
theorem Th13:
  for G2 being _Graph, v,e being object, w being Vertex of G2
  for G1 being addAdjVertex of G2,v,e,w
  st not e in the_Edges_of G2 & not v in the_Vertices_of G2
  holds G1 is non loopfull
proof
  let G2 be _Graph, v,e be object, w be Vertex of G2;
  let G1 be addAdjVertex of G2,v,e,w;
  assume A1: not e in the_Edges_of G2 & not v in the_Vertices_of G2;
  then reconsider v as Vertex of G1 by GLIB_006:130;
  v <> w by A1;
  then for e being object holds not e Joins v,v,G1 by A1, GLIB_006:134;
  hence thesis;
end;
