reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;

theorem
  (LattRel L)~ = LattRel (L.:) & (LattPOSet L)~ = LattPOSet (L.:)
proof
A1: LattRel L = {[p,q]: p [= q} by FILTER_1:def 8;
A2: LattRel (L.:) = {[p9,q9] where p9 is Element of L.:, q9 is Element of L
  .: : p9 [= q9} by FILTER_1:def 8;
A3: L.: = LattStr(#carr(L), met(L), join(L)#) by LATTICE2:def 2;
  thus (LattRel L)~ = LattRel (L.:)
  proof
    let x,y be object;
    thus [x,y] in (LattRel L)~ implies [x,y] in LattRel (L.:)
    proof
      assume [x,y] in (LattRel L)~;
      then [y,x] in LattRel L by RELAT_1:def 7;
      then consider p,q such that
A4:   [y,x] = [p,q] and
A5:   p [= q by A1;
      reconsider p9 = p, q9 = q as Element of L.: by A3;
A6:   x = q by A4,XTUPLE_0:1;
A7:   y = p by A4,XTUPLE_0:1;
      q9 [= p9 by A5,LATTICE2:38;
      hence thesis by A2,A6,A7;
    end;
    assume [x,y] in LattRel (L.:);
    then consider p9, q9 being Element of L.: such that
A8: [x,y] = [p9,q9] and
A9: p9 [= q9 by A2;
    reconsider p = p9, q = q9 as Element of L by A3;
A10: x = p by A8,XTUPLE_0:1;
A11: y = q by A8,XTUPLE_0:1;
    q [= p by A9,LATTICE2:39;
    then [y,x] in LattRel L by A1,A10,A11;
    hence thesis by RELAT_1:def 7;
  end;
  hence thesis by A3;
end;
