 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   :: Lemma 3.15. (2)
  x "\/" bottom L = x
  proof
    bottom L "/\" x = bottom L by GADL0; then
a1: bottom L [= x by LATTICES:def 8;
    set b1 = bottom L;
    x "/\" b1 = x "/\" (b1 "/\" x) by GADL0
             .= (x "/\" b1) "/\" x by LATTICES:def 7
             .= (b1 "/\" x) "/\" x by Lem310
             .= b1 "/\" x by GADL0; then
    (b1 "/\" x) "\/" b1 = b1 by LATTICES:def 8;
    hence thesis by a1,Th3716;
  end;
