 reserve W for WA-Lattice;
 reserve a,b,c for Element of W;
 reserve W for pcs-Compatible pcs-tol-reflexive pcs-tol-symmetric WAP-Lattice;
 reserve a,b for Element of W;
 reserve L for WA_Lattice;

theorem Idem2: ::: WA version of LATTAD_1:27
  for L being WA_Lattice,
      x, y being Element of L holds
    [x,y] in LatOrder L iff x [= y
  proof
    let L be WA_Lattice,
        x, y be Element of L;
    hereby assume [x,y] in LatOrder L; then
      consider a1, b1 being Element of L such that
A1:   [a1,b1] = [x,y] & a1 [= b1;
      a1 = x & b1 = y by A1,XTUPLE_0:1;
      hence x [= y by A1;
    end;
    assume x [= y;
    hence thesis;
  end;
