 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 3.11. (4) <=> (7)
  L is join-commutative iff ThetaOrder L is being_partial-order
  proof
    set R = ThetaOrder L;
    thus L is join-commutative implies ThetaOrder L is being_partial-order
    proof
      assume L is join-commutative; then
      R is antisymmetric by Th31146;
      hence thesis by ORDERS_1:def 5;
    end;
    assume ThetaOrder L is being_partial-order; then
    R is antisymmetric by ORDERS_1:def 5;
    hence thesis by Th31146;
  end;
