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

theorem
  for L1,L2 being B_Lattice st the LattStr of L1 = the LattStr of L2 for
  a being Element of L1, b being Element of L2 st a = b holds a` = b`
proof
  let L1,L2 be B_Lattice such that
A1: the LattStr of L1 = the LattStr of L2;
  let a be Element of L1, b be Element of L2 such that
A2: a = b;
  reconsider b9 = a` as Element of L2 by A1;
  a` is_a_complement_of a by LATTICES:def 21;
  then b9 is_a_complement_of b by A1,A2,Th11;
  hence thesis by LATTICES:def 21;
end;
