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
  a <> b implies ex FB st FB is being_ultrafilter & (a in FB & not b in
  FB or not a in FB & b in FB)
proof
  assume a <> b;
  then not a [= b or not b [= a by LATTICES:8;
  then a"/\"b` <> Bottom B or b"/\"a` <> Bottom B by Th43;
  then (ex FB st a"/\"b` in FB & FB is being_ultrafilter) or ex FB st b"/\"a`
  in FB & FB is being_ultrafilter by Th20;
  then consider FB such that
A1: FB is being_ultrafilter and
A2: a"/\"b` in FB or b"/\"a` in FB;
  take FB;
  thus FB is being_ultrafilter by A1;
  a in FB & b` in FB or b in FB & a` in FB by A2,Th8;
  hence thesis by A1,Th46;
end;
