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
  a <> b implies ex IB st IB is max-ideal & (a in IB & not b in IB or
  not a in IB & b in IB)
proof
  assume a <> b;
  then consider FB being Filter of B.: such that
A1: FB is being_ultrafilter and
A2: a.: in FB & not b.: in FB or not a.: in FB & b.: in FB by FILTER_0:47;
  take IB = .:(FB qua Filter of (B qua Lattice).:);
  IB.: = FB;
  hence IB is max-ideal by A1,Th32;
  thus thesis by A2;
end;
