 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 OrdLat:
  [x,y] in LatOrder L iff x [= y
  proof
    hereby assume [x,y] in LatOrder L; then
      consider a1,b1 being Element of L such that
A1:   [x,y] = [a1,b1] & a1 [= b1;
      thus x [= y by A1,XTUPLE_0:1;
    end;
    assume x [= y;
    hence thesis;
  end;
