reserve a,b,c for set;

theorem
  for T being TopSpace, F being Subset-Family of T for I being non empty
  Subset-Family of F st for G being set st G in I holds F\G is finite holds Cl
union F = union clf F \/ meet {Cl union G where G is Subset-Family of T: G in I
  }
proof
  let T be TopSpace;
  let F be Subset-Family of T;
  let I be non empty Subset-Family of F;
  set G0 = the Element of I;
  reconsider G0 as Subset-Family of T by XBOOLE_1:1;
  set Z = {Cl union G where G is Subset-Family of T: G in I};
A1: Cl union G0 in Z;
  then reconsider Z9 = Z as non empty set;
  assume
A2: for G being set st G in I holds F\G is finite;
  thus Cl union F c= union clf F \/ meet Z
  proof
    let a be object;
    assume that
A3: a in Cl union F and
A4: not a in union clf F \/ meet Z;
    reconsider a as Point of T by A3;
    not a in meet Z9 by A4,XBOOLE_0:def 3;
    then consider b such that
A5: b in Z and
A6: not a in b by SETFAM_1:def 1;
    consider G being Subset-Family of T such that
A7: b = Cl union G and
A8: G in I by A5;
A9: T is non empty by A3;
    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 A8,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 A3,A6,A7,XBOOLE_0:def 3;
    then a in union clf (F\G) by A2,A8,A9,PCOMPS_1:16;
    hence contradiction by A4,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
A14: a in meet Z9;
    union G0 c= union F by ZFMISC_1:77;
    then
A15: Cl union G0 c= Cl union F by PRE_TOPC:19;
    a in Cl union G0 by A1,A14,SETFAM_1:def 1;
    hence thesis by A15;
  end;
end;
