
theorem Th1:
  for K,L being Lattice st the LattStr of K = the LattStr of L
  holds LattPOSet K = LattPOSet L
proof
  let K, L be Lattice;
  assume
A1: the LattStr of K = the LattStr of L;
  for p, q being Element of K holds [p,q] in LattRel K iff [p,q] in LattRel L
  proof
    let p,q be Element of K;
    reconsider p9 = p, q9 = q as Element of L by A1;
    hereby
      assume [p,q] in LattRel K;
      then p [= q by FILTER_1:31;
      then p "\/" q = q;
      then p9 "\/" q9 = q9 by A1;
      then p9 [= q9;
      hence [p,q] in LattRel L by FILTER_1:31;
    end;
    assume [p,q] in LattRel L;
    then p9 [= q9 by FILTER_1:31;
    then p9 "\/" q9 = q9;
    then p "\/" q = q by A1;
    then p [= q;
    hence thesis by FILTER_1:31;
  end;
  hence thesis by A1,RELSET_1:def 2;
end;
