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;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;

theorem Th44:
  FB is being_ultrafilter iff FB <> the carrier of B & for a holds
  a in FB or a` in FB
proof
  thus FB is being_ultrafilter implies FB <> the carrier of B & for a holds a
  in FB or a` in FB
  proof
    reconsider I = B as I_Lattice;
    assume that
A1: FB <> the carrier of B and
A2: for HB st FB c= HB & HB <> the carrier of B holds FB = HB;
    thus FB <> the carrier of B by A1;
    let a such that
A3: not a in FB;
A4: a in <.a.);
A5: FB \/ <.a.) c= <.FB \/ <.a.).) by Def4;
    <.a.) c= FB \/ <.a.) by XBOOLE_1:7;
    then <.a.) c= <.FB \/ <.a.).) by A5;
    then FB c= FB \/ <.a.) & FB <> <.FB \/ <.a.).) by A3,A4,XBOOLE_1:7;
    then <.FB \/ <.a.).) = the carrier of B by A2,A5,XBOOLE_1:1;
    then
A6: a` in <.FB \/ <.a.).);
    reconsider a9 = a as Element of I;
    reconsider FI = FB as Filter of I;
A7: a => a` = a`"\/"a` by Th42;
    <.{a}.) = <.a.) by Th24;
    then
A8: a9` in <.FI \/ {a9}.) by A6,Th34;
    FB = <.FB.) by Th21;
    then a => a` in FB by A8,Th40;
    hence thesis by A7;
  end;
  assume that
A9: FB <> the carrier of B and
A10: for a holds a in FB or a` in FB;
  thus FB <> the carrier of B by A9;
  let HB;
  assume that
A11: FB c= HB and
A12: HB <> the carrier of B and
A13: FB <> HB;
  ex x being object st not (x in FB iff x in HB) by A13,TARSKI:2;
  then consider x such that
A14: x in HB and
A15: not x in FB by A11;
  reconsider x as Element of HB by A14;
  x` in FB by A10,A15;
  then x`"/\"x in HB by A11,Th8;
  then
A16: Bottom B in HB by LATTICES:20;
  HB = <.HB.) by Th21;
  hence contradiction by A12,A16,Th25;
end;
