reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);
reserve L for D_Lattice;
reserve F for Filter of L;
reserve a,b for Element of L;
reserve x,X,X1,X2,Y,Z for set;

theorem Th18:
  for Z st Z <> {} & Z c= SF_have b \ SF_have a & Z is c=-linear
  ex Y st Y in SF_have b \ SF_have a & for X1 st X1 in Z holds X1 c= Y
proof
  let Z;
  assume that
A1: Z <> {} and
A2: Z c= SF_have b \ SF_have a and
A3: Z is c=-linear;
  reconsider Z as Subset-Family of L by A2,XBOOLE_1:1;
  take Y = union Z;
  Y in SF_have b \ SF_have a
  proof
    set X = the Element of Z;
A4: not a in Y
    proof
      assume a in Y;
      then ex X st a in X & X in Z by TARSKI:def 4;
      hence contradiction by A2,Th17;
    end;
    X in SF_have b \ SF_have a by A1,A2;
    then
A5: b in X by Th17;
    then
A6: b in Y by A1,TARSKI:def 4;
    reconsider Y as non empty Subset of L by A1,A5,TARSKI:def 4;
    now
      let p,q be Element of L;
      thus p in Y & q in Y implies p "/\" q in Y
      proof
        assume p in Y;
        then consider X1 such that
A7:     p in X1 and
A8:     X1 in Z by TARSKI:def 4;
A9:     X1 is Filter of L by A2,A8,Th17;
        assume q in Y;
        then consider X2 such that
A10:    q in X2 and
A11:    X2 in Z by TARSKI:def 4;
        X1,X2 are_c=-comparable by A3,A8,A11,ORDINAL1:def 8;
        then
A12:    X1 c= X2 or X2 c= X1;
        X2 is Filter of L by A2,A11,Th17;
        then p "/\" q in X1 or p "/\" q in X2 by A7,A10,A9,A12,FILTER_0:8;
        hence thesis by A8,A11,TARSKI:def 4;
      end;
      assume p "/\" q in Y;
      then consider X1 such that
A13:  p "/\" q in X1 and
A14:  X1 in Z by TARSKI:def 4;
A15:  X1 is Filter of L by A2,A14,Th17;
      then
A16:  q in X1 by A13,FILTER_0:8;
      p in X1 by A13,A15,FILTER_0:8;
      hence p in Y & q in Y by A14,A16,TARSKI:def 4;
    end;
    then Y is Filter of L by FILTER_0:8;
    hence thesis by A4,A6,Lm1;
  end;
  hence thesis by ZFMISC_1:74;
end;
