 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 Eq2:
  for L being WA-Lattice,
      a,b being Element of L,
      aa,bb being Element of wlatt L st a = aa & b = bb holds
    aa [= bb iff a <= b
  proof
    let L be WA-Lattice,
        a,b be Element of L,
        aa,bb be Element of wlatt L;
    assume
AA: a = aa & b = bb;
WW: LatRelStr wlatt L = the RelStr of L by WLatDef;
    thus aa [= bb implies a <= b
    proof
      assume aa [= bb; then
      consider xx,yy being Element of wlatt L such that
A1:   [aa,bb] = [xx,yy] & xx [= yy;
      [a,b] in { [x,y] where x,y is Element of wlatt L : x [= y }
        by AA,A1;
      hence thesis by WW,ORDERS_2:def 5;
    end;
    assume a <= b; then
    [a,b] in the InternalRel of L by ORDERS_2:def 5; then
    consider xx,yy being Element of wlatt L such that
A1: [a,b] = [xx,yy] & xx [= yy by WW;
    a = xx & b = yy by XTUPLE_0:1,A1;
    hence thesis by A1,AA;
  end;
