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

theorem Th65:
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V, W being Walk of G1
  st W.vertices() c= the_Vertices_of G2
  holds W.edges() c= the_Edges_of G2
proof
  let G2,v,V;
  let G1 be addAdjVertexAll of G2,v,V;
  let W be Walk of G1;
  assume A1: W.vertices() c= the_Vertices_of G2;
  per cases;
  suppose A2: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
    then not v in W.vertices() by A1;
    hence thesis by A2, Th63;
  end;
  suppose not (V c= the_Vertices_of G2 & not v in the_Vertices_of G2);
    then G1 == G2 by Def4;
    then the_Edges_of G1 = the_Edges_of G2 by GLIB_000:def 34;
    hence thesis;
  end;
end;
