
theorem Th46:
  for G being _Graph, e,v,w being object st e Joins v,w,G & v <> w
  holds not e in G.loops()
proof
  let G be _Graph, e,v,w be object;
  assume A1: e Joins v,w,G & v <> w;
  assume e in G.loops();
  then consider u being object such that
    A2: e Joins u,u,G by Def2;
  u = v & u = w by A1, A2, GLIB_000:15;
  hence contradiction by A1;
end;
