reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;
reserve B for B_Lattice,
  IB,JB for Ideal of B,
  a,b for Element of B;
reserve a9 for Element of (B qua Lattice).:;

theorem
  IB <> (.B.> & IB is prime iff IB is max-ideal
proof
  reconsider F = IB.: as Filter of B.:;
  <.B.:.) = (.B.>;
  then F <> (.B.> & F is prime iff F is being_ultrafilter by FILTER_0:45;
  hence thesis by Th32,Th43;
end;
