reserve a,b,c for set;

theorem
  for T being TopSpace, F being Subset-Family of T holds Cl union F =
union clf F \/ meet {Cl union G where G is Subset-Family of T: G c= F & F\G is
  finite}
proof
  let T be TopSpace;
  let F be Subset-Family of T;
  set Z = {Cl union G where G is Subset-Family of T: G c= F & F\G is finite};
  F\F = {} by XBOOLE_1:37;
  then
A1: Cl union F in Z;
  then reconsider Z9 = Z as non empty set;
  thus Cl union F c= union clf F \/ meet Z
  proof
    let a be object;
    assume that
A2: a in Cl union F and
A3: not a in union clf F \/ meet Z;
    reconsider a as Point of T by A2;
    not a in meet Z9 by A3,XBOOLE_0:def 3;
    then consider b such that
A4: b in Z and
A5: not a in b by SETFAM_1:def 1;
    consider G being Subset-Family of T such that
A6: b = Cl union G and
A7: G c= F and
A8: F\G is finite by A4;
A9: T is non empty by A2;
    then clf (F\G) c= clf F by PCOMPS_1:14,XBOOLE_1:36;
    then
A10: union clf (F\G) c= union clf F by ZFMISC_1:77;
    F = G \/ (F\G) by A7,XBOOLE_1:45;
    then union F = union G \/ union (F\G) by ZFMISC_1:78;
    then Cl union F = Cl union G \/ Cl union (F\G) by PRE_TOPC:20;
    then a in Cl union (F\G) by A2,A5,A6,XBOOLE_0:def 3;
    then a in union clf (F\G) by A8,A9,PCOMPS_1:16;
    hence contradiction by A3,A10,XBOOLE_0:def 3;
  end;
  let a be object;
  assume
A11: a in union clf F \/ meet Z;
  per cases by A11,XBOOLE_0:def 3;
  suppose
    a in union clf F;
    then consider b such that
A12: a in b and
A13: b in clf F by TARSKI:def 4;
    ex W being Subset of T st b = Cl W & W in F by A13,PCOMPS_1:def 2;
    then b c= Cl union F by PRE_TOPC:19,ZFMISC_1:74;
    hence thesis by A12;
  end;
  suppose
    a in meet Z9;
    hence thesis by A1,SETFAM_1:def 1;
  end;
end;
