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

theorem Th47:
  for G2, v for V being non empty set, G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds the_Edges_of G1 <> {}
proof
  let G2, v;
  let V be non empty set, G1 be addAdjVertexAll of G2,v,V;
  assume V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  then consider E being set such that
    A1: card V = 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 V 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 is non empty by A1;
  then consider e being object such that
    A3: e in E by XBOOLE_0:def 1;
  thus thesis by A2, A3;
end;
