 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 Th31146:  :: Theorem 3.11. (4) <=> (6)
  L is join-commutative iff ThetaOrder L is antisymmetric
  proof
    set R = ThetaOrder L;
    thus L is join-commutative implies ThetaOrder L is antisymmetric
    proof
      assume L is join-commutative; then
A0:   L is meet-commutative by Th31145;
      for x,y being object st
        [x,y] in R & [y,x] in R holds x = y
      proof
        let x,y be object;
        assume
A1:     [x,y] in R & [y,x] in R; then
        consider xx,yy being Element of L such that
A3:     [x,y] = [xx,yy] & xx "/\" yy = yy;
A4:     x = xx & y = yy by A3,XTUPLE_0:1;
        xx "/\" yy = yy & yy "/\" xx = xx by ThetaOrdLat,A1,A4;
        hence thesis by A4,A0;
      end;
      hence thesis by PREFER_1:31;
    end;
    assume
z1: ThetaOrder L is antisymmetric;
    for x, y being Element of L holds x "/\" y = y "/\" x
    proof
      let x,y be Element of L;
B1:   (x "/\" y) "/\" (y "/\" x) =
        (y "/\" x) "/\" (y "/\" x) by Lem310
              .= y "/\" x by IMeet;
B2:   (y "/\" x) "/\" (x "/\" y) =
        (x "/\" y) "/\" (x "/\" y) by Lem310
              .= x "/\" y by IMeet;
B3:   [x "/\" y,y "/\" x] in R by B1;
      [y "/\" x,x "/\" y] in R by B2;
      hence thesis by z1,B3,PREFER_1:31;
    end; then
    L is meet-commutative;
    hence thesis by Th31145;
  end;
