 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 Eq1:
  for L being with_suprema with_infima Poset,
      a,b being Element of L,
      aa,bb being Element of latt L st
   a = aa & b = bb holds
    aa [= bb iff a <= b
  proof
    let L being with_suprema with_infima Poset,
        a,b being Element of L,
        aa,bb being Element of latt L;
    assume
AA: a = aa & b = bb;
WW: LattPOSet latt L = the RelStr of L by LATTICE3:def 15;
    thus aa [= bb implies a <= b
    proof
      assume aa [= bb; then
      [aa,bb] in LatOrder latt L; then
      [a,b] in LattRel latt L by AA,Eq0;
      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
    [a,b] in { [x,y] where x,y is Element of latt L : x [= y }
      by WW,FILTER_1:def 8; then
    consider xx,yy being Element of latt L such that
A1: [a,b] = [xx,yy] & xx [= yy;
    a = xx & b = yy by XTUPLE_0:1,A1;
    hence thesis by A1,AA;
  end;
