in 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 F by A2,Def1;
    ex P being set st x in P & P in Int Cl Int F
    proof
      take Int Cl Int B;
      Int B in Int F by A4,Def1;
      then
A5:   Cl Int B in Cl Int F by PCOMPS_1:def 2;
      A c= Int Cl Int B by A3,TDLAT_1:4;
      hence thesis by A1,A5,Def1;
    end;
    hence x in union(Int Cl Int F) by TARSKI:def 4;
end;
