reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th40:
  BasicDF(L) is onto
proof
  set X = the carrier of L, f = BasicDF(L);
  for w being object st w in X ex z being object st z in [:X,X:] & w = f.z
  proof
    let w be object;
    assume
A1: w in X;
    then reconsider w9 = w as Element of L;
    reconsider w99 = w as Element of L by A1;
    per cases;
    suppose
A2:   w = Bottom L;
       reconsider z = [w,w] as set;
      take z;
      thus z in [:X,X:] by A1,ZFMISC_1:87;
      thus f.z = f.(w9,w9) .= w by A2,Def21;
    end;
    suppose
A3:   w <> Bottom L;
       reconsider z = [Bottom L,w] as set;
      take z;
      thus z in [:X,X:] by A1,ZFMISC_1:87;
      thus f.z = f.(Bottom L,w9) .= Bottom L "\/" w99 by A3,Def21
        .= w by WAYBEL_1:3;
    end;
  end;
  then rng f = the carrier of L by FUNCT_2:10;
  hence thesis by FUNCT_2:def 3;
end;
