reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);
reserve x,y,X for set;
reserve BL for non trivial B_Lattice,
  a,b,c,p,q for Element of BL,
  UF,F,F0,F1,F2 for Filter of BL;

theorem Th28:
  for L being non trivial B_Lattice, D being non empty Subset of L
  st Bottom L in <.D.) ex B being non empty Element of Fin the carrier of L
  st B c= D & FinMeet(B) = Bottom L
proof
  let L be non trivial B_Lattice, D be non empty Subset of L such that
A1: Bottom L in <.D.);
  set A = { FinMeet(C) where C is Element of Fin the carrier of L:
  C c= D & C <> {} };
  set AA = { a where a is Element of L:
  ex c being Element of L st c in A & c [= a };
A2: AA c= the carrier of L
  proof
    let x be object;
    assume x in AA;
    then ex a being Element of L st ( a= x)&( ex c being Element of L
    st c in A & c [= a);
    hence thesis;
  end;
  AA is non empty
  proof
    consider C being Element of Fin D such that
A3: C is non empty by Th27;
A4: C is Subset of D by FINSUB_1:16;
    then C c= the carrier of L by XBOOLE_1:1;
    then C is Element of Fin the carrier of L by FINSUB_1:def 5;
    then consider C being Element of Fin the carrier of L such that
A5: C <> {} and
A6: C c= D by A3,A4;
    reconsider a = FinMeet(C) as Element of L;
    a in A by A5,A6;
    then a in AA;
    hence thesis;
  end;
  then reconsider AA as non empty Subset of L by A2;
A7: for p,q being Element of L st p in AA & q in AA holds p "/\" q in AA
  proof
    let p,q be Element of L;
    assume that
A8: p in AA and
A9: q in AA;
    consider a being Element of L such that
A10: a= p and
A11: ex c being Element of L st c in A & c [= a by A8;
    consider c being Element of L such that
A12: c in A and
A13: c [= a by A11;
    consider b being Element of L such that
A14: b= q and
A15: ex d being Element of L st d in A & d [= b by A9;
    consider d being Element of L such that
A16: d in A and
A17: d [= b by A15;
A18: c "/\" d [= a "/\" b by A13,A17,FILTER_0:5;
    consider C being Element of Fin the carrier of L such that
A19: c = FinMeet(C) and
A20: C c= D and
A21: C <> {} by A12;
    consider E being Element of Fin the carrier of L such that
A22: d = FinMeet(E) and
A23: E c= D and E <> {} by A16;
A24: c "/\" d = FinMeet(C \/ E) by A19,A22,LATTICE4:23;
    C \/ E c= D by A20,A23,XBOOLE_1:8;
    then c "/\" d in A by A21,A24;
    hence thesis by A10,A14,A18;
  end;
  for p,q being Element of L st p in AA & p [= q holds q in AA
  proof
    let p,q be Element of L;
    assume that
A25: p in AA and
A26: p [= q;
A27: ex a being Element of L st ( a= p)&( ex c being Element of L
    st c in A & c [= a) by A25;

    ex b being Element of L st ( b= q)&( ex c being Element of L
    st c in A & c [= b) by A26,A27,LATTICES:7;
    hence thesis;
  end;
  then
A28: AA is Filter of L by A7,FILTER_0:9;
  D c= AA
  proof
    let x be object;
    assume
A29: x in D;
    then
A30: { x } c= D by ZFMISC_1:31;
    { x } c= the carrier of L by A29,ZFMISC_1:31;
then reconsider C = { x } as Element of Fin the carrier of L by FINSUB_1:def 5;
A31: x = (id the carrier of L).x by A29,FUNCT_1:18
      .= (the L_meet of L)$$(C,id the carrier of L) by A29,SETWISEO:17
      .= FinMeet(C,id L) by LATTICE2:def 4
      .= FinMeet(C) by LATTICE4:def 9;
    reconsider a = FinMeet(C) as Element of L;
    a in A by A30;
    hence thesis by A31;
  end;
  then <.D.) c= AA by A28,FILTER_0:def 4;
  then Bottom L in AA by A1;
  then ex d being Element of L st d = Bottom L &
  ex c being Element of L st c in A & c [= d;
  then consider c being Element of L such that
A32: c in A and
A33: c [= Bottom L;
  Bottom L [= c;
  then Bottom L in A by A32,A33,LATTICES:8;
  then ex C being Element of Fin the carrier of L
  st Bottom L = FinMeet(C) & C c= D & C <> {};
  hence thesis;
end;
