 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 LemmaEqual0:
  for L being WA_Lattice
  for x,y being Element of L,
      xx,yy being Element of LatRelStr L st
    x = xx & y = yy holds
  x [= y iff xx <= yy
  proof
    let L be WA_Lattice;
    let x,y be Element of L,
        xx,yy be Element of LatRelStr L;
    assume
A1: x = xx & y = yy;
    thus x [= y implies xx <= yy
    proof
      assume x [= y; then
      [xx,yy] in LatOrder L by A1;
      hence thesis by ORDERS_2:def 5;
    end;
    assume xx <= yy; then
    [xx,yy] in the InternalRel of LatRelStr L by ORDERS_2:def 5;
    hence thesis by A1,Idem2;
  end;
