
theorem Th26:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is Disomorphism ex f being directed PVertexMapping of G1, G2
    st F_V = f & f is Disomorphism
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is Disomorphism;
  then reconsider f = F_V as directed PVertexMapping of G1, G2 by Th21;
  take f;
  thus F_V = f;
  A2: dom f = the_Vertices_of G1 by A1, GLIB_010:def 11;
  hence f is total by PARTFUN1:def 2;
  thus f is one-to-one by A1;
  rng f = the_Vertices_of G2 by A1, GLIB_010:def 12;
  hence f is onto by FUNCT_2:def 3;
  let v,w be Vertex of G1;
  A3: card G1.edgesDBetween({v},{w})
     = card G2.edgesDBetween(F_V.:{v},F_V.:{w}) by A1, GLIB_010:88
    .= card G2.edgesDBetween(F_V.:{v},{F_V.w}) by A2, Lm1
    .= card G2.edgesDBetween({f.v},{f.w}) by A2, Lm1;
  card G1.edgesDBetween({w},{v})
     = card G2.edgesDBetween(F_V.:{w},F_V.:{v}) by A1, GLIB_010:88
    .= card G2.edgesDBetween(F_V.:{w},{F_V.v}) by A2, Lm1
    .= card G2.edgesDBetween({f.w},{f.v}) by A2, Lm1;
  hence thesis by A3;
end;
