 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. (3) => (1)
  (for a,b,c being Element of L holds
      (a "\/" b) "/\" c = (b "\/" a) "/\" c) implies
    for a,b,c being Element of L holds
       (a "\/" b) "/\" c = (a "/\" c) "\/" (b "/\" c)
  proof
    assume
A1: for a,b,c being Element of L holds
      (a "\/" b) "/\" c = (b "\/" a) "/\" c;
    let a,b,c be Element of L;
    (a "/\" c) "\/" (b "/\" c) = ((a "/\" c) "\/" b) "/\" ((a "/\" c) "\/" c)
                     by DefLDS
                  .= ((a "/\" c) "\/" b) "/\" c by LATTICES:def 8
                  .= (b "\/" (a "/\" c)) "/\" c by A1
                  .= ((b "\/" a) "/\" (b "\/" c)) "/\" c by DefLDS
                  .= (((b "\/" a) "/\" b) "\/" ((b "\/" a) "/\" c)) "/\" c
                    by LATTICES:def 11
                  .= (b "\/" ((b "\/" a) "/\" c)) "/\" c by DefA2
                  .= ((b "\/" (b "\/" a)) "/\" (b "\/" c)) "/\" c
                    by DefLDS
                  .= (b "\/" (b "\/" a)) "/\" ((b "\/" c) "/\" c)
                    by LATTICES:def 7
                  .= (b "\/" (b "\/" a)) "/\" ((c "\/" b) "/\" c) by A1
                  .= (b "\/" (b "\/" a)) "/\" c by DefA2
                  .= (b "\/" a) "/\" c by Lem36X
                  .= (a "\/" b) "/\" c by A1;
    hence thesis;
  end;
