
theorem Th35:
  for G1, G2 being _Graph, F being PGraphMapping 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 PGraphMapping of G1, G2;
  assume A1: F is total;
  hereby
    assume A2: G2 is loopless;
    not ex e being object st e in the_Edges_of G1 &
      (the_Source_of G1).e = (the_Target_of G1).e
    proof
      given e being object such that
        A3: e in the_Edges_of G1 and
        A4: (the_Source_of G1).e = (the_Target_of G1).e;
      set v = (the_Source_of G1).e;
      A5: e Joins v,v,G1 by A3, A4, GLIB_000:def 13;
      then v in dom F_V by A1, GLIB_000:13;
      then F_E.e Joins F_V.v,F_V.v,G2 by A1, A3, A5, Th4;
      hence contradiction by A2, GLIB_000:18;
    end;
    hence G1 is loopless by GLIB_000:def 18;
  end;
  thus G2 is edgeless implies G1 is edgeless by A1;
end;
