 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 Th32:
  dom ThetaOrder L = the carrier of L &
    rng ThetaOrder L = the carrier of L &
    field ThetaOrder L = the carrier of L
proof
  now
    let x be object;
    thus x in the carrier of L implies
    ex y being object st [x,y] in ThetaOrder L
    proof
      assume x in the carrier of L;
      then reconsider p = x as Element of L;
      p "/\" p = p by IMeet; then
      [p,p] in ThetaOrder L;
      hence thesis;
    end;
    given y being object such that
A1: [x,y] in ThetaOrder L;
    consider p,q being Element of L such that
A2: [x,y] = [p,q] and
    p "/\" q = q by A1;
    x = p by A2,XTUPLE_0:1;
    hence x in the carrier of L;
  end;
  hence
A3: dom ThetaOrder L = the carrier of L by XTUPLE_0:def 12;
T1: now
    let x be object;
    thus x in the carrier of L implies
      ex y being object st [y,x] in ThetaOrder L
    proof
      assume x in the carrier of L;
      then reconsider p = x as Element of L;
      p "/\" p = p by IMeet; then
      [p,p] in ThetaOrder L;
      hence thesis;
    end;
    given y being object such that
A4: [y,x] in ThetaOrder L;
    consider p,q being Element of L such that
A5: [y,x] = [p,q] and
    p "/\" q = q by A4;
    x = q by A5,XTUPLE_0:1;
    hence x in the carrier of L;
  end;
  hence rng ThetaOrder L = the carrier of L by XTUPLE_0:def 13;
  thus field ThetaOrder L = (the carrier of L) \/ the carrier of L by A3,
     XTUPLE_0:def 13,T1
    .= the carrier of L;
end;
