 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 Lem310:  :: Lemma 3.10.
  x "/\" y "/\" z = y "/\" x "/\" z
  proof
A1: x "/\" z [= z by LATTICES:def 8;
    y "/\" z [= z by LATTICES:def 8; then
    (x "/\" z) "\/" (y "/\" z) = (y "/\" z) "\/" (x "/\" z) by DefB2,A1; then
    (x "/\" z) "/\" (y "/\" z) = (y "/\" z) "/\" (x "/\" z) by IffComm; then
    (x "/\" z) "/\" y "/\" z = (y "/\" z) "/\" (x "/\" z)
      by LATTICES:def 7; then
    x "/\" z "/\" y "/\" z = y "/\" z "/\" x "/\" z by LATTICES:def 7; then
    x "/\" z "/\" (y "/\" z) = y "/\" z "/\" x "/\" z by LATTICES:def 7; then
    x "/\" (z "/\" (y "/\" z)) = y "/\" z "/\" x "/\" z by LATTICES:def 7; then
    x "/\" (y "/\" z) = y "/\" z "/\" x "/\" z by Lem36c; then
    x "/\" y "/\" z = y "/\" z "/\" x "/\" z by LATTICES:def 7; then
    x "/\" y "/\" z = y "/\" z "/\" (x "/\" z) by LATTICES:def 7; then
    x "/\" y "/\" z = y "/\" (z "/\" (x "/\" z)) by LATTICES:def 7; then
    x "/\" y "/\" z = y "/\" (x "/\" z) by Lem36c;
    hence thesis by LATTICES:def 7;
  end;
