
theorem
  for G1, G2 being _Graph, f being continuous PVertexMapping of G1, G2
  st f is onto holds
    (G1 is loopless implies G2 is loopless) &
    (G1 is edgeless implies G2 is edgeless)
proof
  let G1, G2 be _Graph, f be continuous PVertexMapping of G1, G2;
  assume A1: f is onto;
  hereby
    assume A2: G1 is loopless;
    assume not G2 is loopless;
    then consider v being object such that
      A3: ex e being object st e Joins v,v,G2 by GLIB_000:18;
    consider e being object such that
      A4: e Joins v,v,G2 by A3;
    v in the_Vertices_of G2 by A4, GLIB_000:13;
    then v in rng f by A1, FUNCT_2:def 3;
    then consider v0 being object such that
      A5: v0 in dom f & f.v0 = v by FUNCT_1:def 3;
    consider e0 being object such that
      A6: e0 Joins v0,v0,G1 by A4, A5, Th2;
    thus contradiction by A2, A6, GLIB_000:18;
  end;
  hereby
    assume A7: G1 is edgeless;
    assume not G2 is edgeless;
    then consider e9 being object such that
      A8: e9 in the_Edges_of G2 by XBOOLE_0:def 1;
    set v = (the_Source_of G2).e9, w = (the_Target_of G2).e9;
    A9: e9 Joins v,w,G2 by A8, GLIB_000:def 13;
    then v in the_Vertices_of G2 & w in the_Vertices_of G2 by GLIB_000:13;
    then A10: v in rng f & w in rng f by A1, FUNCT_2:def 3;
    then consider v0 being object such that
      A11: v0 in dom f & f.v0 = v by FUNCT_1:def 3;
    consider w0 being object such that
      A12: w0 in dom f & f.w0 = w by A10, FUNCT_1:def 3;
    consider e being object such that
      A13: e Joins v0,w0,G1 by A9, A11, A12, Th2;
    e in the_Edges_of G1 by A13, GLIB_000:def 13;
    hence contradiction by A7;
  end;
end;
