 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;

theorem TransLat:
  x [= y & y [= z implies x [= z
  proof
    assume
A0: x [= y & y [= z; then
    x "/\" y = x & y "/\" z = y by LATTICES:4; then
A1: x "/\" (y "/\" z) = x "/\" z by LATTICES:def 7;
    x "/\" (y "/\" z) = x "/\" y by LATTICES:def 9,A0
       .= x by A0,LATTICES:4;
    hence x [= z by LATTICES:4,A1;
  end;
