
theorem
  for G1, G2 being _Graph, f being PVertexMapping of G1, G2
  st f is total holds
    (G2 is loopless implies G1 is loopless) &
    (G2 is edgeless implies G1 is edgeless)
proof
  let G1, G2 be _Graph, f be PVertexMapping of G1, G2;
  assume A1: f is total;
  hereby
    assume A2: G2 is loopless;
    assume not G1 is loopless;
    then consider v being object such that
      A3: ex e being object st e Joins v,v,G1 by GLIB_000:18;
    consider e being object such that
      A4: e Joins v,v,G1 by A3;
    v in the_Vertices_of G1 by A4, GLIB_000:13;
    then v in dom f by A1, PARTFUN1:def 2;
    then consider e9 being object such that
      A5: e9 Joins f.v,f.v,G2 by A4, Th1;
    thus contradiction by A2, A5, GLIB_000:18;
  end;
  hereby
    assume A6: G2 is edgeless;
    assume not G1 is edgeless;
    then consider e being object such that
      A7: e in the_Edges_of G1 by XBOOLE_0:def 1;
    set v = (the_Source_of G1).e, w = (the_Target_of G1).e;
    A8: e Joins v,w,G1 by A7, GLIB_000:def 13;
    then v in the_Vertices_of G1 & w in the_Vertices_of G1 by GLIB_000:13;
    then v in dom f & w in dom f by A1, FUNCT_2:def 1;
    then consider e9 being object such that
      A9: e9 Joins f.v,f.w,G2 by A8, Th1;
    e9 in the_Edges_of G2 by A9, GLIB_000:def 13;
    hence contradiction by A6;
  end;
end;
