 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   :: Theorem 3.13. (2) => (3)
  (for a,b being Element of L holds (a "\/" b) "/\" b = b) implies
    for a,b,c being Element of L holds
      (a "\/" b) "/\" c = (b "\/" a) "/\" c
  proof
    assume
AA: for a,b being Element of L holds (a "\/" b) "/\" b = b;
    let a,b,c be Element of L;
aa: (b "\/" a) "/\" a = a by AA;
S1: ex d being Element of L st (a "\/" b) "/\" c [= d &
       (b "\/" a) "/\" c [= d
    proof
      take d = c;
      thus thesis by LATTICES:def 8;
    end;
    ((a "\/" b) "/\" c) "\/" ((b "\/" a) "/\" c) =
      ((b "\/" a) "/\" c) "\/" ((a "\/" b) "/\" c) by S1,DefB2; then
    ((a "\/" b) "/\" c) "/\" ((b "\/" a) "/\" c) =
      ((b "\/" a) "/\" c) "/\" ((a "\/" b) "/\" c) by IffComm; then
    (a "\/" b) "/\" (c "/\" ((b "\/" a) "/\" c)) =
      ((b "\/" a) "/\" c) "/\" ((a "\/" b) "/\" c) by LATTICES:def 7; then
    (a "\/" b) "/\" (c "/\" ((b "\/" a) "/\" c)) =
      (b "\/" a) "/\" (c "/\" ((a "\/" b) "/\" c)) by LATTICES:def 7; then
    (a "\/" b) "/\" (((b "\/" a) "/\" c)) =
      (b "\/" a) "/\" (c "/\" ((a "\/" b) "/\" c)) by Lem36c; then
    (a "\/" b) "/\" (((b "\/" a) "/\" c)) =
      (b "\/" a) "/\" (((a "\/" b) "/\" c)) by Lem36c; then
    (a "\/" b) "/\" ((b "\/" a) "/\" c "/\" c) =
      ((b "\/" a) "/\" ((a "\/" b) "/\" c)) "/\" c by LATTICES:def 7; then
    (a "\/" b) "/\" ((b "\/" a) "/\" (c "/\" c)) =
      ((b "\/" a) "/\" ((a "\/" b) "/\" c)) "/\" c by LATTICES:def 7; then
    (a "\/" b) "/\" ((b "\/" a) "/\" c) =
      ((b "\/" a) "/\" ((a "\/" b) "/\" c)) "/\" c by IMeet; then
    (a "\/" b) "/\" ((b "\/" a) "/\" c) =
      (b "\/" a) "/\" (((a "\/" b) "/\" c) "/\" c) by LATTICES:def 7; then
    (a "\/" b) "/\" ((b "\/" a) "/\" c) =
      (b "\/" a) "/\" ((a "\/" b) "/\" (c "/\" c)) by LATTICES:def 7; then
    (a "\/" b) "/\" ((b "\/" a) "/\" c) =
      (b "\/" a) "/\" ((a "\/" b) "/\" c) by IMeet; then
    ((a "\/" b) "/\" (b "\/" a)) "/\" c =
      (b "\/" a) "/\" ((a "\/" b) "/\" c) by LATTICES:def 7; then
    ((a "\/" b) "/\" (b "\/" a)) "/\" c =
      ((b "\/" a) "/\" (a "\/" b)) "/\" c by LATTICES:def 7; then
    (((a "\/" b) "/\" b) "\/" ((a "\/" b) "/\" a)) "/\" c =
      ((b "\/" a) "/\" (a "\/" b)) "/\" c by LATTICES:def 11; then
    (b "\/" ((a "\/" b) "/\" a)) "/\" c =
      ((b "\/" a) "/\" (a "\/" b)) "/\" c by AA; then
    (b "\/" a) "/\" c =
      ((b "\/" a) "/\" (a "\/" b)) "/\" c by DefA2; then
    (b "\/" a) "/\" c =
      (((b "\/" a) "/\" a) "\/" ((b "\/" a) "/\" b)) "/\" c
        by LATTICES:def 11;
    hence thesis by DefA2,aa;
  end;
