reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;

theorem Th18:
  L is lower-bounded implies for F st F <> the carrier of L ex H
  st F c= H & H is being_ultrafilter
proof
  given r such that
A1: for p holds r "/\" p = r & p "/\" r = r;
A2: r in H implies H = the carrier of L
  proof
    assume
A3: r in H;
    thus H c= the carrier of L;
    let x be object;
    assume x in the carrier of L;
    then reconsider p = x as Element of L;
    r "/\" p = r by A1;
    then r [= p by LATTICES:4;
    hence thesis by A3,Th9;
  end;
  let F such that
A4: F <> the carrier of L;
  set X = { A where A is Subset of L : F c= A & A is Filter of L & A <> the
  carrier of L };
A5: x in X implies x is Subset of L & x is Filter of L
  proof
    assume x in X;
    then
    ex A being Subset of L st x = A & F c= A & A is Filter of L & A <> the
    carrier of L;
    hence thesis;
  end;
A6: X1 in X implies F,X1 are_c=-comparable & X1 <> the carrier of L
  proof
    assume X1 in X;
    then ex A being Subset of L st X1 = A & F c= A & A is Filter of L & A <>
    the carrier of L;
    hence thesis;
  end;
A7: for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z
  holds X1 c= Y
  proof
    set x = the Element of F;
    let Z such that
A8: Z c= X and
A9: Z is c=-linear;
    take Y = union (Z \/ {F});
    F in {F} by TARSKI:def 1;
    then
A10: F in Z \/ {F} by XBOOLE_0:def 3;
    x in F;
    then reconsider V = Y as non empty set by A10,TARSKI:def 4;
    V c= the carrier of L
    proof
      let x be object;
      assume x in V;
      then consider X1 such that
A11:  x in X1 and
A12:  X1 in Z \/ {F} by TARSKI:def 4;
      X1 in Z or X1 in {F} by A12,XBOOLE_0:def 3;
      then X1 is Subset of L by A5,A8;
      hence thesis by A11;
    end;
    then reconsider V as non empty Subset of L;
    now
      let p,q;
      thus p in V & q in V implies p "/\" q in V
      proof
        assume p in V;
        then consider X1 such that
A13:    p in X1 and
A14:    X1 in Z \/ {F} by TARSKI:def 4;
A15:    X1 in Z or X1 in {F} by A14,XBOOLE_0:def 3;
        then
A16:    X1 in X & X1 in Z or X1 = F by A8,TARSKI:def 1;
        assume q in V;
        then consider X2 such that
A17:    q in X2 and
A18:    X2 in Z \/ {F} by TARSKI:def 4;
A19:    X2 in Z or X2 in {F} by A18,XBOOLE_0:def 3;
        then X2 in X & X2 in Z or X2 = F by A8,TARSKI:def 1;
        then X1,X2 are_c=-comparable by A6,A9,A16,ORDINAL1:def 8;
        then
A20:    X1 c= X2 or X2 c= X1;
A21:    X1 is Filter of L by A5,A8,A15,TARSKI:def 1;
        X2 is Filter of L by A5,A8,A19,TARSKI:def 1;
        then p "/\" q in X1 or p "/\" q in X2 by A13,A17,A20,A21,Th9;
        hence thesis by A14,A18,TARSKI:def 4;
      end;
      assume p "/\" q in V;
      then consider X1 such that
A22:  p "/\" q in X1 and
A23:  X1 in Z \/ {F} by TARSKI:def 4;
      X1 in Z or X1 in {F} by A23,XBOOLE_0:def 3;
      then X1 is Filter of L by A5,A8,TARSKI:def 1;
      then p in X1 & q in X1 by A22,Th8;
      hence p in V & q in V by A23,TARSKI:def 4;
    end;
    then reconsider V as Filter of L by Th8;
    now
      assume r in V;
      then consider X1 such that
A24:  r in X1 and
A25:  X1 in Z \/ {F} by TARSKI:def 4;
      X1 in Z or X1 in {F} by A25,XBOOLE_0:def 3;
      then X1 in X or X1 = F by A8,TARSKI:def 1;
      then
      ex A being Subset of L st X1 = A & F c= A & A is Filter of L & A <>
      the carrier of L by A4;
      hence contradiction by A2,A24;
    end;
    then
A26: V <> the carrier of L;
    F c= V
    by A10,TARSKI:def 4;
    hence Y in X by A26;
    let X1;
    assume X1 in Z;
    then X1 in Z \/ {F} by XBOOLE_0:def 3;
    hence thesis by ZFMISC_1:74;
  end;
  F in X by A4;
  then consider Y such that
A27: Y in X and
A28: for Z st Z in X & Z <> Y holds not Y c= Z by A7,ORDERS_1:65;
  consider H being Subset of L such that
A29: Y = H and
A30: F c= H and
A31: H is Filter of L and
A32: H <> the carrier of L by A27;
  reconsider H as Filter of L by A31;
  take H;
  thus F c= H & H <> the carrier of L by A30,A32;
  let G be Filter of L;
  assume that
A33: H c= G and
A34: G <> the carrier of L;
  F c= G by A30,A33;
  then G in X by A34;
  hence thesis by A28,A29,A33;
end;
