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).:;
reserve P for non empty ClosedSubset of L,
  o1,o2 for BinOp of P;

theorem
  for L1,L2 being Lattice for F1 being Filter of L1, F2 being Filter of
  L2 st the LattStr of L1 = the LattStr of L2 & F1 = F2 holds latt F1 = latt F2
proof
  let L1,L2 be Lattice, F1 be Filter of L1, F2 be Filter of L2;
  ex o1,o2 being BinOp of F1 st o1 = (the L_join of L1)||F1 & o2 = (the
  L_meet of L1)||F1 & latt F1 = LattStr (#F1, o1, o2#) by FILTER_0:def 9;
  hence thesis by FILTER_0:def 9;
end;
