
theorem Th41:
  for G1, G2 being non-Dmulti _Graph, f being directed PVertexMapping of G1, G2
  st f is Dcontinuous holds DPVM2PGM(f) is Dcontinuous
proof
  let G1, G2 be non-Dmulti _Graph, f be directed PVertexMapping of G1, G2;
  assume A1: f is Dcontinuous;
  now
    let e9,v,w be object;
    assume A2: v in dom (DPVM2PGM f)_V & w in dom (DPVM2PGM f)_V;
    then reconsider v0 = v, w0 = w as Vertex of G1;
    assume A3: e9 DJoins (DPVM2PGM f)_V.v,(DPVM2PGM f)_V.w,G2;
    then consider e being object such that
      A4: e DJoins v,w,G1 by A1, A2;
    take e;
    thus e DJoins v,w,G1 by A4;
    e Joins v,w,G1 by A4, GLIB_000:16;
    then e in G1.edgesBetween(dom (DPVM2PGM f)_V) by A2, GLIB_000:32;
    hence e in dom (DPVM2PGM f)_E by Def11;
    then (DPVM2PGM f)_E.e DJoins (DPVM2PGM f)_V.v,(DPVM2PGM f)_V.w,G2
      by A2, A4, GLIB_010:def 14;
    hence (DPVM2PGM f)_E.e = e9 by A3, GLIB_000:def 21;
  end;
  hence thesis by GLIB_010:def 18;
end;
