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 & X misses X` & Y misses Y` implies X = Y` & Y = X`
proof
  assume that
A1: X` "\/" Y` = X "\/" Y and
A2: X misses X` and
A3: Y misses Y`;
A4: X "/\" X`= Bottom L by A2;
A5: Y "/\" Y` = Bottom L by A3;
  then
A6: Y` "/\" (X` "\/" Y`) = (Y` "/\" X) "\/" Bottom L by A1,LATTICES:def 11;
  (Y "/\" X`) "\/" (Y "/\" Y`) = Y "/\" (X "\/" Y) by A1,LATTICES:def 11;
  then Y "/\" X` = Y "/\" (X "\/" Y) by A5;
  then
A7: Y "/\" X` = Y by LATTICES:def 9;
  (X "/\" X`) "\/" (X "/\" Y`) = X "/\" (X "\/" Y) by A1,LATTICES:def 11;
  then X "/\" Y` = X "/\" (X "\/" Y) by A4
    .= X by LATTICES:def 9;
  hence X = Y` by A6,LATTICES:def 9;
  X` "/\" (X` "\/" Y`) = Bottom L "\/" (X` "/\" Y) by A1,A4,LATTICES:def 11;
  hence thesis by A7,LATTICES:def 9;
end;
