 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 Eq0:
  for L being Lattice holds
    LattRel L = LatOrder L
  proof
    let L be Lattice;
    LattRel L = LatOrder L
    proof
T1:   LattRel L c= LatOrder L
      proof
        let x,y be object;
        assume [x,y] in LattRel L; then
        [x,y] in { [xx,yy] where xx,yy is Element of L : xx [= yy }
          by FILTER_1:def 8; then
        consider xx,yy being Element of L such that
ZZ:     [xx,yy] = [x,y] & xx [= yy;
        thus thesis by ZZ;
      end;
      LatOrder L c= LattRel L
      proof
        let x,y be object;
        assume [x,y] in LatOrder L; then
        consider xx,yy being Element of L such that
ZZ:     [xx,yy] = [x,y] & xx [= yy;
        [x,y] in { [x1,y1] where x1,y1 is Element of L : x1 [= y1 }
          by ZZ;
        hence thesis by FILTER_1:def 8;
      end;
      hence thesis by T1,XBOOLE_0:def 10;
    end;
    hence thesis;
  end;
