
theorem Th15:
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  holds the_Vertices_of G1 = the_Vertices_of G2
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  per cases;
  suppose V c= the_Vertices_of G2;
    hence thesis by Def5;
  end;
  suppose not V c= the_Vertices_of G2;
    then G1 == G2 by Def5;
    hence thesis by GLIB_000:def 34;
  end;
end;
