reserve L for Lattice;
reserve X,Y,Z,V for Element of L;
reserve L for D_Lattice;
reserve X,Y,Z for Element of L;
reserve L for 0_Lattice;
reserve X,Y,Z for Element of L;
reserve L for B_Lattice;
reserve X,Y,Z,V for Element of L;

theorem
  X \ Y = Y \ X implies X = Y
proof
  assume
A1: X \ Y = Y \ X;
  then (X "/\" Y`) "/\" X = Y "/\" (X` "/\" X) by LATTICES:def 7
    .= Y "/\" Bottom L by LATTICES:20
    .= Bottom L;
  then (X "/\" X) "/\" Y` = Bottom L by LATTICES:def 7;
  then (X "/\" Y`) "\/" X`= (X "/\" X`) "\/" X` by LATTICES:20
    .= X` by LATTICES:def 8;
  then (X "\/" X`) "/\" (Y` "\/" X`) = X` by LATTICES:11;
  then Top L "/\" (Y` "\/" X`) = X` by LATTICES:21;
  then Y` "/\" X` = Y` by LATTICES:def 9;
  then X`` [= Y`` by LATTICES:4,26;
  then X [= Y``;
  then
A2: X [= Y;
  X "/\" (Y` "/\" Y) = (Y "/\" X`) "/\" Y by A1,LATTICES:def 7;
  then X "/\" Bottom L = (Y "/\" X`) "/\" Y by LATTICES:20;
  then Bottom L
    = X` "/\" (Y "/\" Y) by LATTICES:def 7
    .= X` "/\" Y;
  then Y [= X`` by LATTICES:25;
  then Y [= X;
  hence thesis by A2;
end;
