
theorem Th11:
  for G2 being _Graph, V being set, G1 being addVertices of G2, V
  st V \ the_Vertices_of G2 <> {} holds G1 is non loopfull
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  assume V \ the_Vertices_of G2 <> {};
  then consider v being object such that
    A1: v in V \ the_Vertices_of G2 by XBOOLE_0:def 1;
  reconsider v as Vertex of G1 by A1, GLIB_006:86;
  v.edgesInOut() = {} by A1, GLIB_006:88, GLIB_000:def 49;
  then for e being object holds not e Joins v,v,G1 by GLIB_000:62;
  hence thesis;
end;
