 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 Th31143:  :: Theorem 3.11. (4) <=> (3)
  L is join-commutative iff L is ADL-absorbing
  proof
    thus L is join-commutative implies L is ADL-absorbing
    proof
      assume
A1:   L is join-commutative;
      let a,b be Element of L;
      a "\/" b = b "\/" a by A1;
      hence thesis by LATTICES:def 9;
    end;
    assume
B1: L is ADL-absorbing;
    let a,b be Element of L;
    a "/\" (b "\/" a) = a by B1;
    hence thesis by Th3726;
  end;
