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` = X "\/" Y & Y misses X` & X misses Y` implies X = X` & Y = Y`
proof
  assume that
A1: X` "\/" Y` = X "\/" Y and
A2: Y misses X` and
A3: X misses Y`;
A4: Y "/\" X` = Bottom L by A2;
  then (X "\/" Y) "/\" (X "\/" X`) = X "\/" Bottom L by LATTICES:11;
  then (X "\/" Y) "/\" Top L = X by LATTICES:21;
  then (Y "/\" X)` [= Y` by LATTICES:def 9;
  then
A5: X "\/" Y [= Y` by A1,LATTICES:23;
A6: X "/\" Y` = Bottom L by A3;
  then (Y "\/" X) "/\" (Y "\/" Y`) = Y "\/" Bottom L by LATTICES:11;
  then (Y "\/" X) "/\" Top L = Y by LATTICES:21;
  then (X "/\" Y)` [= X` by LATTICES:def 9;
  then
A7: X "\/" Y [= X` by A1,LATTICES:23;
  (Y` "\/" Y) "/\" (Y` "\/" X`) = Y` "\/" Bottom L by A4,LATTICES:11;
  then Top L "/\" (Y` "\/" X`) = Y` by LATTICES:21;
  then (X` "/\" Y`)` [= X`` by LATTICES:def 9;
  then (X` "/\" Y`)` [= X;
  then X`` "\/" Y`` [= X by LATTICES:23;
  then X "\/" Y`` [= X;
  then
A8: X` "\/" Y` [= X by A1;
  X` [= X` "\/" Y` by LATTICES:5;
  then
A9: X` [= X by A8,LATTICES:7;
  (X` "\/" X) "/\" (X` "\/" Y`) = X` "\/" Bottom L by A6,LATTICES:11;
  then Top L "/\" (X` "\/" Y`)= X` by LATTICES:21;
  then (Y` "/\" X`)` [= Y`` by LATTICES:def 9;
  then Y`` "\/" X`` [= Y`` by LATTICES:23;
  then Y`` "\/" X`` [= Y;
  then Y`` "\/" X [= Y;
  then
A10: X` "\/" Y` [= Y by A1;
  Y` [= X` "\/" Y` by LATTICES:5;
  then
A11: Y` [= Y by A10,LATTICES:7;
  X [= X "\/" Y by LATTICES:5;
  then
A12: X [= X` by A7,LATTICES:7;
  Y [= X "\/" Y by LATTICES:5;
  then Y [= Y` by A5,LATTICES:7;
  hence thesis by A11,A9,A12;
end;
