reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th46:
  for G2, v for G1 being addAdjVertexAll of G2,v,{}
  holds the_Edges_of G2 = the_Edges_of G1
proof
  let G2, v;
  let G1 be addAdjVertexAll of G2,v,{};
  per cases;
  suppose {} c= the_Vertices_of G2 & not v in the_Vertices_of G2;
    then consider E being set such that
      A1: card {} = card E & E misses the_Edges_of G2 and
      A2: the_Edges_of G1 = the_Edges_of G2 \/ E and
      for v1 being object st v1 in {} ex e1 being object st e1 in E &
        e1 Joins v1,v,G1 &
        for e2 being object st e2 Joins v1,v,G1 holds e1 = e2
      by Def4;
    E = {} by A1;
    hence thesis by A2;
  end;
  suppose not ({} c= the_Vertices_of G2 & not v in the_Vertices_of G2);
    then G2 == G1 by Def4;
    hence the_Edges_of G2 = the_Edges_of G1 by GLIB_000:def 34;
  end;
end;
