 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) <=> (2)
  L is join-commutative iff LatRelStr L is directed
  proof
    thus L is join-commutative implies LatRelStr L is directed
    proof
      assume
A1:   L is join-commutative;
      set X = [#]LatRelStr L;
      for x,y being Element of LatRelStr L st x in X & y in X
       ex z being Element of LatRelStr L st z in X & x <= z & y <= z
      proof
        let x,y be Element of LatRelStr L;
        assume x in X & y in X;
        reconsider xx = x, yy = y as Element of L;
        xx "\/" yy = yy "\/" xx by A1; then
        consider z being Element of L such that
B1:     xx [= z & yy [= z by Th3823,LemX3;
        reconsider zz = z as Element of LatRelStr L;
C1:     x <= zz by OrdLat,ORDERS_2:def 5,B1;
        y <= zz by OrdLat,ORDERS_2:def 5,B1;
        hence thesis by C1;
      end;
      hence thesis by WAYBEL_0:def 6,WAYBEL_0:def 1;
    end;
    assume
a1: LatRelStr L is directed;
    for a,b being Element of L holds a "\/" b = b "\/" a
    proof
      let a,b be Element of L;
      reconsider aa = a, bb = b as Element of LatRelStr L;
      set X = [#]LatRelStr L;
      consider z being Element of LatRelStr L such that
B2:   z in X & aa <= z & bb <= z by WAYBEL_0:def 1,a1,WAYBEL_0:def 6;
      reconsider zz = z as Element of L;
B3:   a [= zz by B2,OrdLat,ORDERS_2:def 5;
      b [= zz by B2,OrdLat,ORDERS_2:def 5;
      hence thesis by DefB2,B3;
    end;
    hence thesis;
  end;
