 reserve L for AD_Lattice;
 reserve x,y,z for Element of L;
 reserve L for GAD_Lattice;
 reserve x,y,z for Element of L;
reserve L for with_zero GAD_Lattice,
        x,y for Element of L;

theorem Lem316:  :: Lemma 3.16.
  x "/\" y = bottom L iff y "/\" x = bottom L
  proof
    set b1 = bottom L;
    thus x "/\" y = bottom L implies y "/\" x = bottom L
    proof
      assume
Z1:   x "/\" y = b1;
      y "/\" x = y "/\" (x "/\" x) by IMeet
              .= (y "/\" x) "/\" x by LATTICES:def 7
              .= (x "/\" y) "/\" x by Lem310
              .= b1 by GADL0,Z1;
      hence y "/\" x = bottom L;
    end;
    assume
Z1: y "/\" x = bottom L;
    x "/\" y = x "/\" (y "/\" y) by IMeet
            .= (x "/\" y) "/\" y by LATTICES:def 7
            .= (y "/\" x) "/\" y by Lem310
            .= b1 by GADL0,Z1;
    hence thesis;
  end;
