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

theorem
  for G, v, V for G1, G2 being addAdjVertexAll of G,v,V
  holds the_Vertices_of G1 = the_Vertices_of G2 &
    for v1 being Vertex of G1 holds v1 is Vertex of G2
proof
  let G,v,V;
  let G1,G2 be addAdjVertexAll of G,v,V;
  thus the_Vertices_of G1 = the_Vertices_of G2
  proof
    per cases;
    suppose V c= the_Vertices_of G & not v in the_Vertices_of G;
      then the_Vertices_of G1 = the_Vertices_of G \/ {v} &
        the_Vertices_of G2 = the_Vertices_of G \/ {v} by Def4;
      hence thesis;
    end;
    suppose not (V c= the_Vertices_of G & not v in the_Vertices_of G);
      then G1 == G & G2 == G by Def4;
      then G1 == G2 by GLIB_000:85;
      hence thesis by GLIB_000:def 34;
    end;
  end;
  hence thesis;
end;
