
theorem Th5:
  for G being simple _Graph ex f being one-to-one Function
  st dom f = the_Edges_of G & rng f c= 2Set 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 simple _Graph;
  consider f being one-to-one Function such that
    A1: dom f = the_Edges_of G and
    A2: rng f c= 2Set the_Vertices_of G \/ singletons the_Vertices_of G and
    A3: for e being object st e in dom f holds
      f.e = {(the_Source_of G).e,(the_Target_of G).e} by Th4;
  take f;
  rng f /\ singletons the_Vertices_of G = {}
  proof
    assume rng f /\ singletons the_Vertices_of G <> {};
    then consider y being object such that
      A4: y in rng f /\ singletons the_Vertices_of G by XBOOLE_0:def 1;
    A5: y in rng f & y in singletons the_Vertices_of G by A4, XBOOLE_0:def 4;
    then consider e being object such that
      A6: e in dom f & f.e = y by FUNCT_1:def 3;
    consider Y being Subset of the_Vertices_of G such that
      A7: y = Y & Y is 1-element by A5;
    card Y = 1 by A7, CARD_1:def 7
      .= card {the object} by CARD_1:30;
    then consider v being object such that
      A8: Y = {v} by CARD_1:29;
    y = {(the_Source_of G).e,(the_Target_of G).e} by A3, A6;
    hence contradiction by A1, A6, A7, A8, ZFMISC_1:5, GLIB_000:def 18;
  end;
  hence thesis by A1, A2, A3, XBOOLE_1:73, XBOOLE_0:def 7;
end;
