
theorem Th36:
  for G1, G2 being non-multi _Graph, f being PVertexMapping of G1, G2
  st f is onto continuous holds PVM2PGM(f) is onto
proof
  let G1, G2 be non-multi _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is onto continuous;
  then A2: rng (PVM2PGM f)_V = the_Vertices_of G2 by FUNCT_2:def 3;
  set g = (PVM2PGM 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;
    A6: e Joins f.v1,f.w1,G2 by A3, A4, A5, GLIB_000:def 13;
    then consider e0 being object such that
      A7: e0 Joins v1,w1,G1 by A1, A4, A5, Th2;
    e0 in G1.edgesBetween(dom f) by A4, A5, A7, GLIB_000:32;
    then A8: e0 in dom g by Def10;
    then g.e0 Joins (PVM2PGM f)_V.v1,(PVM2PGM f)_V.w1,G2
      by A4, A5, A7, GLIB_010:4;
    then g.e0 = e by A6, GLIB_000:def 20;
    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;
