reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;

theorem Th16:
  for L1,L2 being Lattice st the LattStr of L1 = the LattStr of L2
  for x st x is Ideal of L1 holds x is Ideal of L2
proof
  let L1,L2 be Lattice such that
A1: the LattStr of L1 = the LattStr of L2;
  let x;
  assume x is Ideal of L1;
  then reconsider F = x as Ideal of L1;
  now
    let a,b be Element of L2;
    reconsider a9 = a, b9 = b as Element of L1 by A1;
    a"\/"b = a9"\/"b9 by A1;
    hence a in F & b in F iff a"\/"b in F by Lm1;
  end;
  hence thesis by A1,Lm1;
end;
