reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

theorem Th32:
  dom LattRel L = the carrier of L & rng LattRel L = the carrier
  of L & field LattRel L = the carrier of L
proof
  now
    let x be object;
    thus x in the carrier of L implies ex y being object st [x,y] in LattRel L
    proof
      assume x in the carrier of L;
      then reconsider p = x as Element of L;
      [p,p] in LattRel L;
      hence thesis;
    end;
    given y being object such that
A1: [x,y] in LattRel L;
    consider p,q such that
A2: [x,y] = [p,q] and
    p [= q by A1;
    x = p by A2,XTUPLE_0:1;
    hence x in the carrier of L;
  end;
  hence
A3: dom LattRel L = the carrier of L by XTUPLE_0:def 12;
  now
    let x be object;
    thus x in the carrier of L implies ex y being object st [y,x] in LattRel L
    proof
      assume x in the carrier of L;
      then reconsider p = x as Element of L;
      [p,p] in LattRel L;
      hence thesis;
    end;
    given y being object such that
A4: [y,x] in LattRel L;
    consider p,q such that
A5: [y,x] = [p,q] and
    p [= q by A4;
    x = q by A5,XTUPLE_0:1;
    hence x in the carrier of L;
  end;
  hence rng LattRel L = the carrier of L by XTUPLE_0:def 13;
  hence field LattRel L = (the carrier of L) \/ the carrier of L by A3,
RELAT_1:def 6
    .= the carrier of L;
end;
