
theorem Th37:
  for G1, G2 being non-Dmulti _Graph, f being directed PVertexMapping of G1, G2
  st f is onto Dcontinuous holds DPVM2PGM(f) is onto
proof
  let G1, G2 be non-Dmulti _Graph, f be directed PVertexMapping of G1, G2;
  assume A1: f is onto Dcontinuous;
  then A2: rng (DPVM2PGM f)_V = the_Vertices_of G2 by FUNCT_2:def 3;
  set g = (DPVM2PGM f)_E;
  for e being object st e in the_Edges_of G2 holds e in rng g
  proof
    let e be object;
    set v2 = (the_Source_of G2).e, w2 = (the_Target_of G2).e;
    assume e in the_Edges_of G2;
    then e in G2.edgesBetween(the_Vertices_of G2) by GLIB_000:34;
    then e in G2.edgesBetween(rng f) by A1, FUNCT_2:def 3;
    then A3: e in the_Edges_of G2 & v2 in rng f & w2 in rng f by GLIB_000:31;
    consider v1 being object such that
      A4: v1 in dom f & f.v1 = v2 by A3, FUNCT_1:def 3;
    consider w1 being object such that
      A5: w1 in dom f & f.w1 = w2 by A3, FUNCT_1:def 3;
    e DJoins v2,w2,G2 by A3, GLIB_000:def 14;
    then A6: e DJoins f.v1,f.w1,G2 by A4, A5;
    then consider e0 being object such that
      A7: e0 DJoins v1,w1,G1 by A1, A4, A5;
    e0 Joins v1,w1, G1 by A7, GLIB_000:16;
    then e0 in G1.edgesBetween(dom f) by A4, A5, GLIB_000:32;
    then A8: e0 in dom g by Def11;
    then g.e0 DJoins (DPVM2PGM f)_V.v1,(DPVM2PGM f)_V.w1,G2
      by A4, A5, A7, GLIB_010:def 14;
    then g.e0 = e by A6, GLIB_000:def 21;
    hence e in rng g by A8, FUNCT_1:def 3;
  end;
  then the_Edges_of G2 c= rng g by TARSKI:def 3;
  then rng g = the_Edges_of G2 by XBOOLE_0:def 10;
  hence thesis by A2, GLIB_010:def 12;
end;
