
theorem Th3:
  for G being Dsimple _Graph ex f being one-to-one Function
  st dom f = the_Edges_of G &
    rng f c= [: the_Vertices_of G, the_Vertices_of G :]\id the_Vertices_of G &
    for e being object st e in dom f holds
      f.e = [(the_Source_of G).e,(the_Target_of G).e]
proof
  let G be Dsimple _Graph;
  consider f being one-to-one Function such that
    A1: dom f = the_Edges_of G &
      rng f c= [: the_Vertices_of G, the_Vertices_of G :] and
    A2: for e being object st e in dom f holds
      f.e = [(the_Source_of G).e,(the_Target_of G).e] by Th1;
  take f;
  rng f misses id the_Vertices_of G
  proof
    assume rng f meets id the_Vertices_of G;
    then consider y being object such that
      A3: y in rng f & y in id the_Vertices_of G by XBOOLE_0:3;
    consider x being object such that
      A4: x in dom f & f.x = y by A3, FUNCT_1:def 3;
    y = [(the_Source_of G).x,(the_Target_of G).x] by A2, A4;
    then (the_Source_of G).x = (the_Target_of G).x by A3, RELAT_1:def 10;
    hence contradiction by A1, A4, GLIB_000:def 18;
  end;
  hence thesis by A1, A2, XBOOLE_1:86;
end;
