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 Th46:
  FB is being_ultrafilter implies for a holds a in FB iff not a` in FB
proof
  assume
A1: FB is being_ultrafilter;
  let a;
  thus a in FB implies not a` in FB
  proof
    assume a in FB & a` in FB;
    then a"/\"a` in FB by Th8;
    then Bottom B in FB by LATTICES:20;
    then FB = the carrier of B by Th26;
    hence contradiction by A1;
  end;
  thus thesis by A1,Th44;
end;
