
theorem Th12:
  for G2 being _Graph, v being Vertex of G2, e,w being object
  for G1 being addAdjVertex of G2,v,e,w
  st not e in the_Edges_of G2 & not w in the_Vertices_of G2
  holds G1 is non loopfull
proof
  let G2 be _Graph, v be Vertex of G2, e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  assume A1: not e in the_Edges_of G2 & not w in the_Vertices_of G2;
  then reconsider w as Vertex of G1 by GLIB_006:129;
  v <> w by A1;
  then for e being object holds not e Joins w,w,G1 by A1, GLIB_006:133;
  hence thesis;
end;
